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<strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong><strong>and</strong> <strong>Gravity</strong><strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong> Advancesin the Search for a Unified FrameworkFelix FinsterOlaf MüllerMarc NardmannJürgen TolksdorfEberhard ZeidlerEditors
Edito rsFelix FinsterLehrstuhl für MathematikUniversität RegensburgGermanyMarc NardmannFachbereich MathematikUniversität HamburgGermanyOlaf MüllerLehrstuhl für MathematikUniversität RegensburgGermanyJürgen TolksdorfMax-Planck-Institut fürMathematik in den NaturwissenschaftenLeipzig, GermanyEberhard ZeidlerMax-Planck-Institut fürMathematik in den NaturwissenschaftenLeipzig, GermanyISBN 978-3-0348-0042- 6 e-ISBN 978-3-0348-0043-3DOI 10.1007/978-3-0348-0043-3Springer Basel Dordrecht Heidelberg London New YorkLibrary of Congress Control Number: 2012930975<strong>Mathematical</strong> Subject Classification (2010): 81-06, 83-06© Springer Basel AG 2012This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting,reproduction on microfilms or in other ways, <strong>and</strong> storage in data banks. For any kind of use, permissionof the copyright owner must be obtained.Printed on acid-free paperSpringer Basel AG is part of Springer Science + Business Mediawww.birkhauser-science.com
ContentsPreface .................................................................viiClaus Kiefer<strong>Quantum</strong> <strong>Gravity</strong>: Whence, Whither? ..............................1Klaus Fredenhagen, Katarzyna RejznerLocal Covariance <strong>and</strong> Background Independence ...................15Blake TempleThe “Big Wave” <strong>Theory</strong> for Dark Energy ..........................25Steffen Gielen, Daniele OritiDiscrete <strong>and</strong> Continuum Third Quantization of <strong>Gravity</strong> ............41Andreas Döring, Rui Soares BarbosaUnsharp Values, Domains <strong>and</strong> Topoi ..............................65José Luis Flores, Jónatan Herrera, Miguel SánchezCausal Boundary of Spacetimes:Revision <strong>and</strong> Applications to AdS/CFT Correspondence ...........97Dietrich HäfnerSome <strong>Mathematical</strong> Aspects ofthe Hawking Effect for Rotating Black Holes .....................121Robert OecklObservables in the General Boundary Formulation . . . .............137Felix Finster, Andreas Grotz, Daniela SchiefenederCausal Fermion Systems:A <strong>Quantum</strong> Space-Time Emerging From an Action Principle . . . . . 157Christian Bär, Nicolas GinouxCCR- versus CAR-Quantization on Curved Spacetimes ...........183Christopher J. FewsterOn the Notion of ‘the Same Physics in All Spacetimes’ ...........207Rainer VerchLocal Covariance, Renormalization Ambiguity,<strong>and</strong> Local Thermal Equilibrium in Cosmology ....................229Julian BarbourShape Dynamics. An Introduction ................................257v
viContentsMichael K.-H. KiesslingOn the Motion of Point Defects in Relativistic <strong>Field</strong>s . . . ..........299Stefan Holl<strong>and</strong>sHow Unique Are Higher-dimensional Black Holes? ................337Domenico GiuliniEquivalence Principle,<strong>Quantum</strong> Mechanics, <strong>and</strong> Atom-interferometric Tests .............345Index .................................................................371
PrefaceThe present volume arose from the conference on “<strong>Quantum</strong> field theory <strong>and</strong>gravity – <strong>Conceptual</strong> <strong>and</strong> mathematical advances in the search for a unifiedframework”, held at the University of Regensburg (Germany) from September28 to October 1, 2010. This conference was the successor of similar conferenceswhich took place at the Heinrich Fabri Institut in Blaubeuren in 2003<strong>and</strong> 2005 <strong>and</strong> at the Max Planck Institute for Mathematics in the Sciencesin Leipzig in 2007. The intention of this series of conferences is to bring togethermathematicians <strong>and</strong> physicists to discuss profound questions withinthe non-empty intersection of mathematics <strong>and</strong> physics. More specifically,the series aims at discussing conceptual ideas behind different mathematical<strong>and</strong> physical approaches to quantum field theory <strong>and</strong> (quantum) gravity.As its title states, the Regensburg conference was devoted to the searchfor a unified framework of quantum field theory <strong>and</strong> general relativity. Onthe one h<strong>and</strong>, the st<strong>and</strong>ard model of particle physics – which describes allphysical interactions except gravitation – is formulated as a quantum fieldtheory on a fixed Minkowski-space background. The affine structure of thisbackground makes it possible for instance to interpret interacting quantumfields as asymptotically “free particles”. On the other h<strong>and</strong>, the gravitationalinteraction has the peculiar property that all kinds of energy couple to it.Furthermore, since Einstein developed general relativity theory, gravity isconsidered as a dynamical property of space-time itself. Hence space-timedoes not provide a fixed background, <strong>and</strong> a back-reaction of quantum fieldsto gravity, i.e. to the curvature of space-time, must be taken into account.It is widely believed that such a back-reaction can be described consistentlyonly by a (yet to be found) quantum version of general relativity, commonlycalled quantum gravity. <strong>Quantum</strong> gravity is expected to radically change ourideas about the structure of space-time. To find this theory, it might even benecessary to question the basic principles of quantum theory as well.Similar to the third conference of this series, the intention of the conferenceheld at the University of Regensburg was to provide a forum to discussdifferent mathematical <strong>and</strong> conceptual approaches to a quantum (field) theoryincluding gravitational back-reactions. Besides the two well-known pathslaid out by string theory <strong>and</strong> loop quantum gravity, also other ideas were presented.In particular, various functorial approaches were discussed, as well asthe possibility that space-time emerges from discrete structures.vii
viiiPrefaceThe present volume provides an appropriate cross-section of the conference.The refereed articles are intended to appeal to experts working indifferent fields of mathematics <strong>and</strong> physics who are interested in the subjectof quantum field theory <strong>and</strong> (quantum) gravity. Together they give the readersome overview of new approaches to develop a quantum (field) theory takinga dynamical background into account.As a complement to the invited talks which the articles in this volumeare based on, discussion sessions were held on the second <strong>and</strong> the last day ofthe conference. We list some of the questions raised in these sessions:1. Can we expect to obtain a quantum theory of gravity by purely mathematicalconsiderations? What are the physical requirements to expectfrom a unified field theory? How can these be formulated mathematically?Are the present mathematical notions sufficient to formulatequantum gravity, or are new mathematical concepts needed? Are thecriteria of mathematical consistency <strong>and</strong> simplicity promising guidingprinciples for finding a physical theory? Considering the wide varietyof existing approaches, the use of gedanken experiments as guidingparadigms seems indispensable even for pure mathematicians in thefield.2. Evolution or revolution? Should we expect progress rather by smallsteps or by big steps? By “small steps” we mean a conservative approachtowards a unified theory where one tries to keep the conventional terminologyas far as possible. In contrast, proceeding in “big steps” oftenentails to replace the usual terminology <strong>and</strong> the conventional physicalobjects by completely new ones.In the discussion, the possibilities for giving up the following conventionalstructures were considered:• Causality: In what sense should it hold in quantum gravity?• Superposition principle: Should it hold in a unified field theory?More specifically, do we have to give up the Hilbert space formalism<strong>and</strong> its probabilistic interpretation?A related question is:3. Can we quantize gravity separately? That is, does it make physical senseto formulate a quantum theory of pure gravity? Can such a formulationbe mathematically consistent? Or is it necessary to include all otherinteractions to obtain a consistent theory?4. Background independence: How essential is it, <strong>and</strong> which of the presentapproaches implement it? Which basic mathematical structure wouldbe physically acceptable as implementing background independence?5. What are the relevant open problems in classical field theory? One problemis the concept of charged point particles in classical electrodynamics(infinite self-energy). Other problems concern the notion of quasi-localmass in general relativity <strong>and</strong> the cosmic censorship conjectures.
Prefaceix6. (How) can we test quantum gravity? Can one hope to test quantumgravity in experiments whose initial conditions are controlled by humans,similar to tests of the st<strong>and</strong>ard model in particle accelerators?Or does one need to rely on astronomical observations (of events likesupernovae or black hole mergers)?Having listed some of the basic questions, we will now give brief summariesof the articles in this volume. They are presented in chronologicalorder of the corresponding conference talks. Unfortunately, not all the topicsdiscussed at the conference are covered in this volume, because a few speakerswere unable to contribute; see also pp. xii–xiii below.The volume begins with an overview by Claus Kiefer on the main roadstowards quantum gravity. After a brief motivation why one should search fora quantum theory of gravitation, he discusses canonical approaches, covariantapproaches like loop quantum gravity, <strong>and</strong> string theory. As two mainproblems that a theory of quantum gravity should solve, he singles out a statisticalexplanation of the Bekenstein–Hawking entropy <strong>and</strong> a description ofthe final stage of black-hole evaporation. He summarizes what the previouslydiscussed approaches have found out about the first question so far.Locally covariant quantum field theory is a framework proposed by Brunetti–Fredenhagen–Verchthat replaces the Haag–Kastler axioms for a quantumfield theory on a fixed Minkowski background, by axioms for a functorwhich describes the theory on a large class of curved backgrounds simultaneously.After reviewing this framework, Klaus Fredenhagen ∗ 1 <strong>and</strong> KatarzynaRejzner suggest that quantum gravity can be obtained from it via perturbativerenormalization à la Epstein–Glaser of the Einstein–Hilbert action. Oneof the technical problems one encounters is the need for a global version ofBRST cohomology related to diffeomorphism invariance. As a preliminarystep, the authors discuss the classical analog of this quantum problem interms of infinite-dimensional differential geometry.Based on his work with Joel Smoller, Blake Temple suggests an alternativereason for the observed increase in the expansion rate of the universe,which in the st<strong>and</strong>ard model of cosmology is explained in terms of “dark energy”<strong>and</strong> usually assumed to be caused by a positive cosmological constant.He argues that since the moment when radiation decoupled from matter379000 years after the big bang, the universe should be modelled by a wavelikeperturbation of a Friedmann–Robertson–Walker space-time, accordingto the mathematical theory of Lax–Glimm on how solutions of conservationlaws decay to self-similar wave patterns. The possible perturbations form a1-parameter family. Temple proposes that a suitable member of this familydescribes the observed anomalous acceleration of the galaxies (without invokinga cosmological constant). He points out that his hypothesis makestestable predictions.1 In the cases where articles have several authors, the star marks the author who deliveredthe corresponding talk at the conference.
xPrefaceThe term “third quantization” refers to the idea of quantum gravity asa quantum field theory on the space of geometries (rather than on spacetime),which includes a dynamical description of topology change. SteffenGielen <strong>and</strong> Daniele Oriti ∗ explain how matrix models implement the thirdquantizationprogram for 2-dimensional Riemannian quantum gravity, via arigorous continuum limit of discretized geometries. Group field theory (GFT)models, which originated in loop quantum gravity (LQG) but are also relevantin other contexts, implement third quantization for 3-dimensional Riemannianquantum gravity – but only in the discrete setting, without takinga continuum limit. The authors compare the GFT approach to the LQGmotivatedidea of constructing, at least on a formal level, a continuum thirdquantization on the space of connections rather than geometries. They arguethat the continuum situation should be regarded only as an effectivedescription of a physically more fundamental GFT.Andreas Döring ∗ <strong>and</strong> Rui Soares Barbosa present the topos approach toquantum theory, an attempt to overcome some conceptual problems with theinterpretation of quantum theory by using the language of category theory.One aspect is that physical quantities take their values not simply in the realnumbers; rather, the values are families of real intervals. The authors describea connection between the topos approach, noncommutative operator algebras<strong>and</strong> domain theory.Many problems in general relativity, as well as the formulation of theAdS/CFT correspondence, involve assigning a suitable boundary to a givenspace-time. A popular choice is Penrose’s conformal boundary, but it does notalways exist, <strong>and</strong> it depends on non-canonical data <strong>and</strong> is therefore not alwaysunique. José Luis Flores, Jónatan Herrera <strong>and</strong> Miguel Sánchez ∗ explain theconstruction of a causal boundary of space-time which does not suffer fromthese problems. They describe its properties <strong>and</strong> the relation to the conformalboundary. Several examples are discussed, in particular pp-waves.Dietrich Häfner gives a mathematically rigorous description of the Hawkingeffect for second-quantized spin- 1 2fields in the setting of the collapse ofa rotating charged star. The result, which confirms physical expectations, isstated <strong>and</strong> proved using the language <strong>and</strong> methods of scattering theory.One problem in constructing a background-free quantum theory is thatthe st<strong>and</strong>ard quantum formalism depends on a background metric: its operationalmeaning involves a background time, <strong>and</strong> its ability to describe physicslocally in field theory arises dynamically, via metric concepts like causality<strong>and</strong> cluster decomposition. In his general boundary formulation (GBF) ofquantum theory, Robert Oeckl triestoovercomethisproblembyusing,insteadof spacelike hypersurfaces, boundaries of arbitrary spacetime regionsas carriers of quantum states. His article lists the basic GBF objects <strong>and</strong> theaxioms they have to satisfy, <strong>and</strong> describes how the usual quantum states,observables <strong>and</strong> probabilities are recovered from a GBF setting. He proposesvarious quantization schemes to produce GBF theories from classical theories.
PrefacexiFelix Finster, Andreas Grotz ∗ <strong>and</strong> Daniela Schiefeneder introduce causalfermion systems as a general mathematical framework for formulating relativisticquantum theory. A particular feature is that space-time is a secondaryobject which emerges by minimizing an action for the so-called universalmeasure. The setup provides a proposal for a “quantum geometry” in theLorentzian setting. Moreover, numerical <strong>and</strong> analytical results on the supportof minimizers of causal variational principles are reviewed which reveala “quantization effect” resulting in a discreteness of space-time. A brief surveyis given on the correspondence to quantum field theory <strong>and</strong> gauge theories.Christian Bär ∗ <strong>and</strong> Nicolas Ginoux present a systematic constructionof bosonic <strong>and</strong> fermionic locally covariant quantum field theories on curvedbackgrounds in the case of free fields. In particular, they give precise mathematicalconditions under which bosonic resp. fermionic quantization is possible.It turns out that fermionic quantization requires much more restrictiveassumptions than bosonic quantization.ChristopherJ.Fewsterasks whether every locally covariant quantumfield theory (cf. the article by Fredenhagen <strong>and</strong> Rejzner described above)represents “the same physics in all space-times”. In order to give this phrasea rigorous meaning, he defines the “SPASs” property for families of locallycovariant QFTs, which intuitively should hold whenever each member of thefamily represents the same physics in all space-times. But not every family oflocally covariant QFTs has the SPASs property. However, for a “dynamicallocality” condition saying that kinematical <strong>and</strong> dynamical descriptions oflocal physics coincide, every family of dynamically local locally covariantQFTs has SPASs.Rainer Verch extends the concept of local thermal equilibrium (LTE)states, i.e. quantum states which are not in global thermal equilibrium butpossess local thermodynamical parameters like temperature, to quantum fieldtheory on curved space-times. He describes the ambiguities <strong>and</strong> anomaliesthat afflict the definition of the stress-energy tensor of QFT on curved spacetimes<strong>and</strong> reviews the work of Dappiaggi–Fredenhagen–Pinamonti which, inthe setting of the semi-classical Einstein equation, relates a certain fixing ofthese ambiguities to cosmology. In this context, he applies LTE states <strong>and</strong>shows that the temperature behavior of a massless scalar quantum field inthe very early history of the universe is more singular than the behavior ofthe usually considered model of classical radiation.Inspired by a version of Mach’s principle, Julian Barbour presents aframework for the construction of background-independent theories whichaims at quantum gravity, but whose present culmination is a theory of classicalgravitation called shape dynamics. Its dynamical variables are the elementsof the set of compact 3-dimensional Riemannian manifolds divided byisometries <strong>and</strong> volume-preserving conformal transformations. It “eliminatestime”, involves a procedure called conformal best matching, <strong>and</strong> is equivalentto general relativity for space-times which admit a foliation by compactspacelike hypersurfaces of constant mean curvature.
xiiPrefaceMichael K.-H. Kiessling considers the old problem of finding the correctlaws of motion for a joint evolution of electromagnetic fields <strong>and</strong> theirpoint-charge sources. After reviewing the long history of proposals, he reportson recent steps towards a solution by coupling the Einstein–Maxwell–Born–Infeld theory for an electromagnetic space-time with point defects toa Hamilton–Jacobi theory of motion for these defects. He also discusses howto construct a “first quantization with spin” of the sources in this classicaltheory by replacing the Hamilton–Jacobi law with a de Broglie–Bohm–Diracquantum law of motion.Several theories related to quantum gravity postulate (large- or smallsized)extra dimensions of space-time. Stefan Holl<strong>and</strong>s’ contribution investigatesa consequence of such scenarios, the possible existence of higherdimensionalblack holes, in particular of stationary ones. Because of theirlarge number, the possible types of such stationary black holes are muchharder to classify than their 4-dimensional analogs. Holl<strong>and</strong>s reviews somepartial uniqueness results.Since properties of general relativity, for instance the Einstein equivalenceprinciple(EEP),could conceivably fail to apply to quantum systems,experimental tests of these properties are important. Domenico Giulini’s articleexplains carefully which subprinciples constitute the EEP, how they applyto quantum systems, <strong>and</strong> to which accuracy they have been tested. In 2010,Müller–Peters–Chu claimed that the least well-tested of the EEP subprinciples,the universality of gravitational redshift, had already been verified withvery high precision in some older atom-interferometry experiments. Giuliniargues that this claim is unwarranted.Besides the talks summarized above there were also presentations coveringthe “main roads” to quantum gravity <strong>and</strong> other topics related to quantumtheory <strong>and</strong> gravity. PDF files of these presentations can be found at www.uniregensburg.de/qft2010.Dieter Lüst (LMU München) gave a talk with the title The l<strong>and</strong>scape ofmultiverses <strong>and</strong> strings: Is string theory testable?. He argued that, despite thehuge number of vacua that superstring/M-theory produces after compactification,it might still yield experimentally testable predictions. If the stringmass scale, which can a priori assume arbitrary values in brane-world scenarios,is not much larger than 5 TeV, then effects like string Regge excitationswill be seen at the Large Hadron Collider.Christian Fleischhack from the University of Paderborn gave an overviewof loop quantum gravity, emphasizing its achievements – e.g. the constructionof geometric operators for area <strong>and</strong> volume, <strong>and</strong> the derivation ofblack hole entropy – but also its problems, in particular the still widely unknowndynamics of the quantum theory.In her talk New ‘best hope’ for quantum gravity, Renate Loll from theUniversity of Utrecht presented the motivation, the status <strong>and</strong> perspectives of“<strong>Quantum</strong> <strong>Gravity</strong> from Causal Dynamical Triangulation (CDT)” <strong>and</strong> how
Prefacexiiiit is related to other approaches to a non-perturbative <strong>and</strong> mathematicallyrigorous formulation of quantum gravity.Mu-Tao Wang from Columbia University gave a talk On the notion ofquasilocal mass in general relativity. After explaining why it is difficult todefine a satisfying notion of quasilocal mass, he presented a new proposaldue to him <strong>and</strong> Shing-Tung Yau. This mass is defined via isometric embeddingsinto Minkowski space <strong>and</strong> has several desired properties, in particulara vanishing property that previous definitions were lacking.Motivated by the question – asked by ’t Hooft <strong>and</strong> others – whetherquantum mechanics could be an emergent phenomenon that occurs on lengthscales sufficiently larger than the Planck scale but arises from different dynamicsat shorter scales, Thomas Elze from the University of Pisa discussed inthe talk General linear dynamics: quantum, classical or hybrid a path-integralrepresentation of classical Hamiltonian dynamics which allows to consider directcouplings of classical <strong>and</strong> quantum objects. <strong>Quantum</strong> dynamics turns outto be rather special within the class of such general linear evolution laws.In his talk on Massive quantum gauge models without Higgs mechanism,Michael Dütsch explained how to construct the S-matrix of a non-abeliangauge theory in Epstein–Glaser style, via the requirements of renormalizability<strong>and</strong> causal gauge invariance. These properties imply already the occurrenceof Higgs fields in massive non-abelian models; the Higgs fields do nothave to be put in by h<strong>and</strong>. He discussed the relation of this approach tomodel building via spontaneous symmetry breaking.Jerzy Kijowski from the University of Warszawa spoke about <strong>Field</strong> quantizationvia discrete approximations: problems <strong>and</strong> perspectives. He explainedhow the set of discrete approximations of a physical theory is partially ordered,<strong>and</strong> that the observable algebras form an inductive system for thispartially ordered set, whereas the states form a projective system. Then heargued that loop quantum gravity is the best existing proposal for a quantumgravity theory, but suffers from the unphysical property that its states forminstead an inductive system.AcknowledgmentsIt is a great pleasure for us to thank all participants for their contributions,which have made the conference so successful. We are very grateful to thestaff of the Department of Mathematics of the University of Regensburg,especially to Eva Rütz, who managed the administrative work before, during<strong>and</strong> after the conference excellently.We would like to express our deep gratitude to the German ScienceFoundation (DFG), the Leopoldina – National Academy of Sciences, the AlfriedKrupp von Bohlen und Halbach Foundation, the International Associationof <strong>Mathematical</strong> Physics (IAMP), the private Foundation Hans Vielberthof the University of Regensburg, <strong>and</strong> the Institute of Mathematics atthe University of Regensburg for their generous financial support.
xivPrefaceAlso, we would like to thank the Max Planck Institute of Mathematicsin the Sciences, Leipzig, for the financial support of the conference volume.Finally, we thank Barbara Hellriegel <strong>and</strong> her team from Birkhäuser for theexcellent cooperation.Bonn, Leipzig, RegensburgSeptember 2011Felix FinsterOlaf MüllerMarc NardmannJürgen TolksdorfEberhard Zeidler
<strong>Quantum</strong> <strong>Gravity</strong>: Whence, Whither?Claus KieferAbstract. I give a brief summary of the main approaches to quantumgravity <strong>and</strong> highlight some of the recent developments.Mathematics Subject Classification (2010). Primary 83-02; Secondary83C45, 83C47, 83E05, 83E30.Keywords. <strong>Quantum</strong> gravity, string theory, quantum geometrodynamics,loop quantum gravity, black holes, quantum cosmology.1. Why quantum gravity?<strong>Quantum</strong> theory provides a universal framework that encompasses so farall particular interactions – with one exception: gravitation. The questionwhether gravity must also be described by a quantum theory at the mostfundamental level <strong>and</strong>, if yes, how such a theory can be constructed, is perhapsthe deepest unsolved problem of theoretical physics. In my contributionI shall try to give a general motivation <strong>and</strong> a brief overview of the mainapproaches as well as of some recent developments <strong>and</strong> applications. A comprehensivepresentation can be found in [1], where also many references aregiven; an earlier short overview is [2].The main obstacle so far in constructing a theory of quantum gravityis the lack of experimental support. Physics is an empirical science, <strong>and</strong> itis illusory to expect that a new fundamental physical theory can be foundwithout the help of data. This difficulty is connected with the fact that thefundamental quantum-gravity scale – the Planck scale – is far from beingdirectly accessible. The Planck scale (Planck length, Planck time, <strong>and</strong> Planckmass or energy) follows upon combining the gravitational constant, G, theF. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_1,© Springer Basel AG 20121
2 C. Kieferspeed of light, c, <strong>and</strong> the quantum of action, ,√Gl P =c ≈ 1.62 × 3 10−33 cm , (1)t P= l Pc = √Gc 5 ≈ 5.39 × 10−44 s , (2)m P = √cl P c = G ≈ 2.18 × 10−5 g ≈ 1.22 × 10 19 GeV/c 2 . (3)To probe the Planck scale with present technology, for example, one wouldneed a storage ring of galactic size, something beyond any imagination. Sowhy should one be interested in looking for a quantum theory of gravity?The reasons are of conceptual nature. The current edifice of theoreticalphysics cannot be complete. First, Einstein’s theory of general relativity(GR) breaks down in certain situations, as can be inferred from the singularitytheorems. Such situations include the important cases of big bang (ora singularity in the future) <strong>and</strong> the interior of black holes. The hope is thata quantum theory can successfully deal with such situations <strong>and</strong> cure thesingularities. Second, present quantum (field) theory <strong>and</strong> GR use concepts oftime (<strong>and</strong> spacetime) that are incompatible with each other. Whereas currentquantum theory can only be formulated with a rigid external spacetime structure,spacetime in GR is dynamical; in fact, even the simplest features of GR(such as the gravitational redshift implemented e.g. in the GPS system) cannotbe understood without a dynamical spacetime. This is often called theproblem of time, since non-relativistic quantum mechanics is characterizedby the absolute Newtonian time t as opposed to the dynamical configurationspace. A fundamental quantum theory of gravity is therefore assumed to befully background-independent. And third, the hope that all interactions ofNature can be unified into one conceptual framework will only be fulfilled ifthe present hybrid character of the theoretical structure is overcome.In the following, I shall first review the situations where quantum effectsare important in a gravitational context. I shall then give an overview of themain approaches <strong>and</strong> end with some applications.2. Steps towards quantum gravityThe first level of connection between gravity <strong>and</strong> quantum theory is quantummechanics in an external Newtonian gravitational field. This is the onlylevel where experiments exist so far. The quantum-mechanical systems aremostly neutrons or atoms. Neutrons, like any spin-1/2 system, are describedby the Dirac equation, which for the experimental purposes is investigated ina non-relativistic approximation (‘Foldy–Wouthuysen approximation’). Onethereby arrives ati ∂ψ∂t ≈ H FWψ
<strong>Quantum</strong> <strong>Gravity</strong>: Whence, Whither? 3with (in a st<strong>and</strong>ard notation)H FW = βmc 2 β+} {{ } 2m p2} {{ }rest masskinetic energy−ωL}{{}Sagnac effect−−ωS}{{}Mashhoon effectβ8m 3 c 2 p4 + βm(a x)} {{ }} {{ }COWSR correction+ β2m pax c 2 p + β4mc ⃗ 2 Σ(a × p)+O( 1c 3 )The underbraced terms have been experimentally tested directly or indirectly.(‘COW’ st<strong>and</strong>s for the classic neutron interferometry experiment performedby Colella, Overhauser, <strong>and</strong> Werner in 1975.)The next level on the way to quantum gravity is quantum field theoryin an external curved spacetime (or, alternatively, in a non-inertial system inMinkowski spacetime). Although no experimental tests exist so far, there aredefinite predictions.One is the Hawking effect for black holes. Black holes radiate with atemperature proportional to ,T BH =κ2πk B c , (5)where κ is the surface gravity. In the important special case of a Schwarzschildblack hole with mass M, one has for the Hawking temperature,c 3T BH =8πk B GM( )≈ 6.17 × 10 −8 M⊙K .MDue to the smallness of this temperature, the Hawking effect cannot be observedfor astrophysical black holes. One would need for this purpose primordialblack holes or small black holes generated in accelerators.Since black holes are thermodynamical systems, one can associate withthem an entropy, the Bekenstein–Hawking entropyS BH = k BA4l 2 P.(4)Schwarzschild≈ 1.07 × 10 77 k B( MM ⊙) 2. (6)Among the many questions for a quantum theory of gravity is the microscopicfoundation of S BH in the sense of Boltzmann.There exists an effect analogous to (5) in flat spacetime. An observerlinearly accelerated with acceleration a experiences a temperatureT DU =a2πk B c ≈ 4.05 × 10−23 a[ cms 2 ]K , (7)the ‘Unruh’ or ‘Davies–Unruh’ temperature. The analogy to (5) is more thanobvious. An experimental confirmation of (7) is envisaged with higher-power,short-pulse lasers [3].
4 C. KieferThe fact that black holes behave like thermodynamical systems has ledto speculations that the gravitational field might not be fundamental, but isinstead an effective macroscopic variable like in hydrodynamics, see e.g. [4]for a discussion. If this were true, the search for a quantum theory of thegravitational field would be misleading, since one never attempts to quantizeeffective (e.g. hydrodynamic) variables. So far, however, no concrete ‘hydrodynamic’theory of gravity leading to a new prediction has been formulated.The third, <strong>and</strong> highest, level is full quantum gravity. At present, thereexist various approaches about which no consensus is in sight. The mostconservative class of approaches is quantum general relativity, that is, thedirect application of quantization rules to GR. Methodologically, one distinguishesbetween covariant <strong>and</strong> canonical approaches. A more radical approachis string theory (or M-theory), which starts with the assumption that a quantumdescription of gravity can only be obtained within a unified quantumtheory of all interactions. Out of these approaches have grown many otherones, most of them building on discrete structures. Among them are quantumtopology, causal sets, group field theory, spin-foam models, <strong>and</strong> modelsimplementing non-commutative geometry. In the following, I shall restrictmyself to quantum general relativity <strong>and</strong> to string theory. More details ondiscrete approaches can be found in [5] <strong>and</strong> in other contributions to thisvolume.3. Covariant quantum gravityThe first, <strong>and</strong> historically oldest, approach is covariant perturbation theory.For this purpose one exp<strong>and</strong>s the four-dimensional metric g μν around a classicalbackground given by ḡ μν ,√32πGg μν =ḡ μν +c 4 f μν , (8)where f μν denotes the perturbation. This is similar to the treatment of weakgravitational waves in GR. Associated with f μν is a massless ‘particle’ of spin2, the graviton. The strongest observational constraint on the mass of thegraviton comes from investigating gravity over the size of galaxy clusters <strong>and</strong>leads to m g 10 −29 eV, cf. [6] for a discussion of this <strong>and</strong> other constraints.This mass limit would correspond to a Compton wavelength of 2 × 10 22 m.One can now insert the expansion (8) into the Einstein–Hilbert action<strong>and</strong> develop Feynman rules as usual. This can be done [1], but comparedto Yang–Mills theories an important difference occurs: perturbative quantumgravity is non-renormalizable, that is, one would need infinitely manyparameters to absorb the divergences. As has been shown by explicit calculations,the expected divergences indeed occur from two loops on. Recentprogress in this direction was made in the context of N = 8 supergravity[7], see also [8]. N = 8 supergravity, which has maximal supersymmetry, isfinite up to four loops, as was shown by an explicit calculation using powerfulnew methods. There are arguments that it is finite even at five <strong>and</strong> six loops
<strong>Quantum</strong> <strong>Gravity</strong>: Whence, Whither? 5<strong>and</strong> perhaps up to eight loops. If this is true, the question will arise whetherthere exists a hitherto unknown symmetry that prevents the occurrence ofdivergences at all.Independent of this situation, one must emphasize that there exist theoriesat the non-perturbative level that are perturbatively non-renormalizable.One example is the non-linear σ model for dimension D>2, which exhibitsa non-trivial UV fixed point at some coupling g c (‘phase transition’). Anexpansion in D − 2 <strong>and</strong> use of renormalization-group (RG) techniques givesinformation about the behaviour in the vicinity of the non-trivial fixed point.The specific heat exponent of superfluid helium as described by this modelwas measured in a space shuttle experiment, <strong>and</strong> the results are in accordancewith the calculations; the details are described, for example, in [9].Another covariant approach that makes heavy use of RG techniquesis asymptotic safety. A theory is called asymptotically safe if all essentialcoupling parameters g i of the theory approach for k →∞a non-trivial (i.e.non-vanishing) fixed point. This approach has recently attracted a lot ofattention, see, for example, [9, 10] <strong>and</strong> the references therein. The paper[10] puts particular emphasis on the role of background independence in thisapproach.Most modern covariant approaches make use of path integrals. Formally,one has to integrate over all four-dimensional metrics,∫Z[g] = Dg μν (x) e iS[gμν(x)]/ ,<strong>and</strong>, if needed, non-gravitational fields. The expression is formal, since fora rigorous definition one would have to specify the details of the measure<strong>and</strong> the regularization procedure. Except for general manipulations, the pathintegral has therefore been used mainly in a semiclassical expansion or fordiscretized approaches. An example for the first is Hawking’s use of the Euclideanpath integral in quantum cosmology, while examples for the secondapplication are Regge calculus <strong>and</strong> dynamical triangulation. In dynamical triangulation,for example, one decomposes spacetime into simplices whose edgelengths remain fixed. The sum in the path integral is then performed overall possible combinations with equilateral simplices, <strong>and</strong> heavy use of Monte-Carlo simulations is made, see, for example [11] for a review. Among the manyinteresting results of this approach, I want to mention here the fact that the(expected) four-dimensionality of spacetime emerges at macroscopic scales,but that spacetime appears two-dimensional at small scales. Surprisingly,this microscopic two-dimensionality is also a result of the asymptotic-safetyapproach.In spite of being perturbatively non-renormalizable, quantum generalrelativity can be used in the limit of small energies as an effective field theory.One can obtain, for example, one-loop corrections to non-relativistic potentialsfrom the scattering amplitude by calculating the non-analytic terms inthe momentum transfer. In this way one can find one-loop corrections to the
6 C. KieferNewton potential [12],V (r) =− Gm 1m 2ras well as to the Coulomb potential [13],V (r) = Q 1Q 2r(1+3 G(m 1 + m 2 )rc 2 + 41 )G10π r 2 c 3(1+3 G(m 1 + m 2 )+ 6 )Grc 2 π r 2 c 3 + ... ,The first correction terms, which do not contain , describe, in fact, effects ofclassical GR. The quantum gravitational corrections themselves are too smallto be measurable in the laboratory, but they are at least definite predictionsfrom quantum gravity.,4. Canonical quantum gravityCanonical quantum gravity starts from a Hamiltonian formulation for GR<strong>and</strong> uses quantization rules to arrive at a wave functional Ψ that depends onthe configuration space of the theory [1]. A central feature of all canonicaltheories is the presence of constraints,ĤΨ =0, (9)where (9) st<strong>and</strong>s for both the Hamiltonian <strong>and</strong> the diffeomorphism (momentum)constraints, which arise as a consequence of the presence of redundancies(‘coordinate freedom’) in GR. The various canonical versions of GR can bedistinguished by the choice of canonical variables. The main approaches areGeometrodynamics. The canonical variables are the 3-dimensional metrich ab <strong>and</strong> a linear combination p cd of the components of the extrinsiccurvature.Connection dynamics. The canonical variables are a connection A i a <strong>and</strong> acoloured electric field Ei a.Loop dynamics. The canonical variables are a holonomy constructed fromA i a <strong>and</strong> the flux of Eia through a two-dimensional surface.I shall give a brief review of the first <strong>and</strong> the third approach.4.1. <strong>Quantum</strong> geometrodynamics<strong>Quantum</strong> geometrodynamics is a very conservative approach [14]. One arrivesinevitably at the relevant equations if one proceeds analogously toSchrödinger in 1926. In that year Schrödinger found his famous equationby looking for a wave equation that leads to the Hamilton–Jacobi equationin the (as we now say) semiclassical limit. As already discussed by Peresin 1962, the Hamilton–Jacobi equation(s) 1 for GR reads (here presented for1 The second equation states that S be invariant under infinitesimal three-dimensionalcoordinate transformations.
<strong>Quantum</strong> <strong>Gravity</strong>: Whence, Whither? 7simplicity in the vacuum case)√δS δS h16πG G abcd −δh ab δh cd 16πG ( (3) R − 2Λ) = 0 , (10)δSD a = 0 , (11)δh abwhere G abcd is a local function of the three-metric <strong>and</strong> is called ‘DeWittmetric’, since it plays the role of a metric on the space of all three-metrics.The task is now to find a functional wave equation that yields theHamilton–Jacobi equation(s) in the semiclassical limit given by) iΨ[h ab ]=C[h ab ]exp( S[h ab] ,where the variation of the prefactor C with respect to the three-metric ismuch smaller than the corresponding variation of S. From (10) one thenfinds the Wheeler–DeWitt equation (Hamiltonian constraint)ĤΨ ≡(−16πG 2 δ 2G abcd − (16πG) √ −1 h ( (3) R − 2Λ )) Ψ=0, (12)δh ab δh cd<strong>and</strong> from (11) the quantum diffeomorphism (momentum) constraintsˆD a δΨΨ ≡−2∇ b =0. (13)i δh abThe latter equations guarantee that the wave functional Ψ is independent ofinfinitesimal three-dimensional coordinate transformations.A detailed discussion of this equation <strong>and</strong> its applications can be foundin [1]. We emphasize here only a central conceptual issue: the wave functionaldoes not depend on any external time parameter. This is a direct consequenceof the quantization procedure, which treats the three-metric <strong>and</strong> the extrinsiccurvature (which can be imagined as the ‘velocity’ of the three-metric) ascanonically conjugated, similar to position <strong>and</strong> momentum in quantum mechanics.By its local hyperbolic form, however, one can introduce an intrinsictimelike variable that is constructed out of the three-metric itself; in simplequantum cosmological models, the role of intrinsic time is played by the scalefactor a of the Universe.By its very construction, it is obvious that one can recover quantumfield theory in an external spacetime from (12) <strong>and</strong> (13) in an appropriatelimit [1]. The corresponding approximation scheme is similar to the Born–Oppenheimer approximation in molecular physics. In this way one finds theequations (10) <strong>and</strong> (11) together with a functional Schrödinger equation fornon-gravitational fields on the background defined by the Hamilton–Jacobiequation. The time parameter in this Schrödinger equation is a many-fingeredtime <strong>and</strong> emerges from the chosen solution S.The next order in this Born–Oppenheimer approximation gives correctionsto the Hamiltonian Ĥm that occurs in the Schrödinger equation for the
8 C. Kiefernon-gravitational fields. They are of the formĤ m → Ĥm + 1m 2 (various terms) ,Psee [1] for details. From this one can calculate, for example, the quantumgravitational correction to the trace anomaly in de Sitter space. The resultis [15]δɛ ≈−2G2 HdS63(1440) 2 π 3 c 8 .A more recent example is the calculation of a possible contribution to theCMB anisotropy spectrum [16]. The terms lead to an enhancement of powerat small scales; from the non-observation of such an enhancement one canthen get a weak upper limit on the Hubble parameter of inflation, H dS 10 17GeV.One may ask whether there is a connection between the canonical <strong>and</strong>the covariant approach. Such a connection exists at least at a formal level:the path integral satisfies the Wheeler–DeWitt equation <strong>and</strong> the diffeomorphismconstraints. At the one-loop level, this connection was shown in a moreexplicit manner. This means that the full path integral with the Einstein–Hilbert action (if defined rigorously) should be equivalent to the constraintequations of canonical quantum gravity.4.2. Loop quantum gravityAn alternative <strong>and</strong> inequivalent version of canonical quantum gravity is loopquantum gravity [17]. The development started with the introduction ofAshtekar’s New Variables in 1986, which are defined as follows. The newmomentum variable is the densitized version of the triad,E a i (x) := √ h(x)e a i (x) ,<strong>and</strong> the new configuration variable is the connection defined byGA i a(x) :=Γ i a(x)+βK i a(x) ,where Γ i a(x) is the spin connection, <strong>and</strong> K i a(x) is related to the extrinsic curvature.The variable β is called the Barbero–Immirzi parameter <strong>and</strong> constitutesan ambiguity of the theory; its meaning is still mysterious. The variablesare canonically conjugated,{A i a(x),E b j (y)} =8πβδ i jδ b aδ(x, y) ,<strong>and</strong> define the connection representation mentioned above.In loop gravity, one uses instead the following variables derived fromthem. The new configuration variable is the holonomy around a loop (givingthe theory its name),( ∫ )U[A, α] :=P exp G A ,α
<strong>Quantum</strong> <strong>Gravity</strong>: Whence, Whither? 9<strong>and</strong> the new momentum variable is the densitized triad flux through thesurface S enclosed by the loop,∫E i [S] := dσ a Ei a .SIn the quantum theory, these variables obey canonical commutation rules. Itwas possible to prove a theorem analogous to the Stone–von Neumann theoremin quantum mechanics [18]: under some mild assumption, the holonomy–flux representation is unique. The kinematical structure of loop quantumgravity is thus essentially unique. As in quantum geometrodynamics, onefinds a Hamiltonian constraint <strong>and</strong> a diffeomorphism constraint, althoughtheir explicit forms are different from there. In addition, a new constraintappears in connection with the use of triads instead of metrics (‘Gauss constraint’).A thorough presentation of the many formal developments of loop quantumgravity can be found in [17], see also [19] for a critical review. A mainfeature is certainly the discrete spectrum of geometric operators. One canassociate, for example, an operator  with the surface area of a classical twodimensionalsurface S. Within the well-defined <strong>and</strong> essentially unique Hilbertspace structure at the kinematical level one can find the spectrum∑ √Â(S)Ψ S [A] =8πβlP2 jP (j P +1)Ψ S [A] ,P ∈S∩Swhere the j P denote integer multiples of 1/2, <strong>and</strong> P denotes an intersectionpoint between the fundamental discrete structures of the theory (the ‘spinnetworks’) <strong>and</strong> S. Area is thus quantized <strong>and</strong> occurs as a multiple of a fundamentalquantum of area proportional to lP 2 . It must be emphasized thatthis (<strong>and</strong> related) results are found at the kinematical level, that is, beforeall quantum constraints are solved. It is thus an open problem whether theysurvive the solution of the constraints, which would be needed in order toguarantee physical meaning. Moreover, in contrast to quantum geometrodynamics,it is not yet clear whether loop quantum gravity has the correctsemiclassical limit.5. String theoryString theory is fundamentally different from the approaches described above.The aim is not to perform a direct quantization of GR, but to constructa quantum theory of all interactions (a ‘theory of everything’) from wherequantum gravity can be recovered in an appropriate limit. The inclusion ofgravity in string theory is, in fact, unavoidable, since no consistent theorycan be constructed without the presence of the graviton.String theory has many important features such as the presence of gaugeinvariance, supersymmetry, <strong>and</strong> higher dimensions. Its structure is thus muchmore rigid than that of quantum GR which allows but does not dem<strong>and</strong>these features. The hope with string theory is that perturbation theory is
10 C. Kieferfinite at all orders, although the sum diverges. The theory contains only threefundamental dimensionful constants, ,c,l s ,wherel s is the string length. Theexpectation is (or was) that all other parameters (couplings, masses, . . . ) canbe derived from these constants, once the path from the higher-dimensional(10- or 11-dimensional) spacetime to four dimensions is found. Whether thisgoal can ever be reached is far from clear. It is even claimed that there are somany possibilities available that a sensible selection can only be made on thebasis of the anthropic principle. This is the idea of the ‘string l<strong>and</strong>scape’ inwhich at least 10 500 ‘vacua’ corresponding to a possible world are supposedto exist, cf. [20]. If this were true, much of the original motivation for stringtheory would have gone.Since string theory contains GR in some limit, the above argumentsthat lead to the Wheeler–DeWitt equation remain true, that is, this equationshould also follow as an approximate equation in string theory if one is awayfrom the Planck (or string) scale. The disappearance of external time shouldthus also hold in string theory, but has not yet been made explicit.It is not the place here to give a discussion of string theory. An accessibleintroduction is, for example, [21]; some recent developments can befound in [5] as well as in many other sources. In fact, current research focuseson issues such as AdS/CFT correspondence <strong>and</strong> holographic principle,which are motivated by string theory but go far beyond it [22]. Roughly, thiscorrespondence states that non-perturbative string theory in a backgroundspacetime that is asymptotically anti-de Sitter (AdS) is dual to a conformalfield theory (CFT) defined in a flat spacetime of one less dimension, a conjecturemade by Maldacena in 1998. This is often considered as a mostlybackground-independent definition of string theory, since information aboutthe background metric enters only through boundary conditions at infinity. 2AdS/CFT correspondence is considered to be a realization of the holographicprinciple which states that all the information needed for the descriptionof a spacetime region is already contained in its boundary. If theMaldacena conjecture is true, laws including gravity in three space dimensionswill be equivalent to laws excluding gravity in two dimensions. In a sense,space has then vanished, too. It is, however, not clear whether this equivalenceis a statement about the reality of Nature or only (as I suspect) aboutthe formal properties of two descriptions describing a world with gravity.6. Black holes <strong>and</strong> cosmologyAs we have seen, effects of quantum gravity in the laboratory are expectedto be too small to be observable. The main applications of quantum gravityshould thus be found in the astrophysical realm – in cosmology <strong>and</strong> thephysics of black holes.2 But it is also claimed that string field theory is the only truly background-independentapproach to string theory, see the article by Taylor in [5].
<strong>Quantum</strong> <strong>Gravity</strong>: Whence, Whither? 11As for black holes, at least two questions should be answered by quantumgravity. First, it should provide a statistical description of the Bekenstein–Hawkingentropy (6). And second, it should be able to describe thefinal stage of black-hole evaporation when the semiclassical approximationused by Hawking breaks down.The first problem should be easier to tackle because its solution shouldbe possible for black holes of arbitrary size, that is, also for situations wherethe quantum effects of the final evaporation are negligible. In fact, preliminaryresults have been found in all of the above approaches <strong>and</strong> can be summarizedas follows:Loop quantum gravity. The microscopic degrees of freedom are the spinnetworks; S BH only follows for a specific choice of the Barbero–Immirziparameter β: β =0.237532 ... [23].String theory. The microscopic degrees of freedom are the ‘D-branes’; S BHfollows for special (extremal or near-extremal) black holes. More generally,the result follows for black holes characterized by a near-horizonregion with an AdS 3 -factor [24].<strong>Quantum</strong> geometrodynamics. One can find S ∝ A in various models,for example the LTB model describing a self-gravitating sphericallysymmetricdust cloud [25].A crucial feature is the choice of the correct state counting [26]. One musttreat the fundamental degrees of freedom either as distinguishable (e.g. loopquantum gravity) or indistinguishable (e.g. string theory) in order to reproduce(6).The second problem (final evaporation phase) is much more difficult,since the full quantum theory of gravity is in principle needed. At the level ofthe Wheeler–DeWitt equation, one can consider oversimplified models suchas the one presented in [27], but whether the results have anything in commonwith the results of the full theory is open.The second field of application is cosmology. Again, it is not the placehere to give an introduction to quantum cosmology <strong>and</strong> its many applications,see, for example, [1, 28] <strong>and</strong> the references therein. Most work in this field isdone in the context of canonical quantum gravity. For example, the Wheeler–DeWitt equation assumes the following form for a Friedmann universe withscale factor a <strong>and</strong> homogeneous scalar field φ,( (1 G2∂a ∂ )− 2 ∂ 2)2 a 2 ∂a ∂a a 3 ∂φ − 2 G−1 −1 Λa3a + G3 + m2 a 3 φ 2 ψ(a, φ) =0.In loop quantum cosmology, the Wheeler–DeWitt equation is replaced by adifference equation [29]. Important issues include the possibility of singularityavoidance, the semiclassical limit including decoherence, the justification ofan inflationary phase in the early Universe, the possibility of observationalconfirmation, <strong>and</strong> the origin of the arrow of time.
12 C. KieferAcknowledgmentI thank the organizers of this conference for inviting me to a most inspiringmeeting.References[1] C. Kiefer, <strong>Quantum</strong> gravity. 2nd edition, Oxford University Press, Oxford,2007.[2] C. Kiefer, <strong>Quantum</strong> gravity — a short overview, in: <strong>Quantum</strong> gravity, editedbyB.Fauser,J.Tolksdorf,<strong>and</strong>E.Zeidler.Birkhäuser Verlag, Basel, 2006,pp. 1–13.[3] P. G. Thirolf et al., Signatures of the Unruh effect via high-power, short-pulselasers, Eur. Phys. J. D 55, 379–389 (2009).[4] T. Padmanabhan, Thermodynamical aspects of gravity: new insights, Rep.Prog. Phys. 73, 046901 (2010).[5] Approaches to quantum gravity, edited by D. Oriti. Cambridge UniversityPress, Cambridge, 2009.[6] A. S. Goldhaber <strong>and</strong> M. N. Nieto, Photon <strong>and</strong> graviton mass limits, Rev. Mod.Phys. 82, 939–979 (2010).[7] Z. Bern et al., Ultraviolet behavior of N = 8 supergravity at four loops, Phys.Rev. Lett. 103, 081301 (2009).[8] H. Nicolai, Vanquishing infinity, Physics 2, 70 (2009).[9]H.W.Hamber,<strong>Quantum</strong> gravitation – The Feynman path integral approach.Springer, Berlin, 2009.[10] M. Reuter <strong>and</strong> H. Weyer, The role of background independence for asymptoticsafety in quantum Einstein gravity, Gen. Relativ. Gravit. 41, 983–1011 (2009).[11] J. Ambjørn, J. Jurkiewicz, <strong>and</strong> R. Loll, <strong>Quantum</strong> gravity as sum over spacetimes,in: Lect. Notes Phys. 807, 59–124 (2010).[12] N. E. J. Bjerrum-Bohr, J. F. Donoghue, <strong>and</strong> B. R. Holstein, <strong>Quantum</strong> gravitationalcorrections to the nonrelativistic scattering potential of two masses,Phys. Rev. D 67, 084033 (2003).[13] S. Faller, Effective field theory of gravity: leading quantum gravitational correctionsto Newton’s <strong>and</strong> Coulomb’s law, Phys. Rev. D 77, 12409 (2008).[14] C. Kiefer, <strong>Quantum</strong> geometrodynamics: whence, whither?, Gen. Relativ.Gravit. 41, 877–901 (2009).[15] C. Kiefer, <strong>Quantum</strong> gravitational effects in de Sitter space, in: New frontiersin gravitation, editedbyG.A.Sardanashvily.HadronicPress,PalmHarbor,1996, see also arXiv:gr-qc/9501001v1 (1995).[16] C. Kiefer <strong>and</strong> M. Krämer, <strong>Quantum</strong> gravitational contributions to the CMBanisotropy spectrum. arXiv:1103.4967v1 [gr-qc].[17] T. Thiemann, Modern canonical quantum general relativity. Cambridge UniversityPress, Cambridge, 2007.[18] C. Fleischhack, Representations of the Weyl algebra in quantum geometry,Commun. Math. Phys. 285, 67–140 (2009); J. Lew<strong>and</strong>owski et al., Uniquenessof diffeomorphism invariant states on holonomy-flux algebras, Commun. Math.Phys. 267, 703–733 (2006).
<strong>Quantum</strong> <strong>Gravity</strong>: Whence, Whither? 13[19] H. Nicolai, K. Peeters, <strong>and</strong> M. Zamaklar, Loop quantum gravity: an outsideview, Class. <strong>Quantum</strong> Grav. 22, R193–R247 (2005).[20] Universe or multiverse?, edited by B. Carr (Cambridge University Press, Cambridge,2007).[21] B. Zwiebach, A first course in string theory, 2nd Edition. Cambridge UniversityPress, Cambridge, 2009.[22] O. Aharony et al., Large N field theories, string theory <strong>and</strong> gravity, Phys. Rep.323, 183–386 (2000).[23] M. Domagala <strong>and</strong> J. Lew<strong>and</strong>owski, Black-hole entropy from quantum geometry,Class. <strong>Quantum</strong> Grav. 21, 5233–5243 (2004); K. A. Meissner, Black-holeentropy in loop quantum gravity, Class. <strong>Quantum</strong> Grav. 21, 5245–5251 (2004).[24] A. Strominger, Black hole entropy from near-horizon microstates, JHEP 02(1998) 009.[25] C. Vaz, S. Gutti, C. Kiefer, <strong>and</strong> T. P. Singh, <strong>Quantum</strong> gravitational collapse<strong>and</strong> Hawking radiation in 2+1 dimensions, Phys. Rev. D 76, 124021 (2007).[26] C. Kiefer <strong>and</strong> G. Koll<strong>and</strong>, Gibbs’ paradox <strong>and</strong> black-hole entropy, Gen. Relativ.Gravit. 40, 1327–1339 (2008).[27] C. Kiefer, J. Marto, <strong>and</strong> P. V. Moniz, Indefinite oscillators <strong>and</strong> black-holeevaporation, Annalen der Physik 18, 722–735 (2009).[28] M. Bojowald, C. Kiefer, <strong>and</strong> P. V. Moniz, <strong>Quantum</strong> cosmology for the 21stcentury: a debate. arXiv:1005.2471 [gr-qc] (2010).[29] M. Bojowald, Canonical gravity <strong>and</strong> applications. Cambridge University Press,Cambridge, 2011.Claus KieferInstitut für Theoretische PhysikUniversität zu KölnZülpicher Straße 77D 50937 KölnGermanye-mail: kiefer@thp.uni-koeln.de
Local Covariance <strong>and</strong> BackgroundIndependenceKlaus Fredenhagen <strong>and</strong> Katarzyna RejznerAbstract. One of the many conceptual difficulties in the development ofquantum gravity is the role of a background geometry for the structureof quantum field theory. To some extent the problem can be solved bythe principle of local covariance. The principle of local covariance wasoriginally imposed in order to restrict the renormalization freedom forquantum field theories on generic spacetimes. It turned out that it canalso be used to implement the request of background independence.Locally covariant fields then arise as background-independent entities.Mathematics Subject Classification (2010). 83C45, 81T05, 83C47.Keywords. Local covariance principle, quantum gravity, background independence,algebraic quantum field theory.1. IntroductionThe formulation of a theory of quantum gravity is one of the most importantunsolved problems in physics. It faces not only technical but, above all,conceptual problems. The main one arises from the fact that, in quantumphysics, space <strong>and</strong> time are a priori structures which enter the definition ofthe theory as well as its interpretation in a crucial way. On the other h<strong>and</strong>,in general relativity, spacetime is a dynamical object, determined by classicalobservables. To solve this apparent discrepancy, radical new approaches weredeveloped. Among these the best-known are string theory <strong>and</strong> loop quantumgravity. Up to now all these approaches meet the same problem: It isextremely difficult to establish the connection to actual physics.Instead of following the st<strong>and</strong>ard approaches to quantum gravity wepropose a more conservative one. We concentrate on the situation when theinfluence of the gravitational field is weak. This idealization is justified in alarge scope of physical situations. Under this assumption one can approachthe problem of quantum gravity from the field-theoretic side. In the first stepwe consider spacetime to be a given Lorentzian manifold, on which quantumF. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_ 2,© Springer Basel AG 201215
16 K. Fredenhagen <strong>and</strong> K. Rejznerfields live. In the second step gravitation is quantized around a given background.This is where the technical problems start. The resulting theory isnonrenormalizable, in the sense that infinitely many counterterms arise in theprocess of renormalization. Furthermore, the causal structure of the theory isdetermined by the background metric. Before discussing these difficulties wewant to point out that also the first step is by no means trivial. Namely, thest<strong>and</strong>ard formalism of quantum field theory is based on the symmetries ofMinkowski space. Its generalization even to the most symmetric spacetimes(de Sitter, anti-de Sitter) poses problems. There is no vacuum, no particles,no S-matrix, etc. A solution to these difficulties is provided by concepts ofalgebraic quantum field theory <strong>and</strong> methods from microlocal analysis.One starts with generalizing the Haag-Kastler axioms to generic spacetimes.We consider algebras A(O) of observables which can be measuredwithin the spacetime region O, satisfying the axioms of isotony, locality (commutativityat spacelike distances) <strong>and</strong> covariance. Stability is formulated asthe existence of a vacuum state (spectrum condition). The existence of a dynamicallaw (field equation) is understood as fulfilling the timeslice axiom(primitive causality) which says that the algebra of a timeslice is alreadythe algebra of the full spacetime. This algebraic framework, when applied togeneric Lorentzian manifolds, still meets a difficulty. The causal structure iswell defined, but the absence of nontrivial symmetries raises the question:What is the meaning of repeating an experiment? This is a crucial point ifone wants to keep the probability interpretation of quantum theory. A relatedissue is the need of a generally covariant version of the spectrum condition.These problems can be solved within locally covariant quantum field theory,a new framework for QFT on generic spacetimes proposed in [8].2. Locally covariant quantum field theoryThe framework of locally covariant quantum field theory was developed in[8, 17, 18]. The idea is to construct the theory simultaneously on all spacetimes(of a given class) in a coherent way. Let M be a globally hyperbolic,oriented, time-oriented Lorentzian 4d spacetime. Global hyperbolicity meansthat M is diffeomorphic to R×Σ, where Σ is a Cauchy surface of M. Betweenspacetimes one considers a class of admissible embeddings. An embeddingχ : N → M is called admissible if it is isometric, time-orientation <strong>and</strong> orientationpreserving, <strong>and</strong> causally convex in the following sense: If γ is a causalcurve in M with endpoints p, q ∈ χ(N), then γ = χ ◦ γ ′ with a causal curveγ ′ in N. A locally covariant QFT is defined by assigning to spacetimes Mcorresponding unital C ∗ -algebras A(M). This assignment has to fulfill a setof axioms, which generalize the Haag-Kastler axioms:1. M ↦→ A(M) unital C ∗ -algebra (local observables).2. If χ : N → M is an admissible embedding, then α χ : A(N) → A(M) isa unit-preserving C ∗ -homomorphism (subsystems).
Local Covariance <strong>and</strong> Background Independence 173. Let χ : N → M, χ ′ : M → L be admissible embeddings; then α χ′ ◦χ =α χ ′ ◦ α χ (covariance).4. If χ 1 : N 1 → M, χ 2 : N 2 → M are admissible embeddings such thatχ 1 (N 1 )<strong>and</strong>χ 2 (N 2 ) are spacelike separated in M, then[α χ1 (A(N 1 )),α χ2 (A(N 2 ))] = 0 (locality).5. If χ(N) contains a Cauchy surface of M, thenα χ (A(N)) = A(M) (timesliceaxiom).Axioms 1-3 have a natural interpretation in the language of category theory.Let Loc be the category of globally hyperbolic Lorentzian spacetimeswith admissible embeddings as morphisms <strong>and</strong> Obs the category of unitalC ∗ -algebras with homomorphisms as morphisms. Then a locally covariantquantum field theory is defined as a covariant functor A between Loc <strong>and</strong>Obs, withAχ := α χ .The fourth axiom is related to the tensorial structure of the underlyingcategories. The one for the category Loc is given in terms of disjoint unions.ItmeansthatobjectsinLoc∐ ∐ ⊗ are all elements M that can be written asM 1 ⊗ ...⊗ M N := M 1 ... Mn with the unit∐provided∐by the empty set∅. The admissible embeddings are maps χ : M 1 ... Mn → M such thateach component satisfies the requirements mentioned above <strong>and</strong> additionallyall images are spacelike to each other, i.e., χ(M 1 ) ⊥ ...⊥ (M n ). The tensorialstructure of the category Obs is a more subtle issue. Since there is no uniquetensor structure on general locally convex vector spaces, one has to eitherrestrict to some subcategory of Obs (for example nuclear spaces) or makea choice of the tensor structure based on some physical requirements. Thefunctor A can be then extended to a functor A ⊗ between the categories Loc ⊗<strong>and</strong> Obs ⊗ . It is a covariant ( tensor functor if the following conditions hold:∐ )A ⊗ M 1 M2 = A(M 1 ) ⊗ A(M 2 ) (2.1)A ⊗ (χ ⊗ χ ′ ) = A ⊗ (χ) ⊗ A ⊗ (χ ′ ) (2.2)A ⊗ (∅) = C (2.3)It can be shown that if A is a tensor functor, then the∐causality follows. To seethis, consider the natural embeddings ι i : M i → M 1 M2 , i =1, 2, for whichAι 1 (A 1 )=A 1 ⊗ 1, Aι 2 (A 2 )=1 ⊗ A 2 , A i ∈ A(M i ). Now let χ i : M i → Mbe admissible embeddings such that the images of χ 1 <strong>and</strong> χ 2∐are causallydisjoint in M. We define now an admissible embedding χ : M 1 M2 → Mby{χ1 (x) , x ∈ Mχ(x) =1. (2.4)χ 2 (x) , x ∈ M 2Since A ⊗ is a covariant tensor functor, it follows that[Aχ 1 (A 1 ), Aχ 2 (A 2 )] = Aχ[Aι 1 (A 1 ), Aι 2 (A 2 )] = Aχ[A 1 ⊗1, 1⊗A 2 ]=0. (2.5)This proves the causality. With a little bit more work it can be shown thatalso the opposite implication holds, i.e., the causality axiom implies that thefunctor A is tensorial.
Local Covariance <strong>and</strong> Background Independence 193. Perturbative quantum gravityAfter the brief introduction to the locally covariant QFT framework we cannow turn back to the problem of quantum gravity seen from the point of viewof perturbation theory. First we split off the metric:g ab = g (0)ab + h ab , (3.1)where g (0) is the background metric, <strong>and</strong> h is a quantum field. Now we canrenormalize the Einstein-Hilbert action by the Epstein-Glaser method (interactionrestricted to a compact region between two Cauchy surfaces) <strong>and</strong>construct the functor A. Next we compute (2.7) for two background metricswhich differ by κ ab compactly supported between two Cauchy surfaces. LetM 1 =(M,g (0) )<strong>and</strong>M 2 =(M,g (0) + κ). Following [8, 7], we assume thatthere are two causally convex neighbourhoods N ± of the Cauchy surfacesΣ ± which can be admissibly embedded both in M 1 <strong>and</strong> M 2 ,<strong>and</strong>thatκ issupported in a compact region between N − <strong>and</strong> N + .Wedenotethecorrespondingembeddings by χ ± i: N ± → M i , i =1, 2. We can now define anautomorphism of M 1 by:β κ := α χ−1◦ α −1χ − 2◦ α χ+2◦ α −1 . (3.2)χ + 1This automorphism corresponds to a change of the background between thetwo Cauchy surfaces. Under the geometrical assumptions given in [8] onecan calculate a functional derivative of β κ with respect to κ. Ifthemetricis not quantized, it was shown in [8] that this derivative corresponds to thecommutator with the stress-energy tensor. In the case of quantum gravity,δβ κδκ ab (x)involves in addition also the Einstein tensor. Therefore the backgroundδβindependence may be formulated as the condition that κδκ ab= 0, i.e., one(x)requires the validity of Einstein’s equation for the quantized fields. This canbe translated into a corresponding renormalization condition.The scheme proposed above meets some technical obstructions. Thefirst of them is the nonrenormalizability. This means that in every ordernew counterterms appear. Nevertheless, if these terms are sufficiently small,we can still have a predictive power of the resulting theory, as an effectivetheory. The next technical difficulty is imposing the constraints related tothe gauge (in this case: diffeomorphism) invariance. In perturbation theorythis can be done using the BRST method [4, 5] (or more generally Batalin-Vilkovisky formalism [3, 1]). Since this framework is based on the conceptof local objects, one encounters another problem. Local BRST cohomologyturns out to be trivial [14], hence one has to generalize the existing methodsto global objects. C<strong>and</strong>idates for global quantities are fields, considered asnatural transformations between the functor of test function spaces D <strong>and</strong>the quantum field theory functor A. A quantum field Φ : D → A correspondstherefore to a family of mappings (Φ M ) M∈Obj(Loc) , such that Φ M (f) ∈ A(M)for f ∈ D(M) <strong>and</strong> given a morphism χ : N → M we have α χ (Φ N (f)) =Φ M (χ ∗ f).
20 K. Fredenhagen <strong>and</strong> K. Rejzner4. BRST cohomology for classical gravityWhile quantum gravity is still elusive, classical gravity is (to some extent)well understood. Therefore one can try to test concepts for quantum gravityin a suitable framework for classical gravity. Such a formalism is providedby the algebraic formulation, where classical field theory occurs as the =0limit of quantum field theory [9, 10, 12, 11]. In this approach (the functionalapproach) one replaces the associative involutive algebras by Poisson algebras.In the case of gravity, to obtain a suitable Poisson structure one has to fix thegauge. In the BRST method this is done by adding a gauge-fixing term <strong>and</strong>a ghost term to the Einstein-Hilbert action. The so-called ghost fields havea geometrical interpretation as Maurer-Cartan forms on the diffeomorphismgroup. This can be made precise in the framework of infinite-dimensionaldifferential geometry. The notion of infinite-dimensional manifolds <strong>and</strong>, inparticular, infinite-dimensional Lie groups is known in mathematics sincethe reviews of Hamilton [16] <strong>and</strong> Milnor [21]. Because one needs to considermanifolds modeled on general locally convex vector spaces, an appropriatecalculus has to be chosen. Unfortunately the choice is not unique when wego beyond Banach spaces. Historically the earliest works concerning suchgeneralizations of calculus are those of Michal [19] (1938) <strong>and</strong> Bastiani [2](1964). At present there are two main frameworks in which the problems ofinfinite-dimensional differential geometry can be approached: the convenientsetting of global analysis [15, 20] <strong>and</strong> the locally convex calculus [16, 22]. Upto now both calculi coincide in the examples which were considered.First we sketch the BRST construction performed on the fixed backgroundM. The basic objects of the classical theory are:• S, a diffeomorphism-invariant action;• field content: configuration space E(M), considered as an infinite-dimensionalmanifold: scalar, vector, tensor <strong>and</strong> spinor fields (including themetric), gauge fields;• ghost fields (fermions): forms on the gauge algebra ΓTM, i.e. elementsof (ΓTM) ∗ ;• antifields (fermions): vector fields ΓT E(M) on the configuration space;• antifields of ghosts (bosons): compactly supported vector fields Γ c TM.The fields listed above constitute the minimal sector of the theory. To imposea concrete gauge, one can also introduce further fields, the so-callednonminimal sector. For the harmonic gauge it consists of Nakanishi-Lautrupfields (bosonic) <strong>and</strong> antighosts (fermionic). The minimal sector of the BRSTextendedfunctional algebra takes the formBV(M) =Sym(Γ c T M) ̂⊗ Λ(ΓT E(M)) ̂⊗ Λ(ΓT M) ∗ , (4.1)where ̂⊗ denotes the sequentially completed tensor product, <strong>and</strong> Sym is thesymmetric algebra. The algebra (4.1) is equipped with a grading called theghost number <strong>and</strong> a graded differential s consisting of two terms s = δ + γ.Both δ <strong>and</strong> γ are graded differentials. The natural action of ΓT M by the
Local Covariance <strong>and</strong> Background Independence 21Lie derivative on E(M) induces in a natural way an action on T E(M). Togetherwith the adjoint action on Γ c T M we obtain an action of ΓT M onSym(Γ c T M) ⊗ Λ(ΓT E(M)) which we denote by ρ. We can now write downhow δ <strong>and</strong> γ act on the basic fields a ∈ Γ c T M, Q ∈ ΓT E(M), ω ∈ (ΓT M) ∗ :•〈γ(a ⊗ Q ⊗ 1),X〉 := ρ X (a ⊗ Q ⊗ 1),•〈γ(a ⊗ Q ⊗ ω),X ∧ Y 〉 :=ρ X (a ⊗ Q ⊗〈ω, Y 〉) − ρ Y (a ⊗ Q ⊗〈ω, X〉) − a ⊗ Q ⊗〈ω, [X, Y ]〉,• δ(1 ⊗ Q ⊗ ω) :=1 ⊗ ∂ Q S ⊗ ω,• δ(a ⊗ 1 ⊗ ω) :=1 ⊗ ρ(a) ⊗ ω.Up to now the construction was done on the fixed spacetime, but it is notdifficult to see that the assignment of the graded algebra BV(M) to spacetimeM can be made into a covariant functor [14].As indicated already, the BRST method when applied to gravity hasto be generalized to global objects. Otherwise the cohomology of the BRSToperator s turns out to be trivial. This corresponds to the well-known factthat there are no local on-shell observables in general relativity. It was recentlyshown in [14] that one can introduce the BRST operator on the levelof natural transformations <strong>and</strong> obtain in this way a nontrivial cohomology.<strong>Field</strong>s are now understood as natural transformations. Let D k be a functorfrom the category Loc to the product category Vec k , that assigns to a manifoldM a k-fold product of the test section spaces D(M) × ...× D(M). LetNat(D k , BV) denote the set of natural transformations from D k to BV. Wedefine the extended algebra of fields asFld =∞⊕Nat(D k , BV) . (4.2)k=0It is equipped with a graded product defined by(ΦΨ) M (f 1 , ..., f p+q )= 1 ∑Φ M (f π(1) , ..., f π(p) )Ψ M (f π(p+1) , ..., f π(p+q) ),p!q!π∈P p+q(4.3)where the product on the right-h<strong>and</strong> side is the product of the algebraBV(M). Let Φ be a field; then the action of the BRST differential on itis defined by(sΦ) M (f) :=s(Φ M (f)) + (−1) |Φ| Φ M (£ (.) f) , (4.4)where |.| denotes the ghost number <strong>and</strong> the action of s on BV(M) isgivenabove. The physical fields are identified with the 0-th cohomology of s onFld. Among them we have for example scalars constructed covariantly fromthe metric.
22 K. Fredenhagen <strong>and</strong> K. Rejzner5. ConclusionsIt was shown that a construction of quantum field theory on generic Lorentzianspacetimes is possible, in accordance with the principle of general covariance.This framework can describe a wide range of physical situations. Also aconsistent incorporation of the quantized gravitational field seems to be possible.Since the theory is invariant under the action of an infinite-dimensionalLie group, the framework of infinite-dimensional differential geometry playsan important role. It provides the mathematical setting in which the BVmethod has a clear geometrical interpretation. The construction of a locallycovariant theory of gravity in the proposed setting was already performed forthe classical theory. Based on the gained insight it seems to be possible toapply this treatment also in the quantum case. One can then investigate therelations to other field-theoretical approaches to quantum gravity (Reuter[23], Bjerrum-Bohr [6], . . . ). As a conclusion we want to stress that quantumfield theory should be taken serious as a third way to quantum gravity.References[1] G. Barnich, F. Br<strong>and</strong>t, M. Henneaux, Phys. Rept. 338 (2000) 439 [arXiv:hepth/0002245].[2] A. Bastiani, J. Anal. Math. 13, (1964) 1-114.[3] I. A. Batalin, G. A. Vilkovisky, Phys. Lett. 102B (1981) 27.[4] C. Becchi, A. Rouet, R. Stora, Commun. Math. Phys. 42 (1975) 127.[5] C. Becchi, A. Rouet, R. Stora, Annals Phys. 98 (1976) 287.[6] N. E. J. Bjerrum-Bohr, Phys. Rev. D67(2003) 084033.[7] R. Brunetti, K. Fredenhagen, proceedings of Workshop on <strong>Mathematical</strong> <strong>and</strong>Physical Aspects of <strong>Quantum</strong> <strong>Gravity</strong>, Blaubeuren, Germany, 28 Jul - 1Aug 2005. In: B. Fauser et al. (eds.), <strong>Quantum</strong> gravity, 151-159 [arXiv:grqc/0603079v3].[8] R. Brunetti, K. Fredenhagen, R. Verch, Commun. Math. Phys. 237 (2003)31-68.[9]R.Brunetti,M.Dütsch, K. Fredenhagen, Adv. Theor. Math. Phys. 13(5)(2009) 1541-1599 [arXiv:math-ph/0901.2038v2].[10] M. Dütsch <strong>and</strong> K. Fredenhagen, Proceedings of the Conference on <strong>Mathematical</strong>Physics in Mathematics <strong>and</strong> Physics, Siena, June 20-25 2000 [arXiv:hepth/0101079].[11] M. Dütsch <strong>and</strong> K. Fredenhagen, Rev. Math. Phys. 16(10) (2004) 1291-1348[arXiv:hep-th/0403213].[12] M. Dütsch <strong>and</strong> K. Fredenhagen, Commun. Math. Phys. 243 (2003) 275[arXiv:hep-th/0211242].[13] K. Fredenhagen, Locally Covariant <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong>, Proceedings ofthe XIVth International Congress on <strong>Mathematical</strong> Physics, Lisbon 2003[arXiv:hep-th/0403007].[14] K. Fredenhagen, K. Rejzner, [arXiv:math-ph/1101.5112].
Local Covariance <strong>and</strong> Background Independence 23[15] A. Frölicher, A. Kriegl, Linear spaces <strong>and</strong> differentiation theory, Pure <strong>and</strong>Applied Mathematics, J. Wiley, Chichester, 1988.[16] R. S. Hamilton, Bull. Amer. Math. Soc. (N.S.) 7(1) (1982) 65-222.[17] S. Holl<strong>and</strong>s, R. Wald, Commun. Math. Phys. 223 (2001) 289.[18] S. Holl<strong>and</strong>s, R. M. Wald, Commun. Math. Phys. 231 (2002) 309.[19] A. D. Michal, Proc. Nat. Acad. Sci. USA 24 (1938) 340-342.[20] A. Kriegl, P. Michor, The convenient setting of global analysis, <strong>Mathematical</strong>Surveys <strong>and</strong> Monographs 53, American <strong>Mathematical</strong> Society, Providence1997. www.ams.org/online_bks/surv53/1.[21] J. Milnor, Remarks on infinite-dimensional Lie groups, In: B. DeWitt, R.Stora (eds.), Les Houches Session XL, Relativity, Groups <strong>and</strong> Topology II,North-Holl<strong>and</strong> (1984), 1007-1057.[22] K.-H. Neeb, Monastir lecture notes on infinite-dimensional Lie groups, www.math.uni-hamburg.de/home/wockel/data/monastir.pdf1.[23] M. Reuter, Phys. Rev. D57(1998) 971 [arXiv:hep-th/9605030].[24] G. Segal, In: Proceedings of the IXth International Congress on <strong>Mathematical</strong>Physics (Bristol, Philadelphia), eds. B. Simon, A. Truman <strong>and</strong> I. M. Davies,IOP Publ. Ltd. (1989), 22-37.[25] C. Torre, M. Varadarajan, Class. Quant. Grav. 16(8) (1999) 2651.[26] R. Verch, Ph.D. Thesis, University of Hamburg, 1996.Klaus Fredenhagen <strong>and</strong> Katarzyna RejznerII Inst. f. Theoretische PhysikUniversität HamburgLuruper Chaussee 149D-22761 HamburgGermanye-mail: klaus.fredenhagen@desy.dekatarzyna.rejzner@desy.de
The “Big Wave” <strong>Theory</strong> for Dark EnergyBlake Temple(Joint work with Joel Smoller)Abstract. We explore the author’s recent proposal that the anomalousacceleration of the galaxies might be due to the displacement of nearbygalaxies by a wave that propagated during the radiation phase of theBig Bang.Mathematics Subject Classification (2010). Primary 83B05; Secondary35L65.Keywords. Anomalous acceleration, self-similar waves, conservation laws.1. IntroductionBy obtaining a linear relation between the recessional velocities of distantgalaxies (redshift) <strong>and</strong> luminosity (distance), the American astronomer EdwinHubble showed in 1927 that the universe is exp<strong>and</strong>ing. This confirmed theso-called st<strong>and</strong>ard model of cosmology, that the universe, on the largest scale,is evolving according to a Friedmann-Robertson-Walker (FRW) spacetime.The starting assumption in this model is the Cosmological Principle—thaton the largest scale, we are not in a special place in the universe—that, in thewords of Robertson <strong>and</strong> Walker, the universe is homogeneous <strong>and</strong> isotropicabout every point like the FRW spacetime. In 1998, more accurate measurementsof the recessional velocity of distant galaxies based on new Type 1asupernova data made the surprising discovery that the universe was actuallyaccelerating relative to the st<strong>and</strong>ard model. This is referred to as theanomalous acceleration of the galaxies. The only way to preserve the FRWframework <strong>and</strong> the Cosmological Principle is to modify the Einstein equationsby adding an extra term called the cosmological constant. Dark Energy,the physical interpretation of the cosmological constant, is then an unknownsource of anti-gravitation that, for the model to be correct, must account forsome 70 percent of the energy density of the universe.Supported by NSF Applied Mathematics Grant Number DMS-070-7532.F. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_ 3,© Springer Basel AG 201225
26 B. TempleIn [14] the authors introduced a family of self-similar exp<strong>and</strong>ing wavesolutions of the Einstein equations of General Relativity (GR) that containthe st<strong>and</strong>ard model during the radiation phase of the Big Bang. Here I discussour cosmological interpretation of this family, <strong>and</strong> explore the possibilitythat waves in the family might account for the anomalous acceleration ofthe galaxies without the cosmological constant or Dark Energy (see [14, 16]for details). In a nutshell, our premise is that the Einstein equations of GRduring the radiation phase form a highly nonlinear system of wave equationsthat support the propagation of waves, <strong>and</strong> [14] is the culmination of ourprogram to discover waves that perturb the uniform background Friedmannuniverse (the setting for the st<strong>and</strong>ard model of cosmology), something likewater waves perturb the surface of a still pond. I also use this as a vehicleto record our unpublished Answers to reporter’s questions which appearedon the author’s website the week our PNAS paper [14] appeared, August 17,2009.In Einstein’s theory of General Relativity, gravitational forces turn outto be just anomalies of spacetime curvature, <strong>and</strong> the propagation of curvaturethrough spacetime is governed by the Einstein equations. The Einsteinequations during the radiation phase (when the equation of state simplifies top = ρc 2 /3) form a highly nonlinear system of conservation laws that supportthe propagation of waves, including compressive shock waves <strong>and</strong> self-similarexpansion waves. Yet since the 1930s, the modern theory of cosmology hasbeen based on the starting assumption of the Copernican Principle, whichrestricts the whole theory to the Friedmann spacetimes, a special class of solutionsof the Einstein equations which describe a uniform three-space of constantcurvature <strong>and</strong> constant density evolving in time. Our approach has beento look for general-relativistic waves that could perturb a uniform Friedmannbackground. The GR self-similar exp<strong>and</strong>ing waves in the family derived in [14]satisfy two important conditions: they perturb the st<strong>and</strong>ard model of cosmology,<strong>and</strong> they are the kind of waves that more complicated solutions shouldsettle down to according to the quantitative theories of Lax <strong>and</strong> Glimm onhow solutions of conservation laws decay in time to self-similar wave patterns.The great accomplishment of Lax <strong>and</strong> Glimm was to explain <strong>and</strong> quantifyhow entropy, shock-wave dissipation <strong>and</strong> time-irreversibility (concepts thatoriginally were understood only in the context of ideal gases) could be givenmeaning in general systems of conservation laws, a setting much more generalthan gas dynamics. (This viewpoint is well expressed in the celebrated works[10, 5, 6].) The conclusion: Shock-waves introduce dissipation <strong>and</strong> increaseof entropy into the dynamics of solutions, <strong>and</strong> this provides a mechanismby which complicated solutions can settle down to orderly self-similar wavepatterns, even when dissipative terms are neglected in the formulation of theequations. A rock thrown into a pond demonstrates how the mechanism cantransform a chaotic “plunk” into a series of orderly outgoing self-similar wavesmoments later. As a result, our new construction of a family of GR self-similarwaves that apply when this decay mechanism should be in place received a
The “Big Wave” <strong>Theory</strong> for Dark Energy 27good deal of media attention when it came out in PNAS, August 2009. (Asampling of press releases <strong>and</strong> articles can be found on my homepage. 1At the value of the acceleration parameter a =1(thefreeparameterin our family of self-similar solutions), the solution reduces exactly to thecritical FRW spacetime of the st<strong>and</strong>ard model with pure radiation sources,<strong>and</strong> solutions look remarkably similar to FRW when a ≠1.Whena ≠1,we prove that the spacetimes in the family are distinct from all the othernon-critical FRW spacetimes, <strong>and</strong> hence it follows that the critical FRWduring the radiation phase is characterized as the unique spacetime lying atthe intersection of these two one-parameter families. Since adjustment of thefree parameter a speeds up or slows down the expansion rate relative to thest<strong>and</strong>ard model, we argue they can account for the leading-order quadraticcorrection to redshift vs luminosity observed in the supernova data, withoutthe need for Dark Energy. I first proposed the idea that the anomalousacceleration might be accounted for by a wave in the talk Numerical ShockwaveCosmology, New Orleans, January 2007, 2 <strong>and</strong> set out to simulate sucha wave numerically. While attempting to set up the numerical simulation,we discovered that the st<strong>and</strong>ard model during the radiation phase admitsa coordinate system (St<strong>and</strong>ard Schwarzschild Coordinates (SSC)) in whichthe Friedmann spacetime is self-similar. That is, it took the form of a noninteractingtime-asymptotic wave pattern according to the theory of Lax <strong>and</strong>Glimm. This was the key. Once we found this, we guessed that the Einsteinequations in these coordinates must close to form a new system of ODEsin the same self-similar variable. After a struggle, we derived this system ofequations, <strong>and</strong> showed that the st<strong>and</strong>ard model was one point in a family ofsolutions parameterized by four initial conditions. Symmetry <strong>and</strong> regularityat the center then reduced the four-parameter family to an implicitly definedone-parameter family, one value of which gives the critical Friedmann spacetimeof the st<strong>and</strong>ard model during the radiation phase of the Big Bang. Ouridea then: an expansion wave that formed during the radiation epoch, whenthe Einstein equations obey a highly nonlinear system of conservation lawsfor which we must expect self-similar non-interacting waves to be the endstate of local fluctuations, could account for the anomalous acceleration ofthe galaxies without Dark Energy. Since we have explicit formulas for suchwaves, it is a verifiable proposition.2. Statement of resultsIn this section we state three theorems which summarize our results in [14,16]. (Unbarred coordinates (t, r) refer to FRW co-moving coordinates, <strong>and</strong>barred coordinates (¯t, ¯r) refer to (SSC).)1 see Media Articles on my homepage http://www.math.ucdavis.edu/~temple/2 the fourth entry under Conference/Seminar Talks on my homepage
28 B. TempleTheorem 2.1. Assume p = 1 3 ρc2 , k =0<strong>and</strong> R(t) = √ t. Then the FRW metricds 2 = −dt 2 + R(t) 2 dr 2 +¯r 2 dΩ 2 ,under the change of coordinates[ ] } 2 R(t)r¯t = ψ 0{1+t, (2.1)2t¯r = R(t)r, (2.2)transforms to the SSC-metricds 2 d¯t 2= −ψ0 2 (1 − v2 (ξ)) + d¯r 21 − v 2 (ξ) +¯r2 dΩ 2 , (2.3)wherev = √ 1 ū 1(2.4)AB ū 0is the SSC velocity, which also satisfiesv = ζ 2 , (2.5)ψ 0 ξ =2v1+v . (2.6)2Theorem 2.2. Let ξ denote the self-similarity variableξ =¯r¯t, (2.7)<strong>and</strong> letG =ξ √AB. (2.8)Assume that A(ξ), G(ξ) <strong>and</strong> v(ξ) solve the ODEs[]4(1 − A)vξA ξ = −(3 + v 2 (2.9))G − 4v{( ) 1 − A 2(1 + v 2 })G − 4vξG ξ = −GA (3 + v 2 )G − 4v − 1 (2.10)( ) { 1 − v2ξv ξ = −(3 + v 2 )G − 4v + 4 ( ) }1−AA {·}N2 {·} D(3 + v 2 , (2.11))G − 4vwhere{·} N= { −2v 2 +2(3− v 2 )vG − (3 − v 4 )G 2} (2.12){·} D= { (3v 2 − 1) − 4vG +(3− v 2 )G 2} , (2.13)<strong>and</strong> define the density byκρ = 3(1 − v2 )(1 − A)G(3 + v 2 )G − 4v1¯r 2 . (2.14)
The “Big Wave” <strong>Theory</strong> for Dark Energy 29Then the metricds 2 = −B(ξ)d¯t 2 + 1A(ξ) d¯r2 +¯r 2 dΩ 2 (2.15)solves the Einstein-Euler equations G = κT with velocity v = v(ξ) <strong>and</strong> equationof state p = 1 3 ρc2 . In particular, the FRW metric (2.3) solves equations(2.9)–(2.11).Note that it is not evident from the FRW metric in st<strong>and</strong>ard co-movingcoordinates that self-similar variables even exist, <strong>and</strong> if they do exist, bywhat ansatz one should extend the metric in those variables to obtain nearbyself-similar solutions that solve the Einstein equations exactly. The mainpoint is that our coordinate mapping to SSC explicitly identifies the selfsimilarvariables as well as the metric ansatz that together accomplish suchan extension of the metric.In [14, 16] we prove that the three-parameter family (2.9)–(2.11) (parameterizedby three initial conditions) reduces to an (implicitly defined)one-parameter family by removing time-scaling invariance <strong>and</strong> imposing regularityat the center. The remaining parameter a changes the expansion rateof the spacetimes in the family, <strong>and</strong> thus we call it the acceleration parameter.Transforming back to (approximate) co-moving coordinates, the resultingone-parameter family of metrics is amenable to the calculation of a redshiftvs luminosity relation, to third order in the redshift factor z, leading to thefollowing theorem which applies during the radiation phase of the expansion,cf. [14, 16]:Theorem 2.3. The redshift vs luminosity relation, as measured by an observerpositioned at the center of the exp<strong>and</strong>ing wave spacetimes (metrics of form(2.15)), is given up to fourth order in redshift factor z by{d l =2ctz + a2 − 12z 2 + (a2 − 1)(3a 2 }+5)z 3 + O(1)|a − 1|z 4 , (2.16)6where d l is luminosity distance, ct is invariant time since the Big Bang,<strong>and</strong> a is the acceleration parameter that distinguishes exp<strong>and</strong>ing waves in thefamily.When a = 1, (2.16) reduces to the correct linear relation of the st<strong>and</strong>ardmodel, [8]. Assuming redshift vs luminosity evolves continuously in time, itfollows that the leading-order part of any (small) anomalous correction tothe redshift vs luminosity relation of the st<strong>and</strong>ard model observed after theradiation phase could be accounted for by suitable adjustment of parametera.3. DiscussionThese results suggest an interpretation that we might call a conservation lawexplanation of the anomalous acceleration of the galaxies. That is, the theoryof Lax <strong>and</strong> Glimm explains how highly interactive oscillatory solutions of
30 B. Templeconservation laws will decay in time to non-interacting waves (shock waves<strong>and</strong> expansion waves), by the mechanisms of wave interaction <strong>and</strong> shockwavedissipation. The subtle point is that even though dissipation terms areneglected in the formulation of the equations, there is a canonical dissipation<strong>and</strong> consequent loss of information due to the nonlinearities, <strong>and</strong>thiscanbemodeled by shock-wave interactions that drive solutions to non-interactingwave patterns. Since the one fact most certain about the st<strong>and</strong>ard model isthat our universe arose from an earlier hot dense epoch in which all sourcesof energy were in the form of radiation, <strong>and</strong> since it is approximately uniformon the largest scale but highly oscillatory on smaller scales 3 , one might reasonablyconjecture that decay to a non-interacting exp<strong>and</strong>ing wave occurredduring the radiation phase of the st<strong>and</strong>ard model, via the highly nonlinearevolution driven by the large sound speed, <strong>and</strong> correspondingly large modulusof genuine nonlinearity 4 , present when p = ρc 2 /3, cf. [11]. Our analysis hasshown that FRW is just one point in a family of non-interacting, self-similarexpansion waves, <strong>and</strong> as a result we conclude that some further explanationis required as to why, on some length scale, decay during the radiation phaseof the st<strong>and</strong>ard model would not proceed to a member of the family satisfyinga ≠1. If decay to a ≠ 1 did occur, then the galaxies that formed frommatter at the end of the radiation phase (some 379, 000 years after the BigBang) would be displaced from their anticipated positions in the st<strong>and</strong>ardmodel at present time, <strong>and</strong> this displacement would lead to a modification ofthe observed redshift vs luminosity relation. In short, the displacement of thefluid particles (i.e., the displacement of the co-moving frames in the radiationfield) by the wave during the radiation epoch leads to a displacement of thegalaxies at a later time. In principle such a mechanism could account for theanomalous acceleration of the galaxies as observed in the supernova data. Ofcourse, if a ≠ 1, then the spacetime within the expansion wave has a center,<strong>and</strong> this would violate the so-called Copernican Principle, a simplifying assumptiongenerally accepted in cosmology, at least on the scale of the wave(cf. the discussions in [17] <strong>and</strong> [1]). Moreover, if our Milky Way galaxy didnot lie within some threshold of the center of expansion, the exp<strong>and</strong>ing wavetheory would imply unobserved angular variations in the expansion rate. Infact, all of these observational issues have already been discussed recentlyin [2, 1, 3] (<strong>and</strong> references therein), which explore the possibility that theanomalous acceleration of the galaxies might be due to a local void or underdensityof galaxies in the vicinity of the Milky Way. 5 Our proposal then is3 In the st<strong>and</strong>ard model, the universe is approximated by uniform density on a scale of abillion light years or so, about a tenth of the radius of the visible universe, [18]. The stars,galaxies <strong>and</strong> clusters of galaxies are then evidence of large oscillations on smaller scales.4 Again, genuine nonlinearity is, in the sense of Lax, a measure of the magnitude of nonlinearcompression that drives decay, cf. [10].5 The size of the center, consistent with the angular dependence that has been observed inthe actual supernova <strong>and</strong> microwave data, has been estimated to be about 15 megaparsecs,approximately the distance between clusters of galaxies, roughly 1/200 the distance acrossthe visible universe, cf. [1, 2, 3].
The “Big Wave” <strong>Theory</strong> for Dark Energy 31that the one-parameter family of general-relativistic self-similar expansionwaves derived here is a family of possible end-states that could result afterdissipation by wave interactions during the radiation phase of the st<strong>and</strong>ardmodel is completed, <strong>and</strong> such waves could thereby account for the appearanceof a local under-density of galaxies at a later time.In any case, the exp<strong>and</strong>ing wave theory is testable. For a first test, wepropose next to evolve the quadratic <strong>and</strong> cubic corrections to redshift vs luminosityrecorded here in relation (2.16), valid at the end of the radiation phase,up through the p ≈ 0 stage to present time in the st<strong>and</strong>ard model, to obtainthe present-time values of the quadratic <strong>and</strong> cubic corrections to redshift vsluminisity implied by the exp<strong>and</strong>ing waves, as a function of the accelerationparameter a. Once accomplished, we can look for a best fit value of a viacomparison of the quadratic correction at present time to the quadratic correctionobserved in the supernova data, leaving the third-order correction atpresent time as a prediction of the theory. That is, in principle, the predictedthird-order correction term could be used to distinguish the exp<strong>and</strong>ing wavetheory from other theories (such as Dark Energy) by the degree to which theymatch an accurate plot of redshift vs luminosity from the supernove data (atopic of the authors’ current research). The idea that the anomalous accelerationmight be accounted for by a local under-density in a neighborhood ofour galaxy was expounded in the recent papers [2, 3]. Our results here mightthen give an accounting for the source of such an under-density.The exp<strong>and</strong>ing wave theory could in principle give an explanation forthe observed anomalous acceleration of the galaxies within classical GeneralRelativity, with classical sources. In the exp<strong>and</strong>ing wave theory, the so-calledanomalous acceleration is not an acceleration at all, but is a correction to thest<strong>and</strong>ard model due to the fact that we are looking outward into an expansionwave. The one-parameter family of non-interacting, self-similar, generalrelativisticexpansion waves derived here contains all possible end-states thatcould result by wave interaction <strong>and</strong> dissipation due to nonlinearities backwhen the universe was filled with pure radiation sources. And when a ≠1,they introduce an anomalous acceleration into the st<strong>and</strong>ard model of cosmology.Unlike the theory of Dark Energy, this provides a possible explanationfor the anomalous acceleration of the galaxies that is not ad hoc in the sensethat it is derivable exactly from physical principles <strong>and</strong> a mathematicallyrigorous theory of general-relativistic expansion waves. In particular, this explanationdoes not require the ad hoc assumption of a universe filled withan as yet unobserved form of energy with anti-gravitational properties (thest<strong>and</strong>ard physical interpretation of the cosmological constant) in order to fitthe data.In summary, these new general-relativistic exp<strong>and</strong>ing waves provide anew paradigm to test against the st<strong>and</strong>ard model. Even if they do not in theend explain the anomalous acceleration of the galaxies, one has to believe theyare present <strong>and</strong> propagating on some scale, <strong>and</strong> their presence represents aninstability in the st<strong>and</strong>ard model in the sense that an explanation is required
32 B. Templeas to why small-scale oscillations have to settle down to large-scale a =1expansions instead of a ≠ 1 expansions (either locally or globally) during theradiation phase of the Big Bang.We now use this proceedings to record the Answers to reporter’s questionswhich appeared on our websites shortly after our PNAS paper cameout in August 2009.4. Answers to reporter’s questions:Blake Temple <strong>and</strong> Joel Smoller, August 17, 2009.To Begin: Let us say at the start that what is definitive about our work isthe construction of a new one-parameter family of exact, self-similar exp<strong>and</strong>ingwave solutions to Einstein’s equations of General Relativity. They applyduring the radiation phase of the Big Bang, <strong>and</strong> approximate the st<strong>and</strong>ardmodel of cosmology arbitrarily well. For this we have complete mathematicalarguments that are not controvertible. Our intuitions that led us to these, <strong>and</strong>their physical significance to the anomalous acceleration problem, are basedon lessons learned from the mathematical theory of nonlinear conservationlaws, <strong>and</strong> only this interpretation is subject to debate.1) Could you explain—in simple terms—what an exp<strong>and</strong>ing wave solutionis <strong>and</strong> what other phenomena in nature can be explained through thismathematics?To best underst<strong>and</strong> what an exp<strong>and</strong>ing wave is, imagine a stone throwninto a pond, making a splash as it hits the water. The initial “plunk” at thestart creates chaotic waves that break every which way, but after a short timethe whole disturbance settles down into orderly concentric circles of wavesthat radiate outward from the center—think of the resulting final sequenceof waves as the “exp<strong>and</strong>ing wave”. In fact, it is the initial breaking of wavesthat dissipates away all of the disorganized motion, until all that is left isthe orderly expansion of waves. For us, the initial “plunk” of the stone is thechaotic Big Bang at the start of the radiation phase, <strong>and</strong> the expansion wave isthe orderly expansion that emerges at the end of the radiation phase. What wehave found is that the st<strong>and</strong>ard model of cosmology is not the only exp<strong>and</strong>ingwave that could emerge from the initial “plunk”. In fact, we constructed awhole family of possible exp<strong>and</strong>ing waves that could emerge; <strong>and</strong> we arguethat which one would emerge depends delicately on the nature of the chaos inthe initial “plunk”. That is, one exp<strong>and</strong>ing wave in the family is equally likelyto emerge as another. Our family depends on a freely assignable number awhich we call the acceleration parameter, such that if we pick a =1,thenweget the st<strong>and</strong>ard model of cosmology, but if a>1 we get an exp<strong>and</strong>ing wavethat looks a lot like the st<strong>and</strong>ard model, but exp<strong>and</strong>s faster, <strong>and</strong> if a1.Summary: By “exp<strong>and</strong>ing wave” we mean a wave that exp<strong>and</strong>s outwardin a “self-similar” orderly way in the sense that at each time the wave looks
The “Big Wave” <strong>Theory</strong> for Dark Energy 33like it did at an earlier time, but more “spread out”. The importance of anexp<strong>and</strong>ing wave is that it is the end state of a chaotic disturbance becauseit is what remains after all the complicated breaking of waves is over. . . onepart of the exp<strong>and</strong>ing solution no longer affects the other parts. Our thesis,then, is that we can account for the anomalous acceleration of the galaxieswithout Dark Energy by taking a>1.2) Could you explain how <strong>and</strong> why you decided to apply exp<strong>and</strong>ing wavesolutions to this particular issue?We (Temple) got the idea that the anomalous acceleration of the galaxiesmight be explained by a secondary expansion wave reflected backwardfrom the shock wave in our earlier construction of a shock wave in the st<strong>and</strong>ardmodel of cosmology, <strong>and</strong> proposed to numerically simulate such a wave.Temple got this idea while giving a public lecture to the National Academyof Sciences in Bangalore India, in 2006. We set out together to simulate thiswave while Temple was Gehring Professor in Ann Arbor in 2007, <strong>and</strong> insetting up the simulation, we subsequently discovered exact formulas for afamily of such waves, without the need for the shock wave model.3) Do you think this provides the strongest evidence yet that Dark Energyis a redundant idea?At this stage we personally feel that this gives the most plausible explanationfor the anomalous acceleration of the galaxies that does not invokeDark Energy. Since we don’t believe in “Dark Energy”. . . [more detail in (12)below].We emphasize that our model implies a verifiable prediction, so it remainsto be seen whether the model fits the red-shift vs luminosity databetter than the Dark Energy theory. (We are working on this now.)4) Is this the first time that exp<strong>and</strong>ing wave solutions of the Einsteinequations have been realized?As far as we know, this is the first time a family of self-similar exp<strong>and</strong>ingwave solutions of the Einstein equations has been constructed forthe radiation phase of the Big Bang, such that the members of the familycan approximate the st<strong>and</strong>ard model of cosmology arbitrarily well. Our mainpoint is not that we have self-similar exp<strong>and</strong>ing waves, but that we have selfsimilarexp<strong>and</strong>ing waves during the radiation phase when (1) decay to suchwaves is possible because p ≠0, <strong>and</strong> (2) they are close to the st<strong>and</strong>ard model.We are not so interested in self-similar waves when p = 0 because we see noreason to believe that self-similar waves during the time when p = ρc 2 /3 ≠0will evolve into exact self-similar waves in the present era when p =0. Thatis, they should evolve into some sort of exp<strong>and</strong>ing spacetime when p =0, butnot a pure (self-similar) expansion wave.
34 B. Temple5) How did you reach the assumption that p =[ρ][c] 2 /3, awiseone?We are mathematicians, <strong>and</strong> in the last several decades, a theory forhow highly nonlinear equations can decay to self-similar waves was workedout by mathematicians, starting with fundamental work of Peter Lax <strong>and</strong> JimGlimm. The theory was worked out for model equations much simpler thanthe Einstein equations. We realized that only during the radiation phase ofthe expansion were the equations “sufficiently nonlinear” to expect sufficientbreaking of waves at the start to create enough dissipation to drive a chaoticdisorganized disturbance into an orderly self-similar expansion wave at theend. The subtle point is that even though no mechanisms for dissipation areput into the model, the nonlinearities alone can cause massive dissipation viathe breaking of waves that would drive a chaotic disturbance into an orderlyexpansion wave.6) How does your suggestion—that the observed anomalous accelerationof the galaxies could be due to our view into an expansion wave—comparewith an idea that I heard Subir Sarkar describe recently: that the Earth couldbe in a void that is exp<strong>and</strong>ing faster than the outer parts of the universe?We became aware of this work in the fall of ’08, <strong>and</strong> forwarded ourpreprints. Our view here is that after the radiation phase is over, <strong>and</strong> thepressure drops to zero, there is no longer any nonlinear mechanism that cancause the breaking of waves that can cause dissipation into an expansionwave. Thus during the recent p = 0 epoch (after some 300,000 years afterthe Big Bang), you might model the evolution of the remnants of such anexp<strong>and</strong>ing wave or under-density (in their terms a local “void”), but there isno mechanism in the p = 0 phase to explain the constraints under which sucha void could form. (When p =0, everything is in “freefall”, <strong>and</strong> there canbe no breaking of waves.) The exp<strong>and</strong>ing wave theory we present provides apossible quantitative explanation for the formation of such a void.7) How do you intend to develop your research from here?Our present paper demonstrates that there is some choice of the numbera (we proved it exists, but still do not know its precise value) such that themember of our family of exp<strong>and</strong>ing waves corresponding to that value of theacceleration parameter a will account for the leading-order correction of theanomalous acceleration. That is, it can account for how the plot of redshiftvs luminosity of the galaxies curves away from a straight line at the center.But once the correct value of a is determined exactly, that value will givea prediction of how the plot should change beyond the first breaking of thecurve. (There are no more free parameters to adjust!) We are currently workingon finding that exact value of a consistent with the observed anomalousacceleration, so that from this we can calculate the next-order correction itpredicts, all with the goal of comparing the exp<strong>and</strong>ing wave prediction tothe observed redshift vs luminosity plot, to see if it does better than theprediction of Dark Energy.
The “Big Wave” <strong>Theory</strong> for Dark Energy 358) What is your view on the relevance of the Copernican Principle tothese new exp<strong>and</strong>ing waves?These self-similar exp<strong>and</strong>ing waves represent possible end states of theexpansion of the Big Bang that we propose could emerge at the end of theradiation phase when there exists a mechanism for their formation. We imaginethat decay to such an exp<strong>and</strong>ing wave could have occurred locally in thevicinity of the Earth, over some length scale, but we can only conjecture as towhat length scale that might be—the wave could extend out to some fractionof the distance across the visible universe or it could extend even beyond,we cannot say, but to explain the anomalous acceleration the Earth must liewithin some proximity of the center. That is, for the a>1wavetoaccountfor the anomalous acceleration observed in the galaxies, we would have tolie in some proximity of the center of such a wave to be consistent with noobserved angular dependence in the redshift vs luminosity plots. (The voidtheory has the same implication.)Now one might argue that our exp<strong>and</strong>ing waves violate the so-calledCopernican or Cosmological Principle which states that on the largest lengthscale the universe looks the same everywhere. This has been a simplifyingassumption taken in cosmology since the mid thirties when Howard Robertson<strong>and</strong> Geoffrey Walker proved that the Friedmann spacetimes of the st<strong>and</strong>ardmodel (constructed by Alex<strong>and</strong>er Friedmann a decade earlier) are theunique spacetimes that are spatially homogeneous <strong>and</strong> isotropic about everypoint—a technical way of saying there is no special place in the universe.The introduction of Dark Energy via the cosmological constant is the onlyway to preserve the Copernican Principle <strong>and</strong> account for the anomalous accelerationon the largest scale, everywhere. The stars, galaxies <strong>and</strong> clustersof galaxies are evidence of small-scale variations that violate the CopernicanPrinciple on smaller length scales. We are arguing that there could be an evenlarger length scale than the clusters of galaxies on which local decay to oneof our exp<strong>and</strong>ing waves has occurred, <strong>and</strong> we happen to be near the centerof one. This would violate the Copernican Principle if these exp<strong>and</strong>ing wavesdescribe the entire universe—but our results allow for the possibility that ona scale even larger than the scale of the exp<strong>and</strong>ing waves, the universe maylook everywhere the same like the st<strong>and</strong>ard model. Thus our view is that theCopernican Principle is really a moot issue here. But it does beg the questionas to how big the effective center can be for the value of a that accountsfor the anomalous acceleration. This is a problem we hope to address in thefuture.Another way to look at this is, if you believe there is no cosmologicalconstant or Dark Energy [see (12) below], then the anomalous accelerationmay really be the first definitive evidence that in fact, by accident, we justhappen to lie near the center of a great expansion wave of cosmic dimensions.We believe our work at this stage gives strong support for this possibility.
36 B. Temple9) How large would the displacement of matter caused by the exp<strong>and</strong>ingwave be, <strong>and</strong> how far out would it extend?For our model, the magnitude of the displacement depends on the valueof the acceleration parameter a. It can be very large or very small, <strong>and</strong> weargue that somewhere in between it can be right on for the first breaking ofthe observed redshift vs luminosity curve near the center. To meet the observations,it has to displace the position of a distant galaxy the right amountto displace the straight line redshift vs luminosity plot of the st<strong>and</strong>ard modelinto the curved graph observed. In their article referenced in our paper (expositionof this appeared as the cover article in Scientific American a fewmonths ago), Clifton <strong>and</strong> Ferreira quote that the bubble of under-densityobserved today should extend out to about one billion lightyears, about atenth of the distance across the visible universe, <strong>and</strong> the size of the centerconsistent with no angular variation is about 15 megaparsecs, about 50 millionlightyears, <strong>and</strong> this is approximately the distance between clusters ofgalaxies, a distance about 1/200 across the visible universe. 610) How do the spacetimes associated with the exp<strong>and</strong>ing waves compareto the spacetime of the st<strong>and</strong>ard model of cosmology?Interestingly, we prove that the spacetimes associated with the exp<strong>and</strong>ingwaves when a ≠ 1 actually have properties surprisingly similar to thest<strong>and</strong>ard model a = 1. Firstly, the exp<strong>and</strong>ing spacetimes (a ≠ 1) look more<strong>and</strong> more like the st<strong>and</strong>ard model a = 1 as you approach the center of expansion.(That is why you have to go far out to see an anomalous acceleration.)6 The following back-of-the-envelope calculation provides a ballpark estimate for what wemight expect the extent of the remnants of one of our exp<strong>and</strong>ing waves might be today.Our thesis is that the self-similar exp<strong>and</strong>ing waves that can exist during the pure radiationphase of the st<strong>and</strong>ard model can emerge at the end of the radiation phase by the dissipationcreated by the strong nonlinearities. Now matter becomes transparent with radiation atabout 300,000 years after the Big Bang, so we might estimate that our wave should haveemerged by about t endrad ≈ 10 5 years after the Big Bang. At this time, the distance oflight-travel since the Big Bang is about 10 5 lightyears. Since the sound speed c/ √ 3 ≈ .58cduring the radiation phase is comparable to the speed of light, we could estimate thatdissipation that drives decay to the exp<strong>and</strong>ing wave might reasonably be operating over ascale of 10 5 lightyears by the end of the radiation phase. Now in the p = 0 expansion thatfollows the radiation phase, the scale factor (that gives the expansion rate) evolves likeR(t) =t 2/3 ,soadistanceof10 5 lightyears at t = t endrad years will exp<strong>and</strong> to a length L at presenttime t present ≈ 10 10 years by a factor of(R(t present ) 10 10 ) 2/3R(t endrad ) ≈ (10 5 ) 2/3 =104.7 ≥ 5 × 10 4 .It follows then that we might expect the scale of the wave at present time to extend overa distance of aboutL =5× 10 5 × 10 4 =5× 10 9 lightyears.This is a third to a fifth of the distance across the visible universe, <strong>and</strong> agrees with theextent of the under-density void region quoted in the Clifton-Ferreira paper, with room tospare.
The “Big Wave” <strong>Theory</strong> for Dark Energy 37Moreover, out to a great distance from the center, say out to about 1/3 to1/2 the distance across the visible universe, (where the anomalous accelerationis apparent), we prove that (to within negligible errors) there is a timecoordinate t such that the 3-space at each fixed t has zero curvature, justlike the st<strong>and</strong>ard model of cosmology, <strong>and</strong> observers fixed in time or at afixed distance from the center will measure distances <strong>and</strong> times exactly thesame as in a Friedmann universe, the spacetime of the st<strong>and</strong>ard model ofcosmology. In technical terms, only line elements changing in space <strong>and</strong> timewill measure dilation of distances <strong>and</strong> times relative to the st<strong>and</strong>ard model.This suggests that it would be easy to mistake one of these exp<strong>and</strong>ing wavesfor the Friedmann spacetime itself until you did a measurement of redshiftvs luminosity far out where the differences are highly apparent (that is, youmeasured the anomalous acceleration).11) Your exp<strong>and</strong>ing wave theory is more complicated than a universefilled with Dark Energy, <strong>and</strong> we have to take into account the Occam’s razorprinciple. What do you think about this assertion?To quote Wikipedia, Occam’s razor states: “The explanation of any phenomenonshould make as few assumptions as possible, eliminating those thatmake no difference in the observable predictions of the explanatory hypothesisor theory.”We could say that our theory does not require the extra hypothesis ofDark Energy or a cosmological constant to explain the anomalous acceleration.Since there is no obvious reason why an expansion wave with one valueof a over another would come out locally at any given location at the end ofthe radiation phase, <strong>and</strong> since we don’t need Dark Energy in the exp<strong>and</strong>ingwave explanation, we could argue that the exp<strong>and</strong>ing wave explanation of theanomalous acceleration is simpler than Dark Energy. But a better answer isthat our theory has an observable prediction, <strong>and</strong> only experiments, not the14th-century principle of Occam, can resolve the physics. Occam’s razor willhave nothing whatsoever to say about whether we are, or are not, near thecenter of a cosmic expansion wave.12) If, as you suggest, Dark Energy doesn’t exist, what is the ingredientof 75% of the mass-energy in our universe?In short, nothing is required to replace it. The term “anomalous acceleration”of the galaxies begs the question “acceleration relative to what?”.The answer is that the anomalous acceleration of the galaxies is an accelerationrelative to the prediction of the st<strong>and</strong>ard model of cosmology. In theexp<strong>and</strong>ing wave theory, we prove that there is no “acceleration” because theanomalous acceleration can be accounted for in redshift vs luminosity by thefact that the galaxies in the exp<strong>and</strong>ing wave are displaced from their anticipatedposition in the st<strong>and</strong>ard model. So the exp<strong>and</strong>ing wave theory requiresonly classical sources of mass-energy for the Einstein equations.
38 B. Temple13) If Dark Energy doesn’t exist, it would be just an invention. What doyou think about Dark Energy theory?Keep in mind that Einstein’s equations have been confirmed without theneed for the cosmological constant or Dark Energy, in every physical settingexcept in cosmology.Dark Energy is the physical interpretation of the cosmological constant.The cosmological constant is a source term with a free parameter (similar tobut different from our a) that can be added to the original Einstein equations<strong>and</strong> still preserve the frame independence, the “general relativity” if you will,of Einstein’s equations. Einstein’s equations express that mass-energy is thesource of spacetime curvature. So if you interpret the cosmological constantas the effect of some exotic mass-energy, then you get Dark Energy. For thevalue of the cosmological constant required to fit the anomalous accelerationobserved in the redshift vs luminosity data, this Dark Energy must accountfor some 73 percent of the mass-energy of the universe, <strong>and</strong> it has to havethe physical property that it anti-gravitates—that is, it gravitationally repelsinstead of attracts. Since no one has ever observed anything that has thisproperty (it would not fall to Earth like an apple, it would fly up like aballoon), it seems rather suspect that such mass-energy could possibly exist.If it does exist, then it also is not like any other mass-energy in that thedensity of it stays constant, stuck there at the same value forever, even asthe universe exp<strong>and</strong>s <strong>and</strong> spreads all the other mass-energy out over larger<strong>and</strong> larger scales—<strong>and</strong> there is no principle that explains why it has the valueit has. 7 On the other h<strong>and</strong>, if you put the cosmological constant on the otherside of the equation with the curvature then there is always some (albeitvery small) baseline curvature permeating spacetime, <strong>and</strong> the zero-curvaturespacetime is no longer possible; that is, the empty-space Minkowski spacetimeof Special Relativity no longer solves the equations. So when the cosmologicalconstant is over on the curvature side of Einstein’s equation, the equationsno longer express the physical principle that led Einstein to discover themin the first place—that mass-energy should be the sole source of spacetimecurvature.Einstein put the cosmological constant into his equations shortly afterhe discovered them in 1915, because this was the only way he could getthe possibility of a static universe. (Anti-gravity holds the static universeup!) After Hubble proved that the universe was exp<strong>and</strong>ing in 1929, Einsteintook back the cosmological constant, declaring it was the greatest blunder ofhis career, as he could have predicted the expansion ahead of time withoutit. At the time, taking out the cosmological constant was interpreted as agreat victory for General Relativity. Since then, cosmologists have becomemore comfortable putting the cosmological constant back in. There are manyrespected scientists who see no problem with Dark Energy.7 In the exp<strong>and</strong>ing wave theory, the principle for determining a is that all values of a neara = 1 should be (roughly) equally likely to appear, <strong>and</strong> one of them did. . .
The “Big Wave” <strong>Theory</strong> for Dark Energy 3914) How does the coincidence in the value of the cosmological constantin the Dark Energy theory compare to the coincidence that the Milky Waymust lie near a local center of expansion in the exp<strong>and</strong>ing wave theory?The Dark Energy explanation of the anomalous acceleration of thegalaxies requires a value of the cosmological constant that accounts for some73 percent of the mass-energy of the universe. That is, to correct for theanomalous acceleration in the supernova data, you need a value of the cosmologicalconstant that is just three times the energy-density of the rest ofthe mass-energy of the universe. Now there is no principle that determinesthe value of the cosmological constant ahead of time, so its value could aprioribe anything. Thus it must be viewed as a great coincidence that it justhappens to be so close to the value of the energy density of the rest of themass-energy of the universe. (Keep in mind that the energy-density of all theclassical sources decreases as the universe spreads out, while the cosmologicalconstant stays constant.) So why does the value of the cosmological constantcome out so close to, just 3 times, the value of the rest of the mass-energyof the universe, instead of 10 10 larger or 10 −10 smaller? This raises a verysuspicious possibility. Since the magnitude of the sources sets the scale forthe overall oomph of the solution, when you need to adjust the equationsby an amount on the order of the sources present in order to fit the data,that smacks of the likelihood that you are really just adding corrections tothe wrong underlying solution. So to us it looks like the coincidence in thevalue of the cosmological constant in the Dark Energy theory may well begreater than the coincidence that we lie near a local center of expansion inthe exp<strong>and</strong>ing wave theory.In summary: Our view is that the Einstein equations make more physicalsense without Dark Energy or the cosmological constant, <strong>and</strong> Dark Energy ismost likely an unphysical fudge factor, if you will, introduced into the theoryto meet the data. But ultimately, whether Dark Energy or an exp<strong>and</strong>ingwave correctly explains the anomalous acceleration of the galaxies can onlybe decided by experiments, not the Copernican Principle or Occam’s razor.References[1] C. Copi, D. Huterer, D.J. Schwarz, G.D. Starkman, On the large-angle anomaliesof the microwave sky, Mon. Not. R. Astron. Soc. 367, 79–102 (2006).[2] T. Clifton, P.G. Ferreira, K. L<strong>and</strong>, Living in a void: testing the Copernicanprinciple with distant supernovae, Phys. Rev. Lett. 101, 131302 (2008).arXiv:0807.1443v2 [astro-ph].[3] T. Clifton <strong>and</strong> P.G. Ferreira, Does dark energy really exist?, Sci. Am., April2009, 48–55.[4] D. Eardley, Self-similar spacetimes: geometry <strong>and</strong> dynamics, Commun. Math.Phys. 37, 287–309 (1974).[5] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,Comm. Pure Appl. Math. 18, 697–715 (1965).
40 B. Temple[6] J. Glimm, P.D. Lax, Decay of solutions of systems of nonlinear hyperbolicconservation laws, Memoirs Amer. Math Soc. 101 (1970).[7] J. Groah <strong>and</strong> B. Temple, Shock-wave solutions of the Einstein equations: existence<strong>and</strong> consistency by a locally inertial Glimm scheme, Memoirs Amer.Math Soc. 172 (2004), no. 813.[8] O. Gron <strong>and</strong> S. Hervik, Einstein’s general theory of relativity with modernapplications in cosmology, Springer, 2007.[9] P. Hartman, A lemma in the structural stability of ordinary differential equations,Proc. Amer. Math. Soc. 11, 610–620 (1960).[10] P.D. Lax, Hyperbolic systems of conservation laws, II, Comm. Pure Appl.Math. 10, 537–566 (1957).[11] J. Smoller <strong>and</strong> B. Temple, Global solutions of the relativistic Euler equations,Comm. Math. Phys. 157, 67–99 (1993).[12] J. Smoller <strong>and</strong> B. Temple, Cosmology, black holes, <strong>and</strong> shock waves beyond theHubble length, Meth. Appl. Anal. 11, 77–132 (2004).[13] J. Smoller <strong>and</strong> B. Temple, Shock-wave cosmology inside a black hole, Proc. Nat.Acad. Sci. 100, 11216–11218 (2003), no. 20.[14] B. Temple <strong>and</strong> J. Smoller, Exp<strong>and</strong>ing wave solutions of the Einstein equationsthat induce an anomalous acceleration into the St<strong>and</strong>ard Model of Cosmology,Proc. Nat. Acad. Sci. 106, 14213–14218 (2009), no. 34.[15] B. Temple <strong>and</strong> J. Smoller, Answers to questions posed by reporters: Temple–Smoller GR exp<strong>and</strong>ing waves, August 19, 2009. http://www.math.ucdavis.edu/~temple/.[16] J. Smoller <strong>and</strong> B. Temple, General relativistic self-similar waves that induce ananomalous acceleration into the St<strong>and</strong>ard Model of Cosmology, Memoirs Amer.Math. Soc., to appear.[17] B. Temple, Numerical refinement of a finite mass shock-wave cosmology,AMS National Meeting, Special Session Numerical Relativity,New Orleans (2007). http://www.math.ucdavis.edu/~temple/talks/NumericalShockWaveCosTalk.pdf.[18] S. Weinberg, Gravitation <strong>and</strong> cosmology: principles <strong>and</strong> applications of the generaltheory of relativity, John Wiley & Sons, 1972.Blake TempleDept of Mathematics1 Shields AvenueDavis, CAUSA 95618e-mail: temple@math.ucdavis.edu
Discrete <strong>and</strong> Continuum Third Quantizationof <strong>Gravity</strong>Steffen Gielen <strong>and</strong> Daniele OritiAbstract. We give a brief introduction to matrix models <strong>and</strong> group fieldtheory (GFT) as realizations of the idea of a third quantization of gravity,<strong>and</strong> present the basic features of a continuum third quantization formalismin terms of a field theory on the space of connections. Buildingup on the results of loop quantum gravity, we explore to what extentone can rigorously define such a field theory. We discuss the relationbetween GFT <strong>and</strong> this formal continuum third-quantized gravity, <strong>and</strong>what it can teach us about the continuum limit of GFTs.Mathematics Subject Classification (2010). 83C45, 83C27, 81T27.Keywords. Group field theory, third quantization, loop quantum gravity,lattice gravity, spin foam models.1. IntroductionSeveral approaches to quantum gravity have been developed over the lastyears [40], with a remarkable convergence, in terms of mathematical structuresused <strong>and</strong> basic ideas shared. Group field theories have been proposed[15, 38, 37, 34] as a kind of second quantization of canonical loop quantumgravity, in the sense that one turns into a dynamical (quantized) field itscanonical wave function. They can be also understood as second quantizationsof simplicial geometry [37]. Because both loop quantum gravity <strong>and</strong>simplicial quantum gravity are supposed to represent the quantization of a(classical) field theory, General Relativity, this brings GFT into the conceptualframework of “third quantization”, a rather inappropriate label for arather appealing idea: a field theory on the space of geometries, rather thanspacetime, which also allows for a dynamical description of topology change.This idea has been brought forward more than 20 years ago in the contextof canonical geometrodynamics, but has never been seriously developed dueto huge mathematical difficulties, nor recast in the language of connectiondynamics, which allowed so much progress in loop quantum gravity.F. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_ 4,© Springer Basel AG 201241
42 S. Gielen <strong>and</strong> D. OritiIn this contribution, we give a brief introduction to the group field theoryformalism, <strong>and</strong> we present in some more detail the idea <strong>and</strong> basic features of acontinuum third-quantization formalism in the space of connections. Buildingup on the results of loop quantum gravity, we explore to what extent one canrigorously define such a field theory. As a concrete example, we shall restrictourselves to the simple case of 3d Riemannian GR. Finally, we discuss therelation between GFT <strong>and</strong> this formal continuum third-quantized gravity,<strong>and</strong> what it can teach us about the continuum limit of GFTs.Our purpose is partly pedagogical, partly technical, partly motivational.The third-quantized framework in the continuum is not a well-defined approachto quantum gravity. Still, its initial motivations remain valid, in ouropinion, <strong>and</strong> it is worth keeping them in mind <strong>and</strong> thus presenting this frameworkin some detail. Most importantly, group field theories can realize thisthird-quantization program, even if at the cost of using a language that isfarther away from that of General Relativity. The GFT formalism itself isstill in its infancy, so that many conceptual aspects are not often stressed.By comparison with continuum third quantization we clarify here some ofthese conceptual issues: the “level of quantization” adopted, the consequentinterpretation of the GFT classical action, the role of topology change, howthe usual canonical quantum theory should be recovered, the difference betweenthe “global” nature of the traditional quantization scheme (dynamicsof a quantum universe) versus the more “local” nature of the GFT approach(quantizing building blocks of the same universe), <strong>and</strong> the implementation ofspacetime symmetries into the GFT context.At the more technical level, our analysis of the third-quantized framework,even if incomplete <strong>and</strong> confined to the classical field theory on the spaceof connections, will be new <strong>and</strong> potentially useful for further developments.Among them, we have in mind both the study of the continuum limit of GFTmodels, which we will discuss in some detail, <strong>and</strong> the construction of simplified“third-quantized” models, in contexts where they can be made amenableto actual calculations, for example in the context of quantum cosmology.Finally, one more goal is to stimulate work on the continuum approximationof GFTs. We will sketch possible lines of research <strong>and</strong> speculate on howthe continuum limit of GFTs can be related to third-quantized continuumgravity, what such a continuum limit may look like, <strong>and</strong> on the emergence ofclassical <strong>and</strong> quantum General Relativity from it. In the end, the hope is tolearn something useful about GFT from the comparison with a more formalframework, but one that is also closer to the language of General Relativity.1.1. Canonical quantization, loop quantum gravity <strong>and</strong> third quantizationThe geometrodynamics program [49] was an early attempt to apply thecanonical quantization procedure to General Relativity, in metric variables,on a manifold of given topology M =Σ× R. The space of “coordinates”is superspace, the space of Riemannian geometries on Σ, i.e. metrics modulospatial diffeomorphisms. One is led to the infamous Wheeler-DeWitt equation
Discrete <strong>and</strong> Continuum Third Quantization of <strong>Gravity</strong> 43[12],[H(x)Ψ[h ij ] ≡ G ijkl δ 2][h](x)δh ij (x)δh kl (x) − (R[h](x) − 2Λ) Ψ[h ij ]=0,(1.1)which is the analogue of the classical Hamiltonian constraint generatingreparametrization of the time coordinate. Here G ijkl = √ hh ij h kl + ... is theDeWitt supermetric, <strong>and</strong> the wave functional Ψ depends on the 3-dimensionalmetric h ij which encodes the geometry of Σ. (1.1) is mathematically illdefinedas an operator equation, <strong>and</strong> suffers from severe interpretational problems.Geometrodynamics therefore never made much progress as a physicaltheory. Nevertheless, quite a lot is known about geometrical <strong>and</strong> topologicalproperties of superspace itself; for a nice review see [23].While from the conceptual point of view superspace is a kind of “metaspace”,or “a space of spaces”, in the sense that each of its “points”, a 3-metric, represents a possible physical space, from the mathematical pointof view it is a manifold in its own right, with a given fixed metric <strong>and</strong> agiven topology. This makes it possible to combine background independencewith respect to physical spacetime required by GR with the use of the background-dependenttools of (almost) ordinary quantum field theory in a thirdquantizedfield theory formalism. This key aspect is shared also by GFTs.With the reformulation of General Relativity in connection variables[1], the canonical quantization program experienced a revival in the form ofloop quantum gravity (LQG) [44, 47]. Superspace is replaced by the spaceof g-connections on Σ, where G is the gauge group of the theory (usually,G = SU(2)). One then takes the following steps towards quantization:• Reparametrize the classical phase space, going from connections to(G-valued) holonomies, exploiting the fact that a connection can be reconstructedif all of its holonomies along paths are known. The conjugate variableto the connection, a triad field, is replaced by its (g-valued) fluxes throughsurfaces; one finds Poisson-commutativity among functions of holonomies <strong>and</strong>non-commutativity among fluxes. In the simplest case, one considers a fixedgraph Γ embedded into Σ <strong>and</strong> takes as elementary variables cylindrical functionsΦ f of holonomies along the edges of the graph <strong>and</strong> fluxes Ee i throughsurfaces intersecting the graph only at a single edge e. One then finds [2]{Φ f , Φ f ′} =0, {E i e,E j e ′ } = δ e,e ′C ij kE k e ,where C ij k are the structure constants of the Lie algebra.One hence has two possible representations, one where wave functions arefunctionals of the connection, <strong>and</strong> a non-commutative flux representation [3].A third representation arises by decomposing a functional of the connectioninto representations of G (these three representations also exist for GFT).• In the connection formulation, one defines a Hilbert space of functionalsof (generalized) g-connections by decomposing such functionals into sumsof cylindrical functions which only depend on a finite number of holonomies,associated to a given graph, each: H ∼ ⊕ Γ H Γ. Holonomy operators act by
44 S. Gielen <strong>and</strong> D. Oritimultiplication <strong>and</strong> flux operators as left-invariant vector fields on G, henceimplementing their non-commutativity.• Define the action of diffeomorphism <strong>and</strong> Gauss constraints on cylindricalfunctions, <strong>and</strong> pass to a reduced Hilbert space of gauge-invariant, (spatially)diffeomorphism-invariant states.While this procedure implements the kinematics of the theory rigorously,the issue of dynamics, i.e. the right definition of the analogue of theHamiltonian constraint (1.1) on the kinematical Hilbert space <strong>and</strong> hence ofthe corresponding space of physical states, is to a large extent still an openissue, also in its covariant sum-over-histories formulation [45].What we have discussed so far is a “first quantization”, where the wavefunction gives probabilities for states of a single hypersurface Σ. One c<strong>and</strong>raw an analogy to the case of a relativistic particle, where the mass-shellconstraint p 2 + m 2 = 0 leads to the wave equation[g μν (x) ∂∂x μ]∂∂x − ν m2 Ψ(x) =0, (1.2)the Klein-Gordon equation. The straightforward interpretation of Ψ(x) asa single-particle wave function fails: In order to define on the kinematicalHilbert space a projection to the solutions of (1.2) which has the correctcomposition properties <strong>and</strong> which on the solutions reduces to a positive definiteinner product (for an overview of possible definitions for inner products<strong>and</strong> their composition laws, see [26]; for analogous expressions in loop quantumcosmology see [10]), one needs to define a splitting of the solutions to(1.2) into positive- <strong>and</strong> negative-frequency solutions. This splitting relies onthe existence of a timelike Killing vector k <strong>and</strong> hence a conserved quantityk · p (“energy”) on Minkowski space; as is well known, for generic metricswithout isometries there is no unambiguous particle concept. This leads to“second quantization” where the particle concept is secondary.The close analogy between (1.1) <strong>and</strong> (1.2) suggests that for a meaningful“one-universe” concept in quantum geometrodynamics, one would need aconserved quantity on all solutions of the constraints. It was argued in [29]that no such quantity exists, so that one has to go to a many-geometriesformalism in which “universes,” 3-manifolds of topology Σ, can be created<strong>and</strong> annihilated, <strong>and</strong> hence to a QFT on superspace. The analogy was pushedfurther, on a purely formal level, by Teitelboim [46], who gave analogues ofthe Feynman propagator, QFT perturbation theory etc. for such a theory.The general idea is to define a (scalar) field theory on superspace S fora given choice of spatial manifold topology Σ, e.g. the 3-sphere, essentiallyturning the wave function of the canonical (first-quantized) theory into anoperator Φ[ 3 h], whose dynamics is defined by an action with a kinetic termof the type∫S free (Φ) = D 3 h Φ[ 3 h]HΦ[ 3 h] , (1.3)Swith H being the Wheeler-DeWitt differential operator of canonical gravity(1.1) here defining the free propagation of the theory. One thinks of a
Discrete <strong>and</strong> Continuum Third Quantization of <strong>Gravity</strong> 45quantum field which is a functional Φ[ 3 h] of the 3-metric defined on Σ; theoperator Φ[ 3 h] creates a 3-manifold of topology Σ with metric h.The quantum theory would be “defined” by the perturbative expansionof the partition function Z = ∫ DΦ e −S(Φ) in “Feynman diagrams”. Addinga term cubic in Φ would give diagrams corresponding to processes such asthe following:+ + ...Thus such a formalism also has the attractive aspect of incorporatingtopology change. In the simplified setting of homogeneous 3-spheres (i.e.a third-quantized minisuperspace model), it was explored by Giddings <strong>and</strong>Strominger in [20] in the hope to find a dynamical mechanism determiningthe value of the cosmological constant, under the name “third quantization.”The Feynman amplitudes are given by the quantum-gravity path integralfor each spacetime topology, identified with a particular interaction ofuniverses, with the one for trivial topology representing a one-particle propagator,a Green function for the Wheeler-DeWitt equation. Other features ofthis very formal setting are: 1) the classical equations of motion for the scalarfield on superspace are a non-linear extension of the Wheeler-DeWitt equation,due to the interaction term in the action, i.e. the inclusion of topologychange; 2) the perturbative third-quantized vacuum of the theory is the “nospace”state, <strong>and</strong> not any state with a semiclassical geometric interpretation.In the third-quantization approach one has to deal with the quantizationof a field theory defined on an infinite-dimensional manifold, clearly a hopelesstask. In a connection formulation, there is more hope of at least definingthe classical theory, due to the work done in LQG. To make more progress,one will have to reduce the complexity of the system. One possibility is topass to a symmetry-reduced sector of GR before quantization, obtaining athird-quantized minisuperspace model, as done in metric variables in [20] <strong>and</strong>for connection variables in [9]. Another possibility is to consider, instead ofa continuous manifold Σ, a discrete structure such as a simplicial complexwhere one is only interested in group elements characterizing the holonomiesalong links. This is the idea behind GFTs. Before getting to the GFT setting,we will show how the third-quantization idea is implemented rigorously in asimpler context, that of 2d quantum gravity, by means of matrix models.2. Matrix models: a success storyA simple context in which the idea of “third quantization” of gravity can berealized rigorously is that of 2d Riemannian quantum gravity. The quantizationis achieved by using matrix models, in the same “go local, go discrete”
46 S. Gielen <strong>and</strong> D. Oritiway that characterizes group field theories, which indeed can be seen as ageneralization of the same formalism.Define a simple action for an N × N Hermitian matrix M, givenbyS(M) = 1 2 TrM 2 − √ g TrM 3 = 1N 2 M i jM j i − √ g M i jM j kM k iN= 1 2 M i jK jkli M k l − √ g(2.1)M i jM m nM k l V jmknliNwith propagator (K −1 ) jkli = K jkli = δ j k δ l i <strong>and</strong> vertex V jmknli = δ j m δ n k δ l i.Feynman diagrams are constructed, representing matrices by two pointscorresponding to their indices, out of: lines of propagation (made of twostr<strong>and</strong>s), non-local “vertices” of interaction (providing a re-routing of str<strong>and</strong>s)<strong>and</strong> faces (closed loops of str<strong>and</strong>s) obtained after index contractions. Thiscombinatorics of indices can be given a simplicial interpretation by viewing amatrix as representing an edge in a 2-dimensional (dual) simplicial complex:MijMijijMjiijMkikMjkThe diagrams used in evaluating the partition function Z = ∫ dM ij e −S(M)correspond to complexes of arbitrary topology, obtained by arbitrary gluingof edges to form triangles (in the interaction) <strong>and</strong> of triangles along commonedges (dictated by the propagator). A discrete spacetime emerges as a virtualconstruction, encoding the interaction processes of fundamental space quanta.The partition function, expressed in terms of Feynman amplitudes, isZ = ∑ Γg V ΓN F Γ− 1 2 V Γ= ∑ Γg V ΓN χ ,where V Γ is the number of vertices <strong>and</strong> F Γ the number of faces of a graph Γ,<strong>and</strong> χ is the Euler characteristic of the simplicial complex.Each Feynman amplitude is associated to a simplicial path integral forgravity discretized on the associated simplicial complex ∫ Δ. The action for 2dGR with cosmological constant on a 2d manifold is 1 G d 2 x √ g (−R(g)+Λ)=− 4π G χ + ΛaGt, where the surface is discretized into t equilateral triangles ofarea a. We can now identify (t Δ = V Γ ,eachvertexisdualtoatriangle)Z = ∑ Γg V ΓN χ ≡ ∑ Δe + 4π G χ(Δ) − aΛ G t Δ, g ≡ e − ΛaG , N ≡ e+ 4π G ,which is a sum over histories of discrete 2d GR, trivial as the only geometricvariable associated to each surface is its area. In addition to this sum overgeometries, we obtain a sum over 2d complexes of all topologies; the matrix
Discrete <strong>and</strong> Continuum Third Quantization of <strong>Gravity</strong> 47model defines a discrete third quantization of 2d GR! We can control bothsums [11, 21] by exp<strong>and</strong>ing in a topological parameter, the genus h,Z = ∑ Δg t ΔN 2−2h = ∑ hN 2−2h Z h (g) =N 2 Z 0 (g) +Z 1 (g) +... .In the limit N →∞, only spherical simplicial complexes (of genus 0) contribute.As N →∞one can also define a continuum limit <strong>and</strong> match theresults of the continuum GR path integral: Exp<strong>and</strong>ing Z 0 (g) inpowersofg,Z 0 (g) = ∑ tt γ−3 ( gg c) t≃ t→∞ (g − g c ) 2−γ ,we see that as t →∞, g → g c (γ>2), Z 0 (g) diverges, signaling a phasetransition. In order to identify it as a continuum limit we compute the expectationvalue for the area: 〈A〉 = a ∂ ∂g ln Z a0(g) ≃ t→∞ g−g c.Thuswecansend a → 0, t→∞, tuning at the same time the coupling constant to g c ,toget finite macroscopic areas. This continuum limit reproduces [11, 21] the resultsfrom a continuum 2d gravity path integral when this can be computed;otherwise it can be taken to define the continuum path integral.One can compute the contribution from all topologies in the continuumlimit, defining a continuum third quantization of 2d gravity, using the socalleddouble-scaling limit [11, 21]. One finds that in the continuum limit∑Z ≃ t→∞ κ 2h−2 f h , κ −1 := N (g − g c ) (2−β)2hfor some constant β. We can then take the limits N →∞,g→ g c , holdingκ fixed. The result is a continuum theory to which all topologies contribute.3. Group field theory: a sketchy introductionOne can construct combinatorial generalizations of matrix models to highertensor models; however such models do not possess any of the nice scalinglimits (large-N limit, continuum <strong>and</strong> double-scaling limit) that allow to controlthe sum over topologies in matrix models <strong>and</strong> to relate the quantumamplitudes to those of continuum gravity. One needs to generalize further bydefining corresponding field theories, replacing the indices of the tensor modelsby continuous variables. For the definition of good models for quantumgravity, the input from other approaches to quantum gravity is crucial.We now describe in some detail the GFT formalism for 3d RiemannianGR, clarifying the general features of the formalism in this specific example.Consider a triangle in R 3 <strong>and</strong> encode its kinematics in a (real) field ϕ,a function on the space of geometries for the triangle, parametrized by threesu(2) elements x i attached to its edges which are interpreted as triad discretizedalong the edges. Using the non-commutative group Fourier transform[17, 18, 5], based on plane waves e g (x) =e i⃗pg·⃗x on g ∼ R n (with coordinates
48 S. Gielen <strong>and</strong> D. Oriti⃗p g on SU(2)), the field can alternatively be seen as a function on SU(2) 3 ,∫ϕ(x 1 ,x 2 ,x 3 )= [dg] 3 ϕ(g 1 ,g 2 ,g 3 ) e g1 (x 1 )e g2 (x 2 )e g3 (x 3 )where the g i ∈ SU(2) are thought of as parallel transports of the gravityconnection along links dual to the edges of the triangle represented by ϕ.In order to define a geometric triangle, the vectors x i have to sum tozero. We thus impose the constraint (⋆ is a non-commutative product reflectingthe non-commutativity of the group multiplication in algebra variables)∫ϕ = C⋆ϕ, C(x 1 ,x 2 ,x 3 )=δ 0 (x 1 +x 2 +x 3 ):= dge g (x 1 + x 2 + x 3 ). (3.1)In terms of the dual field ϕ(g 1 ,g 2 ,g 3 ), the closure constraint (3.1) impliesinvariance under the diagonal (left) action of the group SU(2),∫ϕ(g 1 ,g 2 ,g 3 )=Pϕ(g 1 ,g 2 ,g 3 )= dhφ(hg 1 ,hg 2 ,hg 3 ). (3.2)SU(2)In group variables, the field can be best depicted graphically as a 3-valentvertex with three links, dual to the three edges of the closed triangle (Fig.1). This object will be the GFT building block of our quantum space.A third representation is obtained by decomposition into irreduciblerepresentations (compare with the LQG construction, sect. 5),ϕ(g 1 ,g 2 ,g 3 )=∑ϕ j 1j 2 j 3m 1 m 2 m 3D j 1m 1 n 1(g 1 )D j 2m 2 n 2(g 2 )D j 3m 3 n 3(g 3 ) C j 1j 2 j 3n 1 n 2 n 3,j 1 ,j 2 ,j 3(3.3)where C j 1j 2 j 3n 1 n 2 n 3is the Wigner invariant 3-tensor, the 3j-symbol.Graphically, one can think of the GFT field in any of the three representations(Lie algebra, group, representation), as appropriate:φ ( g g g ) φ( j j j )1 2 3 1 2 3g (j )1 1g (j )2 2g (j )3 3g (j )1 1g (j )2 2g (j )3 3g (j )1 1Figure 1. Different representations of the GFT field ϕ.g (j )2 2g (j )3 3The convolution (in the group or Lie algebra picture) or tracing (in therepresentation picture) of multiple fields with respect to a common argumentrepresents the gluing of triangles along common edges, <strong>and</strong> thus the formationof more complex simplicial structures, or more complex dual graphs:g ( J )4 4g ( J )4 4g ( J )g ( J )1 1 1 1g ( J )2 2g ( J )3 3g ( J )2 2g ( J )3 3J g ( )5 5g ( J )5 5
Discrete <strong>and</strong> Continuum Third Quantization of <strong>Gravity</strong> 49The corresponding field configurations represent extended chunks ofspace; a generic polynomial observable is associated with a particular quantumspace.We now define a classical dynamics for the GFT field. In the interactionterm four geometric triangles should be glued along common edges to forma tetrahedron. The kinetic term should encode the gluing of two tetrahedraalong common triangles. With ϕ 123 := ϕ(x 1 ,x 2 ,x 3 ), we define the actionS = 1 ∫[dx] 3 ϕ 123 ⋆ϕ 123 − λ ∫[dx] 6 ϕ 123 ⋆ϕ 345 ⋆ϕ 526 ⋆ϕ 641 , (3.4)24!where ⋆ relates repeated indices as φ i ⋆φ i :=(φ⋆φ − )(x i ), with φ − (x)=φ(−x).One can generalize the GFT field to a function of n arguments whichwhen satisfying (3.1) or (3.2) can be taken to represent a general n-gon (dualto an n-valent vertex) <strong>and</strong> glued to other fields, giving a polygonized quantumspace just as for triangles. There is also no difficulty in considering a moregeneral action in which, for given ϕ(x i ), one adds other interaction termscorresponding to the gluing of triangles (polygons) to form general polyhedraor more pathological configurations (e.g. with multiple identifications amongtriangles). The only restriction may come from the symmetries of the action.The projection (3.2) takes into account parallel transport between differentframes; from (3.2) <strong>and</strong> (3.4) we can identify a propagator <strong>and</strong> a vertex:∫ 3∏K(x i ,y i )= dh t (δ −xi ⋆e ht )(y i ),i=1∫ ∏V(x i ,y i )= dh tt6∏(δ −xi ⋆e htt ′ )(y i );i=1(3.5)the variables h t <strong>and</strong> h tτ arising from (3.2) are interpreted as parallel transportsthrough the triangle t <strong>and</strong> from the center of the tetrahedron τ totriangle t, respectively. We may represent the propagator <strong>and</strong> vertex as follows:t d(3.6)y 2 y 3 y 1y 1 y 4 y 6ty 3 x 4 x 5t bx 1t a x 2τx 6y 2 t cx 3 y 5x 1 x 2 x 3The integr<strong>and</strong>s in (3.5) factorize into products of functions associatedto str<strong>and</strong>s (one for each field argument), with a geometrical meaning: Thevariables (x i ,y i ) associated to the edge i correspond to the edge vectors in theframes associated to the triangles t, t ′ sharing it; opposite edge orientationsin different triangles <strong>and</strong> a mismatch between reference frames associated tothe same triangle in two different tetrahedra are taken into account.
50 S. Gielen <strong>and</strong> D. OritiUsing the group Fourier transform we obtain the form of Boulatov [8],S[φ] = 1 ∫[dg] 3 ϕ(g 1 ,g 2 ,g 3 )ϕ(g 3 ,g 2 ,g 1 )2− λ ∫[dg] 6 ϕ(g 12 ,g 13 ,g 14 )ϕ(g 14 ,g 24 ,g 34 )ϕ(g 34 ,g 13 ,g 23 )ϕ(g 23 ,g 24 ,g 12 ).4!(3.7)In this group representation, the kinetic <strong>and</strong> vertex functions are∫ 3∏K(g i , ˜g i )= dh δ(g k h˜g −1k ),k=1∫ 4∏(3.8)∏V(g ij ,g ji )= dh i δ(g ij h i h −1j g −1ji ).i=1 i
Discrete <strong>and</strong> Continuum Third Quantization of <strong>Gravity</strong> 51(where propagators correspond to triangles <strong>and</strong> the vertices to tetrahedra):∫Z = Dϕe −S[ϕ] = ∑ λ Nsym[Γ] Z(Γ),Γwhere N is the number of interaction vertices in the Feynman graph Γ, sym[Γ]is a symmetry factor for Γ, <strong>and</strong> Z(Γ) the corresponding Feynman amplitude.In Lie algebra variables, the amplitude for a generic Feynman diagram is [5]∫ ∏ ∏Z(Γ) = dh p dx f e i ∑ f Tr x f H f, (3.11)p fwhere f denotes loops in the graph which bound faces of the 2-complexformed by the graph; such faces are dual to edges in the simplicial complex,<strong>and</strong> are associated a Lie algebra variable x f . For each propagator p we integrateover a group element h p interpreted as a discrete connection. (3.11)corresponds to the simplicial path integral of 3d gravity in first-order formwith action∫S(e, ω) = Tr (e ∧ F (ω)) , (3.12)Mwith su(2)-valued triad e i (x) <strong>and</strong> su(2)-connection ω j (x) with curvatureF (ω). Introducing the simplicial complex Δ <strong>and</strong> its dual cellular complexΓ, we can discretize e i along edges of Δ as x e = ∫ e e(x) =xi τ i ∈ su(2),<strong>and</strong> ω j along links of Γ, dual to triangles of Δ, as h L = e ∫ L ω ∈ SU(2). Thediscrete curvature is given by the holonomy around the face f in Γ dual toan edge e of Δ, obtained as an ordered product H f = ∏ L∈∂f h L ∈ SU(2).The discrete counterpart of (3.12) is then the action ∑ e Tr x eH e in (3.11).The GFT model we have introduced succeeds in at least one point wheretensor models failed: in defining amplitudes for its Feynman diagrams (identifiedwith discrete spacetimes), arising in a perturbative expansion aroundthe “no-space state,” that correctly encode classical <strong>and</strong> quantum simplicialgeometry <strong>and</strong> that can be nicely related to a simplicial gravity action.The Feynman amplitudes can be computed in the other representationswe have at our disposal. In the group picture one obtains a delta functionδ( ∏ L∈∂f h L) in the overall amplitude, making the flatness constraint explicit;the expression of Z(Γ) in terms of representations is an assignment of an irreducibleSU(2) representation j f to each face of Γ, <strong>and</strong> of a group intertwinerto each link of the complex, a spin foam [35, 42, 36]:⎛Z(Γ) = ⎝ ∏ f⎞∑⎠ ∏ (2j f +1) ∏fvj f{ }j1 j 2 j 3.j 4 j 5 j 6This is the Ponzano-Regge model [43] for 3d quantum gravity. The correspondencebetween spin-foam models <strong>and</strong> GFT amplitudes is generic.The above constructions can be generalized to the computation of GFTobservables, in particular n-point functions, which translate into the calculationof Feynman amplitudes for diagrams/simplicial complexes of arbitrary
52 S. Gielen <strong>and</strong> D. Orititopology <strong>and</strong> with boundaries. These in turn take again the form of simplicialgravity path integrals on the corresponding topology.Once more, we have a discrete realization of the third-quantization idea.We refer to the literature (e.g. the recent more extended introduction in[34]) for more details <strong>and</strong> for an account of recent results, in particular forthe definition of GFT models of 4d quantum gravity, for the identificationof (discrete) diffeomorphism symmetry in GFT, <strong>and</strong> for the proof that ageneralization of the large-N limit of matrix models holds true also in (some)GFT models, leading to the suppression of non-trivial topologies, as well asfor more work on the topological properties of the GFT Feynman expansion.4. Continuum third quantization: heuristicsGoing back to the idea of a third quantization of continuum gravity, we wouldlike to define a field theory on the configuration space of GR that reproducesthe constraints of canonical GR through its equations of motion. Since suchconstraints have to be satisfied at each point in Σ, the action should reproduceequations of the form (γ α are the variables of GR, metric or connection)δC i[γ α , ; x Φ[γ α ]=0, (4.1)δγ αwhere we have introduced a field which is classically a functional of γ α , i.e.a canonical quantum gravity wave function. The naive action∫∑S = Dγ α Φ[γ α ] C i[γ α ,i)]δΦ[γ α ] (4.2)δγ αreproduces constraints C i = 0 independent of the point on Σ. This was notedby Giddings <strong>and</strong> Strominger in [20], who took C to be the Hamiltonian constraintintegrated over Σ = S 3 .To get the right number of constraints, one has to increase the numberof degrees of freedom of Φ. One possibility, for only a single constraint perpoint C, would be to add an explicit dependence of Φ on points in Σ:∫ ∫[ )S = Dγ α d D δx Φ[γ α ; x) C γ α , ; x Φ[γ α ; x). (4.3)Σδγ αDefining a formal differential calculus bydΦ[γ α ; x)/dΦ[δ β ; y) =δ[γ α − δ β ]δ(x − y) ,the variation dS/dΦ would reproduce (4.1). However, such a dependence onpoints seems unnatural from the point of view of canonical gravity; pointsin Σ are not variables on the phase space. An alternative <strong>and</strong> more attractivepossibility arises by realizing that in canonical GR one really has anintegrated Hamiltonian involving non-dynamical quantities which are originallypart of the phase space. We thus integrate the constraints (4.1) using
Discrete <strong>and</strong> Continuum Third Quantization of <strong>Gravity</strong> 53Lagrange multipliers, <strong>and</strong> extend Φ to be a functional of all original phasespace variables. An action is then defined as∫(∫))S = Dγ α DΛ i Φ[γ α , Λ i ] d D x Λ i δ(x) C i[γ α , ; x Φ[γ α , Λ i ]. (4.4)Σδγ αVariation with respect to the scalar field Φ then yields the equations of motion[ ](∫))δ0=C γ α , , Λ i Φ[γ α , Λ i δ] ≡; x Φ[γ α , Λ i ].δγ α δγ αd D x Λ i (x) C i[γ α ,Σ(4.5)C is now the Hamiltonian of canonical gravity; since (4.5) has to hold forarbitrary values of Λ i (x), it is equivalent to (4.1). The fact that Λ i are nondynamicalis apparent from the action which does not contain functionalderivatives with respect to them. Introducing smeared constraints is alsowhat one does in LQG since equations of the form (4.1) are highly singularoperator equations, just as (1.1). We are now treating all the constraints ofcanonical gravity on equal footing, <strong>and</strong> expect all of them to result from theequations of motion of the theory; if we are working in metric variables, ourfield is defined on the space of metrics on Σ before any constraint is imposed.Implicit in the very idea of a continuum third quantization of gravity isthe use of a non-trivial kinetic term, corresponding to quantum constraints.On the one h<strong>and</strong> this has to be immediately contrasted with the st<strong>and</strong>ardGFT action which has instead a trivial kinetic term; we will discuss thispoint more extensively in the concluding section. On the other h<strong>and</strong>, sincethe classical action of the field theory on superspace encodes <strong>and</strong> makes useof the quantum constraint operators, choices of operator ordering <strong>and</strong> relatedissues enter prominently also at this level; this should clarify why the classicalGFT action makes use, for example, of ⋆-products in its definition.Onemoreimportantissueisthatoneshouldexpectthesymmetriesof classical (<strong>and</strong> presumably quantum) GR – spatial diffeomorphisms, timereparametrizations etc. – to be manifest as symmetries of (4.4) in some way.The kind of symmetries one normally considers, transformations acting onspacetime that do not explicitly depend on the dynamical fields, will from theviewpoint of the third-quantized framework defined on some form of superspaceappear as global rather than local symmetries 1 . This basic observationis in agreement with recent results in GFT [4], where one can identify a globalsymmetry of the GFT action that has the interpretation of diffeomorphisms.If we now focus on GR in connection variables (γ α = ωiab ), the variable, some function of the frame field, would be represented as aconjugate to ω abifunctional derivative. We then follow the same strategy as in LQG, redefiningvariables on the phase space to obtain a field which depends on holonomiesonly, <strong>and</strong> rewriting the Hamiltonian in terms of holonomies. Issues of operatorordering <strong>and</strong> regularization will be dealt with in the way familiar from LQG.As an example which will allow us to implement this procedure to some extent1 In extensions of st<strong>and</strong>ard GR where one allows for topology change at the classical levelby allowing degenerate frame fields, the situation may be different [27].
54 S. Gielen <strong>and</strong> D. Oriti<strong>and</strong> to make connections to the GFT formalism, we specialize to the case of3d Riemannian GR in first-order formulation (3.12), heuristically defining∫S = Dωiab Dχ a DΩ ab Φ[ωiab ,χ a , Ω ab ]HΦ[ωi ab ,χ a , Ω ab ], (4.6)where H is the Hamiltonian of 3d GR (without cosmological constant),H = − 1 ∫ {}d 2 x ɛ abc χ a ɛ ij Rij bc +Ω ab ∇ (ω)jπ j ab, (4.7)2 Σinvolving Rij bc , the curvature of the su(2) connection ωab i , the conjugate momentumπ j bc ∼ ɛij ɛ abc e a i , the covariant derivative ∇(ω) ,<strong>and</strong>ansu(2)-valuedscalar ɛ abc χ a . Alternatively one could go to a reduced configuration space bydividing out gauge transformations generated by ∇ (ω)∫S =Dωiab Dχ a Φ[ωi ab ,χ a ]jπ j ab(− 1 ∫d 2 xɛ abc χ a (x)ɛ ij Rij bc (x)2 Σ= 0, <strong>and</strong> define)Φ[ωi ab ,χ a ],(4.8)where Φ is now understood as a gauge-invariant functional. At this formallevel, an analogous “construction” would be possible in metric variables, butthe construction of a Hilbert space for GR in connection variables done inLQG will allow us to make more progress towards a rigorous definition of theaction if we work with an SU(2) connection. Let us investigate this in detail.5. Towards a rigorous construction: passing to holonomies,decomposing in graphsConnection variables allow the use of technology developed for LQG (summarizedin [47]) to give a more rigorous meaning to (4.4). One changesphase space variables to holonomies, <strong>and</strong> decomposes functionals on thespace of (generalized) connections into functions depending on a finite numberof holonomies. A measure in (4.4) can be defined from a normalizedmeasure on the gauge group, extended to arbitrary numbers of holonomiesby so-called projective limits; the Hilbert space is defined as the space ofsquare-integrable functionals on the space of (generalized) connections withrespect to this measure. This strategy works well for connections with compactgauge group; here we require functionals that also depend on Lagrangemultipliers. One way of dealing with scalars is by defining “point holonomies”U λ,x (χ) :=exp(λχ(x)) ∈ G where the field χ is in the Lie algebra g (for (4.8),g = su(2)). One can reconstruct χ from its point holonomies, taking derivativesor the limit λ → 0 [47, sect. 12.2.2.2]. The Lagrange multiplier <strong>and</strong>connection can then be treated on the same footing. We will in the followingfocus on 3d Riemannian GR but a similar procedure would be applicable inany setting where one has a connection with compact gauge group.We detail the construction step by step. Consider a graph Γ embeddedinto the surface Σ, consisting of E edges <strong>and</strong> V vertices. Define a functional
Discrete <strong>and</strong> Continuum Third Quantization of <strong>Gravity</strong> 55of a g-connection ω <strong>and</strong> a g-valued scalar χ, called a cylindrical function, by 2Φ Γ,f [ω abi ,χ a ]=f(h e1 (ω),...,h eE (ω),U λ,x1 (χ),...,U λ,xV (χ)), (5.1)where f is a square-integrable function on G E+V whose arguments are theholonomies of the connection along the edges e i of Γ <strong>and</strong> the point holonomiesof χ at the vertices x k . Functional integration of such functions is defined as∫∫Dωiab Dχ a Φ Γ,f [ωi ab ,χ a ]:= [dh] E [dU] V f(h 1 ,...,h E ,U 1 ,...,U V ).(5.2)Now the linear span of the functionals (5.1) is dense in the required spaceof functionals of generalized connections [44]. An orthonormal basis for thefunctions associated to each graph, i.e. for square-integrable functions onG E+V , is provided by the entries of the representation matrices for groupelements in (irreducible) representations (due to the Peter-Weyl theorem),T Γ,je ,C v ,m e ,n e ,p v ,q v[ωi ab ,χ a ] ∼ D j 1m 1n 1(h e1 (ω)) ...D j EmE n E(h eE (ω))× D C 1p 1 q 1(U λ,x1 (χ)) ...D C Vp V q V(U λ,xV (χ)),(5.3)where we omit normalization factors. Note that one associates one such representationj e to each edge <strong>and</strong> another one C v to each vertex in the graph.If we go to gauge-invariant functionals of the connection ωiab <strong>and</strong> thefield χ a , passing to a reduced phase space where one has divided out gaugeorbits, a basis is provided by functions associated to generalized spin networks,graphs whose edges start <strong>and</strong> end in vertices, labeled by irreduciblerepresentations at edges <strong>and</strong> vertices <strong>and</strong> intertwiners at vertices projectingonto singlets. For (4.8), there is only a single SU(2), <strong>and</strong> schematically wehave (an analogous decomposition would be possible at the level of (5.1)already)∑Φ[ωi ab ,χ a ]= Φ(Γ,j e ,C v ,i v ) T Γ,je ,C v ,i v[ωi ab ,χ a ] , (5.4){Γ,j e,C v,i v}where the spin-network functions T Γ,je,C v,i vare (again up to normalization)T Γ,je,C v,i v[ωi ab ,χ a ] ∼ D j 1m 1 n 1(h e1 (ω)) ...D j EmE n E(h eE (ω))D C 1p 1 q 1(U λ,x1 (χ))...D C Vp V q V(U λ,xV (χ))(i 1 ) n 1n 2 ...q 1m 1 m 2 ...p 1...(i V ) q V ...n Ep V ...m E;(5.5)one takes the functions (5.3) in the representations j e <strong>and</strong> C v <strong>and</strong> contractswith intertwiners i v at the vertices. We may represent a spin network asfollows:2 (5.1) bears a close resemblance to “projected cylindrical functions” used in [31] to obtaina Lorentz-covariant 4d formalism, both functions of connection <strong>and</strong> Lagrange multiplier.
56 S. Gielen <strong>and</strong> D. OritiC 2i 2j 2j 1✐•C1✐ •i 1i 4C 4✐•j 3j 7j 4 j 5 j6i 3• ✐C 3Using the expansion (5.4) of the field Φ, (4.8) can be formally written asS = ∑s,s ′ ∈S∫Φ(s)Φ(s ′ )(Dωi ab Dχ a T s [ω, χ] − 1 ∫)d 2 xɛ abc χ a ɛ ij Rijbc T s ′[ω, χ],2 Σ(5.6)whereweuses = {Γ,j e ,C v ,i v } to describe a spin network, <strong>and</strong> S is the set ofall spin networks embedded into Σ. The task is then to determine the actionof the Hamiltonian on a spin-network function <strong>and</strong> to use the orthogonalityproperty 〈T s ,T s ′〉 = δ s,s ′ for normalized spin-network functions to collapsethe sums to a single sum. Let us again be more explicit on the details.The Hamiltonian can be expressed in terms of holonomies, following thestrategy for 3d LQG [33]: For given s <strong>and</strong> s ′ , consider a cellular decompositionof Σ, chosen so that the graphs of s <strong>and</strong> s ′ are contained in the 0-cells <strong>and</strong>1-cells, into plaquettes p with sides of coordinate length ɛ; approximate∫d 2 xɛ abc χ a (x)ɛ ij Rij bc (x) ≈ ∑ ɛ 2 Tr(χ p R p [ω]) ≈ 2 ∑Tr(U λ,p (χ)h p [ω]) ,Σλpp(5.7)where (χ p ) bc = ɛ abc χ a (x p )forsomex p ∈ p <strong>and</strong> (R p ) bc = ɛ ij Rijbc which weapproximate by the point holonomy <strong>and</strong> the holonomy h p around p:U λ,p =1+λχ p + ..., h p =1+ 1 2 ɛ2 R p + ... . (5.8)Instead of treating λ as a parameter taken to zero at the end, one could takeλ = λ(ɛ) <strong>and</strong> then a single limit ɛ → 0. The action becomes, formally,∑∫(S = limɛ→0Φ(s)Φ(s ′ ) DωDχT s [ω, χ] − 1 ∑Tr ( U λ,p (χ)h p [ω] )) T s ′[ω, χ].λλ→0 s,s ′ p(5.9)Next one computes the action of the Hamiltonian on spin-network functions.Since the latter form a basis for gauge-invariant functionals one can exp<strong>and</strong>− 1 ∑Tr(U λ,p (χ)h p [ω])T s ′[ωiab ,χ a ]= ∑c H sλ′ ,s ′′T s ′′[ωab i ,χ a ] . (5.10)ps ′′ ∈S
Discrete <strong>and</strong> Continuum Third Quantization of <strong>Gravity</strong> 57Then one uses the orthogonality property for spin-network functions,∫Dωiab Dχ a T s [ωiab ,χ a ]T s ′′[ωi ab ,χ a ]∫≡ [dh] E [dU] V T s [h i ,U k ]T s ′′[h i ,U k ]=δ s,s ′′ ,(5.11)where integration of spin-network functions is defined as in (5.2); one takesthe union of the graphs Γ <strong>and</strong> Γ ′ used to define s <strong>and</strong> s ′ <strong>and</strong> extends thespin-network functions trivially to the union, integrating over E + V copiesof SU(2) (with normalized Haar measure), where E is the number of edges<strong>and</strong> V the number of vertices in the union of Γ <strong>and</strong> Γ ′ . The action becomes∑S = limɛ→0Φ(s)Φ(s ′ )c H s ,s, (5.12)′λ→0 s,s ′ ∈Swith the action of the Hamiltonian now hidden in the matrix elements c H s ′ ,s ,<strong>and</strong> one has to worry about ɛ → 0<strong>and</strong>λ → 0. (In LQG the only limit isɛ → 0; the Hamiltonian is a finite <strong>and</strong> well-defined operator in this limit [33].)In the traditional LQG formulation [44, 47] the effect of Tr(U λ,p (χ)h p [ω])on a spin network is to add p as a loop with j =1/2, a vertex at x p withC v =1/2 <strong>and</strong> an intertwiner contracting the two SU(2) matrices, so that− 1 ∑λpTr(U λ,p (χ)h p [ω])T s ′[ωiab ,χ a ]=− 1 λ∑pT s ′ ∪p j=Cxp =1/2 [ωab i ,χ a ] ,(5.13)so that c H s ′ ,s = − 1 ∑′′ λ p δ s ′′ ,s ′ ∪p j=Cxp, <strong>and</strong> hence=1/2∑S = limɛ→0− 1 ∑Φ(s ∪ p j=Cxp =1/2)Φ(s). (5.14)λλ→0 s∈S pThere have been suggestions for non-graph-changing holonomy operators,which would mean that c H s ′ ,s = 0 unless ′′s′ <strong>and</strong> s ′′ define the same graph, sothat one would have a (free) action “local in graphs”. Here a more concretedefinition relies on input from canonical quantum gravity (here LQG).The expansion of Φ into spin-network functions, imposed by the structureof the LQG Hilbert space, suggests a change of perspective: While aquantum field Φ[ωiab ,χ a ] would be thought of as creating a hypersurface oftopology Σ with a continuous geometry, the operators Φ(s) would create labeledgraphs of arbitrary valence <strong>and</strong> complexity. One would consider a Fockspace of “many-graph states”, a huge extension of the LQG Hilbert space:H LQG ∼ ⊕ H Γ −→ H 3rd quant. ∼ ⊕ H n LQG ∼ ⊕ ⊕H n Γ. (5.15)ΓnΓ nThis (we stress again, purely formal) picture in terms of graphs is closerto the discrete structures in the interpretation of GFT Feynman graphs, <strong>and</strong>suggests to take the generalization of GFT to fields with arbitrary number ofarguments seriously, as a vertex of arbitrary valence n would be described by aGFT field with n arguments. In (5.12), the graphs forming the spin networksare embedded into Σ, but the next step towards a GFT interpretation in
58 S. Gielen <strong>and</strong> D. Oritiwhich one views them as abstract is conceivably straightforward. A crucialdifference between the continuum third quantization <strong>and</strong> GFT, however, isthat between entire graphs represented by Φ(s), <strong>and</strong> building blocks of graphsrepresented by the GFT field, where entire graphs emerge as Feynman graphsin the quantum theory. We will discuss this issue further in the next section.If one wants to add an interaction term to the action, there is not muchguidance from the canonical theory. Nevertheless, the addition of such a termwould in principle pose no additional difficulty. One would, in adding such aterm, presumably impose conservation laws associated to the geometric interpretationof the quantities imposed, e.g. conservation of the total area/volumerepresented by two graphs merged into one, which when expressed in termsof the canonically conjugate connection would mean requiring “locality” inthe connection. This strategy is proposed for a minisuperspace model in [9].6. Lessons for group field theories <strong>and</strong> some speculationsLet us now conclude with a discussion of the relation between the two formulationsof third-quantized gravity we have presented.One could try to derive the GFT setting from the formal continuumthird-quantization framework, by somehow discretizing (5.12). The relevantquestions are then: What is the limit in which a suitable GFT arises from(5.12)? What is the expression for the action if one adapts the Hamiltonianconstraint to a generic triangulation, dual to the spin network graph? Onepossibility seems to be to assume that Φ(s) = 0 except on a fixed graph γ 0 .One would then hope to be able to identify the nontrivial contributions asɛ → 0, <strong>and</strong> then do a summation over the variables C v to get a GFT model.Here we immediately encounter a basic issue: whatever graph γ 0 chosen, ithas to be one of those appearing in the decomposition of a gauge-invariantfunctional coming from LQG, <strong>and</strong> so it will have to be a closed graph, whilethe fundamental GFT field is typically associated to an open graph.However, not much faith can be placed on the third quantization in thecontinuum as a consistent theory of quantum gravity. It is more sensible toview the problem in the opposite direction, <strong>and</strong> consider such a continuumformulation as an effective description of the dynamics of a more fundamentalGFT in some approximation. It makes more sense, therefore, to tackle thebig issue of the continuum approximation of the GFT dynamics. It may behelpful to use the idea <strong>and</strong> general properties of the formal continuum thirdquantization to gain some insight on how the continuum limit of GFTs shouldbe approached <strong>and</strong> on what it may look like. We see two main possibilities:• The first possibility is to try to define a continuum limit of GFTsat the level of the classical action, trying to obtain the classical action of athird-quantized field on continuum superspace.This would mean recovering, from the GFT field, the LQG functionalas an infinite combination of cylindrical functions each depending on a finite
Discrete <strong>and</strong> Continuum Third Quantization of <strong>Gravity</strong> 59number of group elements. This entails a generalization of the GFT frameworkto allow for all possible numbers of arguments of the GFT field(s),<strong>and</strong> combinatorial patters among them in the interaction term. This generalizationis possible, but will certainly be very difficult to h<strong>and</strong>le. Eventhis huge action would still codify the dynamics of building blocks of graphs,rather than graphs themselves, falling still a step shorter of the continuumthird-quantized action. Let us discuss a few more technical points.First, in (5.14), we have assumed a graph-changing Hamiltonian. Fromthe point of view of a dynamics of vertices, this implies a non-conservationof the same, <strong>and</strong> thus suggests an underlying interacting field theory of suchvertices, with the canonical quantum dynamics encoded in both the kinetic<strong>and</strong> interaction terms of the underlying action, <strong>and</strong> the same interaction termthat produces graph changing also produces topology change. This graph dynamicsis in contrast with the formal continuum one: The continuum thirdquantizedaction features a clear distinction between the topology-preservingcontribution to the dynamics, corresponding to the canonical (quantum)gravitational dynamics, encoded in a non-trivial kinetic term (Hamiltonianconstraint), <strong>and</strong> the topology-changing contribution, encoded in the interactionterm. This distinction would remain true also in the case of a graphpreservingcontinuum Hamiltonian constraint. Only, the kinetic term woulddecompose into a sum of kinetic terms each associated to a different graph.Both the distinction between topology-preserving <strong>and</strong> topology-changing contributionto the dynamics <strong>and</strong> a non-trivial kinetic term are not available in(st<strong>and</strong>ard) GFTs. Concerning the issue of a restriction of the GFT dynamicsto a given (trivial) topology, there have been encouraging results extendingthe large-N limit of matrix models [24, 25], but these clearly relate to thequantum GFT dynamics, <strong>and</strong> a similar approximation would not lead to anysignificant modification at the level of the GFT action. Still, the study of GFTperturbative renormalization [16, 19, 32, 6] is relevant because it may reveal(signs of this are already in [6]) that the GFT action has to be extended to includenon-trivial kinetic terms, in order to achieve renormalizability. Anotherstrategy is the expansion of the GFT action around a non-trivial backgroundconfiguration. What one gets generically [14, 22, 41, 30] is an effective dynamicsfor “perturbations” characterized by a non-trivial kinetic term <strong>and</strong> thus anon-trivial topology- (<strong>and</strong> graph-)preserving dynamics, encoded in the linearpart of the GFT equations of motion, <strong>and</strong> a topology- (<strong>and</strong> graph-)changingdynamics encoded in the non-linearities, i.e. in the GFT interaction. Thisobservation suggests that the GFT dynamics which, in a continuum limit,would correspond to the (quantum) GR dynamics, is to be looked for in adifferent phase, <strong>and</strong> not around the trivial Φ = 0 (Fock, no-space) vacuum.Notwithst<strong>and</strong>ing these considerations, it seems unlikely that a matchingbetween discrete <strong>and</strong> continuum dynamics can be obtained working purely atthe level of the action. The discretization of the continuum third-quantizedframework represented by the GFT formalism seems to operate at a more
60 S. Gielen <strong>and</strong> D. Oritiradical, fundamental level, in terms of the very dynamical objects chosen toconstitute a discrete quantum space: its microscopic building blocks.• The second main possibility is that the continuum limit is to bedefined at the level of the quantum theory, at the level of its transition amplitudes.This seems much more likely, for several reasons. One is simply that,in a quantum GFT, the classical action will be relevant only in some limitedregime. A second one is the analogy with matrix models, where the contactbetween discrete <strong>and</strong> continuum quantization is made at the level of the transitionamplitudes. Moreover, the continuum limit of the theory correspondsto a phase transition in the thermodynamic limit that can not be seen at thelevel of the action. A deeper implication of this is that the emergence of aclassical spacetime is the result of a purely quantum phenomenon.When considering such a limit at the level of the quantum GFT theory,two main mathematical <strong>and</strong> conceptual issues have to be solved, already mentionedabove: the restriction to the dynamics of geometry for given (trivial)topology, <strong>and</strong> the step from a description of the dynamics of (gravitational)degrees of freedom associated to elementary building blocks of graphs, thus ofa quantum space, to that of entire graphs, thus quantum states for the wholeof space (as in LQG). On the first issue, the recent results on the large-Napproximation of the GFT dynamics [24, 25] (for topological models) arecertainly going to be crucial, <strong>and</strong> represent a proof that the needed restrictioncan be achieved. The other example of a procedure giving, in passing,the same restriction is the mean-field approximation of the GFT dynamicsaround a non-trivial background configuration.The second issue is, in a sense, more thorny, <strong>and</strong> does not seem torequire only the ability to solve a mathematical problem, but some new conceptualingredient, some new idea. Let us speculate on what this idea couldbe. Looking back at the LQG set-up, one sees that the continuum nature ofthe theory, with infinitely many degrees of freedom, is encoded in the factthat a generic wave function of the connection decomposes into a sum overcylindrical functions associated to arbitrarily complicated graphs. In termsof graph vertices, the continuum nature of space is captured in the regimeof the theory in which an arbitrary high number of such graph vertices isinteracting. The best way to interpret <strong>and</strong> use the GFT framework, in thisrespect, seems then to be that of many-particle <strong>and</strong> statistical physics. Fromthis point of view, quantum space is a sort of weird condensed-matter system,where the continuum approximation would play the role of the hydrodynamicapproximation, <strong>and</strong> the GFT represents the theory of its “atomic” buildingblocks. This perspective has been advocated already in [39] <strong>and</strong> it resonateswith other ideas about spacetime as a condensate [28, 48] (see also [41] forrecent results in this direction). In practice, this possibility leads immediatelyto the need for ideas <strong>and</strong> techniques from statistical field theory, inparticular to the issue of phase transitions in GFTs, the idea being that the
Discrete <strong>and</strong> Continuum Third Quantization of <strong>Gravity</strong> 61discrete-continuum transition is one such phase transition <strong>and</strong> that continuumspacetime physics applies only in some of the phases of the GFT system.It becomes even more apparent, then, that the study of GFT renormalization[16, 19, 32], both perturbative <strong>and</strong> non-perturbative, will be crucialin the longer run for solving the problem of the continuum. The analogy withmatrix models, where the idea of discrete-continuum phase transition is realizedexplicitly, <strong>and</strong> the generalization of ideas <strong>and</strong> tools developed for themto the GFT context, will be important as well. Probably a crucial role will beplayed by diffeomorphism invariance, in both GFT <strong>and</strong> simplicial gravity, as aguiding principle for recovering a good continuum limit <strong>and</strong> even for devisingthe appropriate procedure to do so [4, 13]. Similarly, classical <strong>and</strong> quantumsimplicial gravity will provide insights about what sort of approximation ofvariables <strong>and</strong> which regime of the quantum dynamics give the continuum.In both cases above, the theory one would recover/define from the GFTdynamics would be a form of quantum gravitational dynamics, either encodedin a classical action which would give equations of motion correspondingto canonical quantum gravity, or in a quantum gravity path integral. Anextra step would then be needed to recover classical gravitational dynamics,presumably some modified form of GR, from it. This is where semi-classicalapproximations would be needed, in a GFT context. Their role will be thatof “de-quantizing” the formalism, i.e. to go from what would still be a thirdquantizedformalism, albeit now in the continuum, to a second-quantized one.This semi-classical approximation, which is a very different approximationwith respect to the continuum one, would be approached most naturally usingWKB techniques or coherent states. Here it is worth distinguishing furtherbetween third-quantized coherent states, coherent states for the quantizedGFT field, within a Fock-space construction for GFTs still to be definedproperly, <strong>and</strong> second-quantized ones, coherent states for the canonical wavefunction <strong>and</strong> defining points in the canonical classical phase space, as defined<strong>and</strong> used in LQG. The last type of coherent states have been used to extractan effective dynamics using mean-field-theory techniques <strong>and</strong> de-quantize thesystem in one stroke in [41]. This issue could also be fruitfully studied in asimplified minisuperspace third quantization; work on such a model, whichcan be seen both as a truncation of the GFT dynamics <strong>and</strong> as a generalizationof the well-developed loop quantum cosmology [7] to include topology change,is in progress [9].AcknowledgmentsThis work is funded by the A. von Humboldt Stiftung, through a Sofja KovalevskajaPrize, which is gratefully acknowledged.
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64 S. Gielen <strong>and</strong> D. Oriti[42] Alej<strong>and</strong>ro Perez. Spin foam models for quantum gravity. Classical <strong>and</strong> <strong>Quantum</strong><strong>Gravity</strong>, 20(6):R43, 2003.[43] G. Ponzano <strong>and</strong> T. Regge. Semiclassical limit of Racah coefficients, in: Spectroscopic<strong>and</strong> Group Theoretical Methods in Physics, F. Block (ed.), 1–58. NewYork: John Wiley <strong>and</strong> Sons, Inc., 1968.[44] Carlo Rovelli. <strong>Quantum</strong> gravity. Cambridge University Press, 2004.[45] Carlo Rovelli. A new look at loop quantum gravity. April 2010.[46] Claudio Teitelboim. <strong>Quantum</strong> mechanics on the gravitational field. Phys. Rev.D, 25:3159–3179, June 1982.[47] Thomas Thiemann. Modern canonical quantum general relativity. CambridgeUniversity Press, 2007.[48] Grigory Volovik. From quantum hydrodynamics to quantum gravity. 2006.[49] J. A. Wheeler. Superspace <strong>and</strong> the nature of quantum geometrodynamics, in:<strong>Quantum</strong> cosmology, L. Z. Fang, R. Ruffini (eds.), 27–92. Singapore: WorldScientific, 1987.Steffen Gielen <strong>and</strong> Daniele OritiMax Planck Institute for Gravitational Physics (Albert Einstein Institute)Am Mühlenberg 1D-14476 GolmGermanye-mail: gielen@aei.mpg.dedoriti@aei.mpg.de
Unsharp Values, Domains <strong>and</strong> TopoiAndreas Döring <strong>and</strong> Rui Soares BarbosaAbstract. The so-called topos approach provides a radical reformulationof quantum theory. Structurally, quantum theory in the topos formulationis very similar to classical physics. There is a state object Σ,analogous to the state space of a classical system, <strong>and</strong> a quantity-valueobject R ↔ , generalising the real numbers. Physical quantities are mapsfrom the state object to the quantity-value object – hence the ‘values’ ofphysical quantities are not just real numbers in this formalism. Rather,they are families of real intervals, interpreted as ‘unsharp values’. Wewill motivate <strong>and</strong> explain these aspects of the topos approach <strong>and</strong> showthat the structure of the quantity-value object R ↔ can be analysed usingtools from domain theory, a branch of order theory that originated intheoretical computer science. Moreover, the base category of the toposassociated with a quantum system turns out to be a domain if the underlyingvon Neumann algebra is a matrix algebra. For general algebras,the base category still is a highly structured poset. This gives a connectionbetween the topos approach, noncommutative operator algebras<strong>and</strong> domain theory. In an outlook, we present some early ideas on howdomains may become useful in the search for new models of (quantum)space <strong>and</strong> space-time.Mathematics Subject Classification (2010). Primary 81P99; Secondary06A11, 18B25, 46L10.Keywords. Topos approach, domain theory, intervals, unsharp values,von Neumann algebras.You cannot depend on your eyeswhen your imagination is out of focus.Mark Twain (1835–1910)1. IntroductionThe search for a theory of quantum gravity is ongoing. There is a range of approaches,all of them differing significantly in scope <strong>and</strong> technical content. Ofcourse, this is suitable for such a difficult field of enquiry. Most approachesF. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_ 5,© Springer Basel AG 201265
66 A. Döring <strong>and</strong> R. Soares Barbosaaccept the Hilbert space formalism of quantum theory <strong>and</strong> try to find extensions,additional structures that would capture gravitational aspects, orreproduce them from the behaviour of underlying, more fundamental entities.Yet, st<strong>and</strong>ard quantum theory is plagued with many conceptual difficultiesitself. Arguably, these problems get more severe when we try to take thestep to quantum gravity <strong>and</strong> quantum cosmology. For example, in the st<strong>and</strong>ardinterpretations of quantum theory, measurements on a quantum systemby an external classical observer play a central rôle. This concept clearlybecomes meaningless if the whole universe is to be treated as a quantumsystem. Moreover, st<strong>and</strong>ard quantum theory <strong>and</strong> quantum field theory arebased on a continuum picture of space-time. <strong>Mathematical</strong>ly, the continuumin the form of the real numbers underlies all structures like Hilbert spaces,operators, manifolds, <strong>and</strong> differential forms (<strong>and</strong> also strings <strong>and</strong> loops). Ifspace-time fundamentally is not a smooth continuum, then it may be wrongto base our mathematical formalism on the mathematical continuum of thereal numbers.These considerations motivated the development of the topos approachto the formulation of physical theories, <strong>and</strong> in particular the topos approachto quantum theory. In a radical reformulation based upon structures in suitable,physically motivated topoi, all aspects of quantum theory – states, physicalquantities, propositions, quantum logic, etc. – are described in a novelway. As one aspect of the picture emerging, physical quantities take theirvalues not simply in the real numbers. Rather, the formalism allows to describe‘unsharp’, generalised values in a systematic way. In this article, wewill show that the mathematical structures used to formalise unsharp valuescan be analysed using techniques from domain theory. We will take some firststeps connecting the topos approach with domain theory.Domain theory is a branch of order theory <strong>and</strong> originated in theoreticalcomputer science, where it has become an important tool. Since domaintheory is not well-known among physicists, we will present all necessary backgroundhere. Recently, domain theory has found some applications in quantumtheory in the work of Coecke <strong>and</strong> Martin [3] <strong>and</strong> in general relativityin the work of Martin <strong>and</strong> Panangaden [27]. Domain theory also has beenconnected with topos theory before, in the form of synthetic domain theory,but this is technically <strong>and</strong> conceptually very different from our specificapplication.The plan of the paper is as follows: in section 2, we will present asketch of the topos approach to quantum theory, with some emphasis on howgeneralised, ‘unsharp’ values for physical quantities arise. In section 3, wepresent some background on domain theory. In section 4, it will be shownthat the structure of the quantity-value object R ↔ , a presheaf whose globalelements are the unsharp values, can be analysed with the help of domaintheoreticaltechniques. Section 5 shows that the base category V(N) of thetopos Set V(N)op associated with a quantum system is a directed complete
Unsharp Values, Domains <strong>and</strong> Topoi 67poset, <strong>and</strong> moreover an algebraic domain if N is a matrix algebra. Physically,V(N) is the collection of all classical perspectives on the quantumsystem. In section 6, we show that the poset V(N) is not continuous, <strong>and</strong>hence not a domain, for non-matrix algebras N , <strong>and</strong> in section 7, we presentsome speculative ideas on how domains may become useful in the constructionof new models of space <strong>and</strong> space-time adequate for quantum theory<strong>and</strong> theories ‘beyond quantum theory’ in the context of the topos approach.Section 8 concludes.2. The topos approach, contexts <strong>and</strong> unsharp valuesBasic ideas. In recent years, the topos approach to the formulation of physicaltheories has been developed by one of us (AD), largely in collaboration withChris Isham [9, 10, 11, 12, 5, 13, 6, 7, 8]. This approach originates fromworks by Isham [21] <strong>and</strong> Isham/Butterfield [22, 23, 24, 25, 26]. L<strong>and</strong>sman etal. have presented a closely related scheme for quantum theory [17, 2, 18, 19],developing some aspects topos-internally, <strong>and</strong> Flori has developed a historyversion [15].The main goal of the topos approach is to provide a framework forthe formulation of ‘neo-realist’ physical theories. Such a theory describes aphysical system by (i) a state space, or more generally, a state object Σ, (ii)a quantity-value object R, where physical quantities take their values, <strong>and</strong>(iii) functions, or more generally, arrows f A ∶ Σ →Rfrom the state object tothe quantity-value object corresponding to physical quantities like position,momentum, energy, spin, etc. Both the state object <strong>and</strong> the quantity-valueobject are objects in a topos, <strong>and</strong> the arrows between them representingphysical quantities are arrows in the topos. Roughly speaking, a topos is amathematical structure, more specifically a category, that can be seen as auniverse of generalised sets <strong>and</strong> generalised functions between them.Each topos has an internal logic that is of intuitionistic type. In fact,one typically has a multivalued, intuitionistic logic <strong>and</strong> not just two-valuedBoolean logic as in the familiar topos Set of sets <strong>and</strong> functions. One mainaspect of the topos approach is that it makes use of the internal logic ofa given topos to provide a logic for a physical system. More specifically,the subobjects of the state object, or a suitable subfamily of these, are therepresentatives of propositions about the values of physical quantities. In thesimplest case, one considers propositions of the form “AεΔ”, which st<strong>and</strong>sfor “the physical quantity A has a value in the (Borel) set Δ of real numbers”.As an example, consider a classical system: the topos is Set, the stateobject is the usual state space, a symplectic manifold S, <strong>and</strong> a physical quantityA is represented by a function f A from S to the set of real numbers R,which in this case is the quantity-value object. The subset T ⊆Srepresentinga proposition “AεΔ” consists of those states s ∈Sfor which f A (s)∈Δholds(i.e., T = fA −1(Δ)).If we assume that the function f A representing the physicalquantity A is (at least) measurable, then the set T is a Borel subset of the
68 A. Döring <strong>and</strong> R. Soares Barbosastate space S. Hence, in classical physics the representatives of propositionsare the Borel subsets of S. Theyformaσ-complete Boolean algebra, whichultimately stems from the fact that the topos Set in which classical physicsis formulated has the familiar two-valued Boolean logic as its internal logic.Of course, we rarely explicitly mention Set <strong>and</strong> its internal logic – it is justthe usual mathematical universe in which we formulate our theories. As anunderlying structure, it usually goes unnoticed.A topos for quantum theory. For non-relativistic quantum theory, anothertopos is being used though. The details are explained elsewhere, see [5, 8]for an introduction to the topos approach <strong>and</strong> [13] for a more detailed description.The main idea is to use presheaves over the set V(N) of abeliansubalgebras of the nonabelian von Neumann algebra N of physical quantities.1 V(N) is partially ordered under inclusion; the topos of presheavesover V(N) is denoted as Set V(N)op . The poset V(N), also called the contextcategory, is interpreted as the collection of all classical perspectives onthe quantum system: each context (abelian subalgebra) V ∈V(N)providesa set of commuting self-adjoint operators, representing compatible physicalquantities. The poset V(N) keeps track of how these classical perspectivesoverlap, i.e., to which degree they are mutually compatible. Presheaves overV(N), which are contravariant functors from V(N) to Set, automaticallyencode this information, too. A presheaf is not a single set, but a ‘varyingset’: a family P =(P V) V ∈V(N ) of sets indexed by elements from V(N),togetherwith functions P (i V ′ V )∶P V→ P V ′ between the sets whenever thereis an inclusion i V ′ V ∶ V ′ → V in V(N).The state object for quantum theory is the so-called spectral presheaf Σthat is given as follows:● To each abelian subalgebra V ∈V(N), one assigns the Gel’f<strong>and</strong> spectrumΣ Vof the algebra V ;● to each inclusion i V ′ V ∶ V ′ → V , one assigns the function Σ(i V ′ V )∶Σ V →Σ V ′ that sends each λ ∈ Σ Vto its restriction λ∣ V ′ ∈ Σ V ′.One can show that propositions of the form “AεΔ” correspond to so-calledclopen subobjects of Σ. The set Sub cl (Σ) of clopen subobjects is the analogueof the set of Borel subsets of the classical state space. Importantly, Sub cl (Σ)is a complete Heyting algebra, which relates to the fact that the internallogic of the presheaf topos Set V(N)op is intuitionistic (<strong>and</strong> not just Boolean).Note that unlike in Birkhoff-von Neumann quantum logic, which is basedon the non-distributive lattice P(N) of projections in the algebra N ,inthetopos scheme propositions are represented by elements in a distributive latticeSub cl (Σ). This allows to give a better interpretation of this form of quantum1 To be precise, we only consider abelian von Neumann subalgebras V of N that have thesame unit element as N . In the usual presentation of this approach, the trivial algebra Cˆ1is excluded from V(N). However, here we will keep the trivial algebra as a bottom elementof V(N). We will occasionally point out which results depend on V(N) having a bottomelement.
Unsharp Values, Domains <strong>and</strong> Topoi 69logic [13, 7]. The map from the P(N) to Sub cl (Σ) is called daseinisation ofprojections. Its properties are discussed in detail in [8].Unsharp values. In this article, we will mostly focus on the quantity-valueobject for quantum theory <strong>and</strong> its properties. Like the state object Σ, thequantity-value object, which will be denoted R ↔ , is an object in the presheaftopos Set V(N)op . The topos approach aims to provide models of quantumsystems that can be interpreted as realist (or as we like to call them, neorealist,because of the richer intuitionistic logic coming from the topos). Oneaspect is that physical quantities should have values at all times, independentof measurements. Of course, this immediately meets with difficulties: inquantum theory, we cannot expect physical quantities to have sharp, definitevalues. In fact, the Kochen-Specker theorem shows that under weak <strong>and</strong>natural assumptions, there is no map from the self-adjoint operators to thereal numbers that could be seen as an assignment of values to the physicalquantities represented by the operators. 2 The Kochen-Specker theorem holdsfor von Neumann algebras [4].Thesimpleideaistouseintervals, interpreted as ‘unsharp values’, insteadof sharp real numbers. The possible (generalised) values of a physicalquantity A are real intervals, or more precisely, real intervals intersected withthe spectrum of the self-adjoint operator  representing A. In our topos approach,each self-adjoint operator  in the algebra N of physical quantitiesis mapped to an arrow ˘δ(Â) from the state object Σ to the quantity-valueobject R ↔ . (The latter object will be defined below.)We will not give the details of the construction of the arrow ˘δ(Â) (see[13, 8]), but we present some physical motivation here. For this, consider twocontexts V,V ′ ∈V(N)such that V ′ ⊂ V ,thatis,V ′ is a smaller contextthan V . ‘Smaller’ means that there are fewer physical quantities availablefrom the classical perspective described by V ′ than from V , hence V ′ givesa more limited access to the quantum system. The step from V to V ′ ⊂ V isinterpreted as a process of coarse-graining.For example, we may be interested in the value of a physical quantityA. Let us assume that the corresponding self-adjoint operator  is containedin a context V , but not in a context V ′ ⊂ V . For simplicity, let us assume furthermorethat the state of the quantum system is an eigenstate of Â. Then,from the perspective of V , we will get a sharp value, namely the eigenvalueof  corresponding to the eigenstate. But from the perspective of V ′ ,theoperator  is not available, so we have to approximate  by self-adjoint operatorsin V ′ . One uses one approximation from below <strong>and</strong> one from above,both taken with respect to the so-called spectral order. These approximationsalways exist. In this way, we obtain two operators δ i (Â) V ′,δo(Â) V ′ inV ′ which, intuitively speaking, contain as much information about  as is2 The conditions are: (a) each self-adjoint operator  is assigned an element of its spectrum;<strong>and</strong> (b) if ˆB = f(Â) for two self-adjoint operators Â, ˆB <strong>and</strong> a Borel function f, then thevalue v( ˆB) =v(f(Â)) assigned to ˆB is f(v(Â)), wherev(Â) is the value assigned to Â.
70 A. Döring <strong>and</strong> R. Soares Barbosaavailable from the more limited classical perspective V ′ . If we now ask for thevalue of A inthegivenstatefromtheperspectiveofV ′ , then we will get tworeal numbers, one from δ i (Â) V ′ <strong>and</strong> one from δo (Â) V ′. By the properties ofthe spectral order, these two numbers lie in the spectrum of Â. We interpretthem as the endpoints of a real interval, which is an ‘unsharp value’ for thephysical quantity A from the perspective of V ′ . Note that we get ‘unsharpvalues’ for each context V ∈V(N)(<strong>and</strong> for some V , we may get sharp values,namely in an eigenstate-eigenvalue situation).In a nutshell, this describes the idea behind daseinisation of self-adjointoperators, which is a map from self-adjoint operators in the nonabelian vonNeumann algebra N of physical quantities to arrows in the presheaf toposSet V(N)op , sending  ∈N sa to ˘δ(Â) ∈Hom(Σ, R↔ ). The arrow ˘δ(Â) is thetopos representative of the physical quantity A. Wewillnowconsidertheconstruction of the quantity-value object R ↔ , the codomain of ˘δ(Â).As a first step, we formalise the idea of unsharp values as real intervals.DefineIR ∶= {[a, b]∣a, b ∈ R, a≤ b}. (2.1)Note that we consider closed intervals <strong>and</strong> that the case a = b is included,which means that the intervals of the form [a, a] are contained in IR. Clearly,these intervals can be identified with the real numbers. In this sense, R ⊂ IR.It is useful to think of the presheaf R ↔ as being given by one copy of IRfor each classical perspective V ∈V(N). Each observer hence has the wholecollection of ‘unsharp’ values available. The task is to fit all these copies ofIR together into a presheaf. In particular, whenever we have V,V ′ ∈V(N)such that V ′ ⊂ V , then we need a function from IR V to IR V ′ (here, we putan index on each copy of IR to show to which context the copy belongs).The simplest idea is to send each interval [a, b]∈IR V to the same interval inIR V ′. But, as we saw, in the topos approach the step from the larger contextV to the smaller context V ′ is seen as a process of coarse-graining. Relatedto that, we expect to get an even more unsharp value, corresponding to abigger interval, in IR V ′ than in IR V in general. In fact, we want to be flexible<strong>and</strong> define a presheaf such that we can map [a, b] ∈IR V either to the sameinterval in IR V ′,ortoanybiggerinterval[c, d] ⊃[a, b], depending on whatis required.We note that as we go from a larger context V to smaller contextsV ′ ⊂ V, V ′′ ⊂ V ′ , ..., the left endpoints of the intervals will get smaller <strong>and</strong>smaller, <strong>and</strong> the right endpoints will get larger <strong>and</strong> larger in general. Theidea is to formalise this by two functions μ V ,ν V ∶↓V → R that give the leftresp. right endpoints of the intervals for all V ′ ⊆ V . Here, ↓V ∶= {V ′ ∈V(N)∣V ′ ⊆ V } denotes the downset of V in V(N). Physically, ↓V is the collectionof all subcontexts of V , that is, all smaller classical perspectives than V .Thisleads to the following definition.Definition 2.1. The quantity-value object R ↔ for quantum theory is given asfollows:
Unsharp Values, Domains <strong>and</strong> Topoi 71● To each V ∈V(N), we assign the setR ↔ V∶= {(μ, ν)∣μ, ν ∶↓V → R,μ order-preserving, νorder-reversing, μ≤ ν};● to each inclusion i V ′ V ∶ V ′ → V , we assign the map(2.2)R ↔ (i V ′ V )∶R ↔ V → R↔ V ′(2.3)(μ, ν)→(μ∣ ↓V ′,ν∣ ↓V ′).A global element γ of the presheaf R ↔ is a choice of one pair of functionsγ V =(μ V ,ν V ) for every context V ∈V(N)such that, whenever V ′ ⊂ V ,onehas γ V ′ =(μ V ′,ν V ′)=(μ V ∣ V ′,ν V ∣ V ′)=γ V ∣ V ′. Clearly, a global element γgives a pair of functions μ, ν ∶V(N)→R such that μ is order-preserving(smaller contexts are assigned smaller real numbers) <strong>and</strong> ν is order-reversing(smaller contexts are assigned larger numbers). Note that μ <strong>and</strong> ν are definedon the whole poset V(N). Conversely, each such pair of functions determinesa global element of R ↔ . Hence we can identify a global element γ with thecorresponding pair of functions (μ, ν). We see that γ =(μ, ν) provides oneclosed interval [μ(V ),ν(V )] for each context V . Moreover, whenever V ′ ⊂ V ,we have [μ(V ′ ),ν(V ′ )] ⊇ [μ(V ),ν(V )], that is, the interval at V ′ is largerthan or equal to the interval at V . We regard a global element γ of R ↔ asone unsharp value. Each interval [μ(V ),ν(V )], V ∈V(N), is one componentof such an unsharp value, associated with a classical perspective/context V .The set of global elements of R ↔ is denoted as ΓR ↔ .In the following, we will show that the set ΓR ↔ of unsharp values forphysical quantities that we obtain from the topos approach is a highly structuredposet, <strong>and</strong> that a subset of ΓR ↔ naturally can be seen as a so-calleddomain if the context category V(N) is a domain (see section 4). Domainsplay an important rôle in theoretical computer science. The context categoryV(N) turns out to be a domain, even an algebraic domain, if N is a matrixalgebra. This leads to a first, simple connection between noncommutativeoperator algebras, the topos approach <strong>and</strong> domain theory (see section 5). Formore general von Neumann algebras N , V(N) is a not a domain, see section6.3. Domain theoryBasics. This section presents some basic concepts of domain theory. St<strong>and</strong>ardreferences are [16, 1]. Since domain theory is not well-known among physicists,we give some definitions <strong>and</strong> motivation. Of course, we barely scratch thesurface of this theory here.The study of domains was initiated by Dana Scott [28, 29], with the aimof finding a denotational semantics for the untyped λ-calculus. Since then,it has undergone significant development <strong>and</strong> has become a mathematicaltheory in its own right, as well as an important tool in theoretical computerscience.
72 A. Döring <strong>and</strong> R. Soares BarbosaDomain theory is a branch of order theory, yet with a strong topologicalflavour, as it captures notions of convergence <strong>and</strong> approximation. Thebasic concepts are easy to grasp: the idea is to regard a partially orderedset as a (qualitative) hierarchy of information content or knowledge. Underthis interpretation, we think of x ⊑ y as meaning ‘y is more specific, carriesmore information than x’. Therefore, non-maximal elements in the posetrepresent incomplete/partial knowledge, while maximal elements representcomplete knowledge. In a more computational perspective, we can see thenon-maximal elements as intermediate results of a computation that proceedstowards calculating some maximal element (or at least a larger elementwith respect to the information order).For the rest of this section, we will mainly be considering a single poset,which we will denote by ⟨P,⊑⟩.Convergence – directed completeness of posets. The first important conceptin domain theory is that of convergence. We start by considering some specialsubsets of P .Definition 3.1. A nonempty subset S ⊆ P is directed if∀x, y ∈ S ∃z ∈ S ∶ x, y ⊑ z. (3.1)A directed set can be seen as a consistent specification of information:the existence of a z ⊒ x, y expresses that x <strong>and</strong> y are compatible, in the sensethat it is possible to find an element which is larger, i.e., contains more information,than both x <strong>and</strong> y. Alternatively, from the computational viewpoint,directed sets describe computations that converge in the sense that for anypair of intermediate results (that can be reached in a finite number of steps),there exists a better joint approximation (that can also be reached in a finitenumber of steps). This is conceptually akin to converging sequences in a metricspace. Hence the natural thing to ask of directed sets is that they possessa suitable kind of limit – that is, an element containing all the informationabout the elements of the set, but not more information. This limit, if itexists, can be seen as the ideal result the computation approximates. Thisleads to the concept of directed-completeness:Definition 3.2. A directed-complete poset (or dcpo) is a poset in which anydirected set has a supremum (least upper bound).Following [1], we shall write ⊔ ↑ S to denote the supremum of a directedset S, instead of simply ⊔S. Note that ⊔ ↑ S means ‘S is a directed set, <strong>and</strong>we are considering its supremum’.The definition of morphisms between dcpos is the evident one:Definition 3.3. A function f ∶ P → P ′ between dcpos ⟨P,⊑⟩ <strong>and</strong> ⟨P ′ , ⊑ ′ ⟩ isScott-continuous if● f is order-preserving (monotone);● for any directed set S ⊆ P , f(⊔ ↑ S)=⊔ ↑ f → (S), wheref → (S) ={f(s) ∣s ∈ S}.
Unsharp Values, Domains <strong>and</strong> Topoi 73Clearly, dcpos with Scott-continuous functions form a category. Thedefinition of a Scott-continuous function can be extended to posets which arenot dcpos, by carefully modifying the second condition to say ‘for any directedset S that has a supremum’. The reference to ‘continuity’ is not fortuitous,asthereistheso-calledScott topology, with respect to which these (<strong>and</strong> onlythese) arrows are continuous. The Scott topology will be defined below.Approximation - continuous posets. The other central notion is sometimescalled approximation. This is captured by the following relation on elementsof P .Definition 3.4. We say that x approximates y or x is way below y, <strong>and</strong>writex
74 A. Döring <strong>and</strong> R. Soares BarbosaRecall that ⊔ ↑ ↡y = y means that ↡y is directed <strong>and</strong> has supremum y.Continuity basically says that the elements ‘much simpler’ than y carry allthe information about y, when taken together. Algebraicity further says thatthe ‘primitive’ (i.e., compact) elements are enough.Bases <strong>and</strong> domains. The continuity <strong>and</strong> algebraicity requirements are oftenexpressed in terms of the notion of a basis. The definition is slightly moreinvolved, but the concept of a basis is useful in its own right.Definition 3.7. A subset B ⊆ P is a basis for P if, for all x ∈ X, ⊔ ↑ (B∩↡x)=x.It is immediate that continuity implies that P itself is a basis. Conversely,the existence of a basis implies continuity.Definition 3.8. A domain (or continuous domain) ⟨D,⊑⟩ isadcpowhichiscontinuous. Equivalently, a domain is a dcpo that has a basis. ⟨D,⊑⟩ is analgebraic domain if it is a domain <strong>and</strong> algebraic, that is, if the set K(D) ofcompact elements is a basis for ⟨D,⊑⟩. Anω-continuous (resp. ω-algebraic)domain is a continuous (resp. algebraic) domain with a countable basis.Note that a domain always captures the notions of convergence <strong>and</strong> ofapproximation as explained above.Bounded complete posets. Later on, we will need another completeness propertythat a poset P may or may not have.Definition 3.9. A poset is bounded complete (or a bc-poset) if all subsets Swith an upper bound have a supremum. It is finitely bounded complete if allfinite subsets with an upper bound have a supremum. It is almost (finitely)bounded complete if all nonempty (finite) subsets with an upper bound havea supremum. 3We state the following result without proof.Proposition 3.10. A(n almost) finitely bounded complete dcpo is (almost)bounded complete. 4Another property, stronger than bounded completeness, will be neededlater on:Definition 3.11. An L-domain is a domain D in which, for each x ∈ D, theprincipal ideal ↓x is a complete lattice.The Scott topology. We now define the appropriate topology on dcpos <strong>and</strong>domains, called the Scott topology, <strong>and</strong> present some useful results. In fact,the Scott topology can be defined on any poset, but we are mostly interestedin dcpos <strong>and</strong> domains.Definition 3.12. Let ⟨P,≤⟩ be a poset. A subset G of P is said to be Scott-openif3 Note that the ‘almost’ versions don’t require a least element ⊥.4 Bounded-complete dcpos are the same as complete semilattices (see [16]).
Unsharp Values, Domains <strong>and</strong> Topoi 75● G is an upper set, that isx ∈ G ∧ x ≤ y ⇒ y ∈ G; (3.4)● G is inaccessible by directed suprema, i.e. for any directed set S with asupremum,⊔ ↑ S ∈ G ⇒∃s ∈ S ∶ s ∈ U. (3.5)The complement of a Scott-open set is called a Scott-closed set. Concretely,this is a lower set closed for all existing directed suprema.The name Scott-open is justified by the following result.Proposition 3.13. The Scott-open subsets of P are the opens of a topology onP , called the Scott topology.Proposition 3.14. If P is a continuous poset, the collectionis a basis for the Scott topology.{↟x ∣ x ∈ P } (3.6)The Scott topology encodes a lot of information about the domaintheoreticalproperties of P relating to convergence <strong>and</strong> (in the case of acontinuous poset) approximation. The following is one of its most importantproperties, relating the algebraic <strong>and</strong> topological aspects of domain theory.Proposition 3.15. Let P <strong>and</strong> Q be two posets. A function f ∶ P → Q isScott-continuous if <strong>and</strong> only if it is (topologically) continuous with respect tothe Scott topologies on P <strong>and</strong> Q.The results above (<strong>and</strong> proofs for the more involved ones) can be foundin [1, §1.2.3]. More advanced results can be found in [1, §4.2.3].As for separation properties, the Scott topology satisfies only a veryweak axiom in all interesting cases.Proposition 3.16. The Scott topology on P gives a T 0 topological space. It isT 2 if <strong>and</strong> only if the order in P is trivial.The real interval domain. As we saw in section 2, the collection of real intervalscan serve as a model for ‘unsharp values’ of physical quantities (atleast if we consider only one classical perspective V on a quantum system).We will now see that the set IR of closed real intervals defined in equation(2.1) actually is a domain, the so-called interval domain. This domain wasintroduced by Scott [30] as a computational model for the real numbers.Definition 3.17. The interval domain is the poset of closed intervals in R(partially) ordered by reverse inclusion,IR ∶= ⟨{[a, b]∣a, b ∈ R}, ⊑∶=⊇⟩ . (3.7)
76 A. Döring <strong>and</strong> R. Soares BarbosaThe intervals are interpreted as approximations to real numbers, hencethe ordering by reverse inclusion: we think of x ⊑ y as ‘y is sharper than x’.Clearly, the maximal elements are the real numbers themselves (or rather,more precisely, the intervals of the form [a, a]).We shall denote by x − <strong>and</strong> x + the left <strong>and</strong> right endpoints of an intervalx ∈ IR. Thatis,wewritex =[x − ,x + ]. Also, if f ∶ X → IR is a functionto the interval domain, we define the functions f − ,f + ∶ X → R given byf ± (x) ∶=(f(x)) ± , so that, for any x ∈ X, f(x) =[f − (x),f + (x)]. Clearly,onealwayshasf − ≤ f + (the order on functions being defined pointwise).Conversely, any two functions g,h ∶ X → R with g ≤ h determine a functionf such that f − = g <strong>and</strong> f + = h.Writing x =[x − ,x + ] amounts to regarding IR as being embedded inR × R (as a set). The decomposition for functions can then be depicted asfollows:R(3.8)f −π 2Xf(f − ≤ f + ) ,IR R × Rπ 1f +R<strong>and</strong> it is nothing more than the universal property of the (categorical) productR×R restricted to IR, the restriction being reflected in the condition f − ≤ f + .This diagram is more useful in underst<strong>and</strong>ing IR than it may seem atfirst sight. Note that we can place this diagram in the category Pos (of posets<strong>and</strong> monotone maps) if we make a judicious choice of order in R 2 .Thisisachieved by equipping the first copy of R with its usual order <strong>and</strong> the secondcopy with the opposite order ≥. Adopting this view, equations 3.9 <strong>and</strong> 3.10below should become apparent.Domain-theoretic structure on IR. The way-below relation in IR is given by:x
Unsharp Values, Domains <strong>and</strong> Topoi 77We now consider the Scott topology on IR. The basic open sets of theScott topology on IR are of the form↟[a, b]={[c, d]∣a < c ≤ d < b} (3.11)for each [a, b]∈IR. Theidentity↟[a, b]={t ∈ IR ∣ t ⊆(a, b)} (3.12)allows us to see ↟[a, b] asakindofopeninterval(a, b). More precisely, itconsists of closed intervals contained in the real interval (a, b).Recall that we can see the poset IR as sitting inside R ≤ × R ≥ . In topologicalterms, this means that the Scott topology is inherited from the Scotttopologies in R ≤ <strong>and</strong> R ≥ . These are simply the usual lower <strong>and</strong> upper semicontinuitytopologies on R, with basic open sets respectively (a, ∞) <strong>and</strong> (−∞,b).Interpreting diagram 3.8 in Top gives the following result.Proposition 3.18. Let X be a topological space <strong>and</strong> f ∶ X → IR be a function.Then f is continuous iff f − is lower semicontinuous <strong>and</strong> f + is uppersemicontinuous.Hence, the subspace topology on IR inherited from R LSC ×R USC ,whereLSC <strong>and</strong> USC st<strong>and</strong> for the lower <strong>and</strong> upper semicontinuity topologies, isthe Scott topology.Generalising R. We want to regard IR as a generalisation of R. Note that theset max IR consists of degenerate intervals [x, x]={x}. This gives an obviousway of embedding the usual continuum R in IR. What is more interesting,this is actually a homeomorphism.Proposition 3.19. R ≅ max IR as topological spaces, where R is equipped withits usual topology <strong>and</strong> max IR with the subspace topology inherited from IR.This result is clear from the following observation that identifies basicopen sets:↟[a, b]∩max IR ={t ∈ IR ∣ t ⊆(a, b)} ∩ max IR ={{t}∣t ∈(a, b)} = (a, b).Another (maybe more informative) way of seeing this is by thinking ofIR as sitting inside R LSC × R USC . The well-known fact that the topology ofR is given as the join of the two semicontinuity topologies can be stated asfollows: the diagonal map diag ∶ R → R LSC × R USC gives an isomorphismbetween R <strong>and</strong> its image Δ. This is because basic opens in Δ areΔ ∩((a, ∞)×(−∞,b)) = (a, ∞)∩(−∞,b)=(a, b). (3.13)The result above just says that this diagonal map factors through IR, withimage max IR.The only separation axiom satisfied by the topology of IR is the T0axiom, whereas R is a T6 space. However, this topology still keeps someproperties of the topology on R. It is second-countable <strong>and</strong> locally compact(the general definition of local compactness for non-Hausdorff spaces is givenin [31]). Note that, obviously, adding a least element ⊥=[−∞, +∞] to thedomain would make it compact.
78 A. Döring <strong>and</strong> R. Soares Barbosa4. The quantity-value object <strong>and</strong> domain theoryWe now study the presheaf R ↔ of (generalised) values that shows up in thetopos approach in the light of domain theory. First of all, one can considereach component R ↔ Vindividually, which is a set. More importantly, we areinterested in the set ΓR ↔ of global elements of R ↔ . We will relate these twosets with the interval domain IR introduced in the previous section.The authors of [10], where the presheaf R ↔ was first introduced as thequantity-value object for quantum theory, were unaware of domain theory atthetime.TheideathatR ↔ (or rather, a closely related co-presheaf) is relatedto the interval domain was first presented by L<strong>and</strong>sman et al. in [17]. Theseauthors considered R ↔ as a topos-internal version of the interval domain.Here, we will focus on topos-external arguments.Rewriting the definition of R ↔ . We start off by slightly rewriting the definitionof the presheaf R ↔ .Recall from the previous section that a function f ∶ X → IR to theinterval domain can be decomposed into two functions f − ,f + ∶ X → R givingthe left <strong>and</strong> right endpoints of intervals. In case X is a poset, it is immediatefrom the definitions that such an f is order-preserving if <strong>and</strong> only if f − isorder-preserving <strong>and</strong> f + is order-reversing with respect to the usual order onthe real numbers. This allows us to rewrite the definition of R ↔ :Definition 4.1. The quantity-value presheaf R ↔ is given as follows:● To each V ∈V(N), we assign the setR ↔ V={f ∶↓V → IR ∣ f order-preserving}; (4.1)● to each inclusion i V ′ V , we assign the functionR ↔ (i V ′ V )∶R ↔ V → R↔ V ′f → f∣ ↓V ′ .This formulation of R ↔ brings it closer to the interval domain.Global elements of R ↔ . In section 2, we stated that in the topos approach,the generalised values of physical quantities are given by global elementsof the presheaf R ↔ . We remark that the global elements of a presheaf are(analogous to) points if the presheaf is regarded as a generalised set. Yet,the set of global elements may not contain the full information about thepresheaf. There are non-trivial presheaves that have no global elements at all– the spectral presheaf Σ is an example. In contrast, R ↔ has many globalelements.We give a slightly modified characterisation of the global elements ofR ↔ (compare end of section 2):Proposition 4.2. Global elements of R ↔ are in bijective correspondence withorder-preserving functions from V(N) to IR.
Unsharp Values, Domains <strong>and</strong> Topoi 79Proof. Let 1 be the terminal object in the topos Set V(N)op , that is, the constantpresheaf with a one-element set {∗} as component for each V ∈V(N).Aglobal element of R ↔ is an arrow in Set V(N)op , i.e., a natural transformationη ∶ 1 → R ↔ . (4.2)For each object V in the base category V(N), this gives a functionη V ∶{⋆}→R ↔ V (4.3)which selects an element γ V ∶= η V (⋆) of R ↔ V . Note that each γ V is an orderpreservingfunction γ V ∶↓V → IR. The naturality condition, expressed bythe diagram1 V = {∗}η V R↔VR ↔ (i V ′ V )1 V ′ ={∗}η V R↔V ′then reads γ V ′ = R ↔ (i V ′ V )(γ V )=f V ∣ ↓V ′. Thus, a global element of R ↔determines a unique function˜γ ∶V(N)→IRV ′ → γ V (V ′ ),where V is some context such that V ′ ⊆ V . The function ˜γ is well-defined: ifwe pick another W ∈V(N)such that V ′ ⊆ W , then the naturality conditionguarantees that γ W (V ′ )=γ V (V ′ ). The monotonicity condition for each γ Vforces ˜γ to be a monotone (order-preserving) function from V(N) to IR.Conversely, given an order-preserving function ˜γ ∶V(N)→IR, weobtaina global element of R ↔ by setting∀V ∈V(N)∶γ V ∶= ˜γ∣ ↓V . (4.4)◻Global elements as a dcpo. So far, we have seen that each R ↔ V , V ∈V(N),<strong>and</strong> the set of global elements ΓR ↔ are sets of order-preserving functionsfrom certain posets to the interval domain IR. Concretely,R ↔ V=OP(↓V,IR), (4.5)ΓR ↔ =OP(V(N), IR), (4.6)where OP(P,IR) denotes the order-preserving functions from the poset P toIR.We now want to apply the following result (for a proof of a more generalresult, see Prop. II-4.20 in [16]):
80 A. Döring <strong>and</strong> R. Soares BarbosaProposition 4.3. Let X be a topological space. If P is a dcpo (resp. boundedcomplete dcpo, resp. almost bounded complete dcpo) equipped with the Scotttopology, then C(X, P) with the pointwise order is a dcpo (resp. boundedcomplete dcpo, resp. almost bounded complete dcpo).The problem is that if we want to apply this result to our situation,then we need continuous functions between the posets, but neither ↓V norV(N) have been equipped with a topology so far. (On IR, we consider theScott topology.) The following result shows what topology to choose:Proposition 4.4. Let P , Q be posets, <strong>and</strong> let f ∶ P → Q be a function. If theposet Q is continuous, the following are equivalent:1. f is order-preserving;2. f is continuous with respect to the upper Alex<strong>and</strong>roff topologies on P<strong>and</strong> Q;3. f is continuous with respect to the upper Alex<strong>and</strong>roff topology on P <strong>and</strong>the Scott topology on Q.Hence, we put the upper Alex<strong>and</strong>roff topology on ↓V <strong>and</strong> V(N) toobtain the following equalities:● For each V ∈V(N),wehaveR ↔ V = C((↓V )UA , IR);● for the global elements of R ↔ ,wehaveΓR ↔ = C(V(N ) UA , IR).By Prop. 4.3, since IR is an almost bounded complete dcpo, both R ↔ V (foreach V ), <strong>and</strong> ΓR ↔ also are almost bounded complete dcpos.A variation of the quantity-value object. The following is a slight variationof the quantity-value presheaf where we allow for completely undeterminedvalues (<strong>and</strong> not just closed intervals [a, b]). This is achieved by including abottom element in the interval domain IR, this bottom element of coursebeing interpreted as the whole real line. Using the L-domain IR ⊥ ,letR ↔ ⊥ bethe presheaf defined as follows:● To each V ∈V(N), we assign the setR ↔ ⊥ ={f ∶↓V → IR ⊥ ∣ f order-preserving}; (4.7)V● to each inclusion i V ′ V , we assign the functionR ↔ ⊥ (i V ′ V )∶R ↔ ⊥ → R ↔V⊥ V ′f → f∣ ↓V ′ .Clearly, we have:● For each V ∈V(N), R ↔ ⊥ = C((↓V ) UA , IR ⊥ );V● ΓR ↔ ⊥ = C(V(N ) UA , IR ⊥ ).Hence, by Prop. 4.3, R ↔ ⊥ (for each V )<strong>and</strong>ΓR ↔ are bounded completeVdcpos. Note that R ↔ is a subpresheaf of R ↔ ⊥ .Also, one can consider the presheaves SR ↔ <strong>and</strong> SR ↔ ⊥ defined analogouslyto R ↔ <strong>and</strong> R ↔ ⊥ , but requiring the functions to be Scott-continuous
Unsharp Values, Domains <strong>and</strong> Topoi 81rather than simply order-preserving. Again, SR ↔ is a subpresheaf of SR ↔ ⊥ .Moreover, note that SR ↔ (resp. SR ↔ ⊥ ) is a subpresheaf of R ↔ (resp. R ↔ ⊥ ).For a finite-dimensional N , since the poset V(N) has finite height, any orderpreservingfunction is Scott-continuous, hence there is no difference betweenR ↔ <strong>and</strong> SR ↔ (resp. R ↔ ⊥ <strong>and</strong> SR ↔ ⊥ ). The presheaves SR ↔ <strong>and</strong> SR ↔ ⊥ using˘δ(Â) Scott-continuous functions are interesting since the arrows of the form∶Σ → R↔ that one obtains from daseinisation of self-adjoint operators[10, 13] actually have image in SR ↔ . We will not prove this result here, sincethis would lead us too far from our current interest. We have:● For each V ∈V(N), SR ↔ V = C((↓V )S , IR) <strong>and</strong> SR ↔ ⊥ = C((↓V ) S , IR ⊥ );V● ΓSR ↔ = C(V(N ) S , IR) <strong>and</strong> ΓSR ↔ ⊥ = C(V(N ) S , IR ⊥ ).Domain-theoretic structure on global sections. We now consider continuity.The following result (from theorem A in [32]) helps to clarify things further:Proposition 4.5. If D is a continuous L-domain <strong>and</strong> X a core-compact space(i.e. its poset of open sets is continuous), then C(X, D) (where D has theScott topology) is a continuous L-domain.In particular, any locally compact space is core-compact ([14]). Notethat V(N) with the Alex<strong>and</strong>roff topology is always locally compact; it iseven compact if we consider V(N) to have a least element. Moreover, wesaw before that IR ⊥ is an L-domain. Hence, we can conclude that the globalsections of R ↔ ⊥ form an L-domain. Similarly, the sections of R ↔ ⊥ V(over ↓V )form an L-domain.For the presheaves R ↔ , SR ↔ ,<strong>and</strong>SR ↔ ⊥ , it is still an open questionwhether their global sections form a domain or not.5. The category of contexts as a dcpoWe now turn our attention to the poset V(N). In this section, we investigateit from the perspective of domain theory. We will show that V(N) is a dcpo,<strong>and</strong> that the assignment N ↦ V(N) gives a functor from von Neumannalgebras to the category of dcpos.From a physical point of view, the fact that V(N) isadcposhowsthatthe information contained in a coherent set of physical contexts is capturedby a larger (limit) context. If N is a finite-dimensional algebra, that is, afinite direct sum of matrix algebras, then V(N) is an algebraic domain. Thiseasy fact will be shown below. We will show in section 6 that, for other typesof von Neumann algebras, V(N) is not continuous.In fact, most of this section is not concerned with V(N) itself, but witha more general kind of posets of which V(N) is but one example.The fact that V(N) is an algebraic domain in the case of matrix algebrasN was suggested to us in private communication by Chris Heunen.
82 A. Döring <strong>and</strong> R. Soares BarbosaDomains of subalgebras. Common examples of algebraic domains are theposets of subalgebras of an algebraic structure, e.g. the poset of subgroupsof a group. (Actually, this is the origin of the term ‘algebraic’.) We startby formalising this statement <strong>and</strong> prove that posets of this kind are indeeddomains. This st<strong>and</strong>ard result will be the point of departure for the followinggeneralisations to posets of subalgebras modulo equations <strong>and</strong> topologicalalgebras.<strong>Mathematical</strong>ly, we will be using some simple universal algebra. A cautionaryremark for the reader familiar with universal algebra: the definitionsof some concepts were simplified. For example, for a fixed algebra A, itssubalgebras are simply defined to be subsets (<strong>and</strong> not algebras in their ownright).Definition 5.1. A signature is a set Σ of so-called operation symbols togetherwith a function ar ∶ Σ → N designating the arity of each symbol.Definition 5.2. AΣ-algebra (or an algebra with signature Σ) is a pair A=⟨A; F⟩ consisting of a set A (the support) <strong>and</strong> a set of operations F={f A ∣f ∈ Σ}, wheref A ∶ A ar(f) → A is said to realise the operation symbol f.Note that the signature describes the operations an algebra is requiredto have. Unless the distinction is necessary, we will omit the superscript <strong>and</strong>therefore not make a distinction between operator symbols <strong>and</strong> operationsthemselves.As an example, a monoid ⟨M; ⋅, 1⟩ is an algebra with two operations, ⋅<strong>and</strong>1,ofaritiesar(⋅) = 2<strong>and</strong>ar(1)=0.Definition 5.3. GivenanalgebraA=⟨A; F⟩,asubalgebra of A is a subset of Awhich is closed under all operations f ∈F. We denote the set of subalgebrasof A by Sub A .Definition 5.4. Let A=⟨A; F⟩ be an algebra <strong>and</strong> G ⊆ A. Thesubalgebra ofA generated by G, denoted ⟨G⟩, is the smallest algebra containing G. Thisisgiven explicitly by the closure of G under the operations, i.e. givenG 0 = G,G k+1 = G k ∪ ⋃ {f(x 1 ,...,x ar(f) )∣x 1 ,...,x ar(f) ∈ G k },f∈Fwe obtain⟨G⟩ = ⋃ G k . (5.1)k∈NA subalgebra B ⊆ A is said to be finitely generated whenever it is generatedby a finite subset of A.We will consider the poset (Sub A , ⊆) in some detail. The following resultsare well-known <strong>and</strong> rather easy to prove:Proposition 5.5. (Sub A , ⊆) is a complete lattice with the operations⋀ S ∶= ⋂ S,⋁ S ∶= ⟨⋃ S⟩.
Unsharp Values, Domains <strong>and</strong> Topoi 83Proposition 5.6. If S⊆Sub A is directed, then ⋁ S=⟨⋃ S⟩ = ⋃ S.Let B ∈ Sub A .WewriteSub fin (B) ∶={C ∈ Sub A ∣ C ⊆ B <strong>and</strong> C is finitely generated}={⟨x 1 ,...,x n ⟩∣n ∈ N, {x 1 ,...,x n }⊆B} (5.2)for the finitely generated subalgebras of B.Lemma 5.7. For all B ∈ Sub A , we have B = ⊔ ↑ Sub fin (B)=⋃ Sub fin (B).We now characterise the way-below relation for the poset Sub A .Lemma 5.8. For B,C ∈ Sub A , one has C
84 A. Döring <strong>and</strong> R. Soares BarbosaWe will now consider the poset ⟨Sub A /E,⊆⟩, which is a subposet of⟨Sub A , ⊆⟩ considered before. Clearly, if A satisfies E, wehaveSub A /E = Sub A ,which is not very interesting. In all other cases Sub A /E has no top element.In fact, it is not even a lattice in most cases. However, we still have someweakened form of completeness:Proposition 5.14. Sub A /E is a bounded-complete algebraic domain, i.e., it isa bc-dcpo <strong>and</strong> it is algebraic.Proof. To prove that it is an algebraic domain, it is enough to show thatSub A /E is a Scott-closed subset of Sub A , i.e., that it is closed for directedsuprema (hence a dcpo) <strong>and</strong> downwards closed (hence, given the first condition,↡x is the same in both posets, <strong>and</strong> algebraicity follows from the sameproperty on Sub A ). Downwards-closedness follows immediately, since a subsetof a set satisfying an equation also satisfies it.For directed completeness, let S be a directed subset of Sub A /E. Wewant to show that its supremum, the union ⋃ S, is still in Sub A /E. Let⟨p, q⟩∈E be an equation over the variables {x 1 ,...,x n } <strong>and</strong> ν ∶{x 1 ,...,x n }→⋃ Sany valuation on ⋃ S. Letuswritea 1 ,...,a n for ν(x 1 ),...,ν(x n ). There existS 1 ,...,S n ∈S such that a 1 ∈ S 1 ,...,a n ∈ S n .BydirectednessofS, thereisan S ∈S such that {a 1 ,...,a n }⊆S. SinceS ∈ Sub A /E <strong>and</strong> the valuation ν isdefined on S, one must have ∣p∣ ν =∣q∣ ν . Therefore, ⋃ S satisfies the equationsin E, <strong>and</strong>soSub A /E is closed under directed suprema.For bounded completeness, note that if S⊆Sub A /E is bounded aboveby a subalgebra S ′ that also satisfies the equations, then the subalgebragenerated by S, ⟨⋃ S⟩, being a subset of S ′ , must also satisfy the equations.So ⟨⋃ S⟩ is in Sub A /E <strong>and</strong> is a supremum for S in this poset.◻Topologically closed subalgebras. The results of the previous section apply toany kind of algebraic structure. In particular, this is enough to conclude that,given a ∗-algebra N , the set of its abelian ∗-subalgebras forms an algebraicdomain. If we restrict attention to the finite-dimensional situation, i.e., matrixalgebras N , or finite direct sums of matrix algebras, then the result shows thatthe poset V(N) of abelian von Neumann subalgebras of a matrix algebra Nis an algebraic domain, since every algebraically closed abelian ∗-subalgebrais also weakly closed (<strong>and</strong> hence a von Neumann algebra) in this case.But for a general von Neumann algebra N , not all abelian ∗-subalgebrasneed to be abelian von Neumann subalgebras. We need to consider the extracondition that each given subalgebra is closed with respect to a certain topology,namely the weak operator topology (or the strong operator topology, orthe σ-weak topology for that matter).Again, we follow a general path, proving what assertions can be madeabout posets of subalgebras of any kind of algebraic structures, where itssubalgebras are also topologically closed.For the rest of this section, we only consider Hausdorff topologicalspaces. This condition is necessary for our proofs to work. The reason is
Unsharp Values, Domains <strong>and</strong> Topoi 85that, in a Hausdorff space, a net (a i ) i∈I converges to at most one point a.We shall write this as (a i ) i∈I → a.Definition 5.15. A topological algebra is an algebra A=⟨A; F⟩ where A isequipped with a Hausdorff topology.We single out the substructures of interest:Definition 5.16. Given a topological algebra A=⟨A; F⟩,aclosed subalgebrais a subset B of A which is simultaneously a subalgebra <strong>and</strong> a topologicallyclosed set. The set of closed subalgebras of A is denoted by CSub A .Moreover,for a system of polynomial equations E over A, CSub A /E denotes CSub A ∩Sub A /E.Our goal is to extend the results about Sub A /E to CSub A /E. For this,we need to impose a topological condition on the behaviour of the operationsof A. Usually, one requires the algebraic operations to be continuous, whichwould allow us to prove results regarding completeness which are similarto those in previous subsections. An example would then be the poset of(abelian) closed subgroups of a topological group. However, multiplication ina von Neumann algebra is not continuous with respect to the weak operatortopology, only separately continuous in each argument. Thus, in order tocapture the case of V(N), we need to weaken the continuity assumption.The following condition will suffice:Definition 5.17. Let A=⟨A; F⟩ be a topological algebra <strong>and</strong> f ∈F an operationwith arity n.Wesaythatf is separately continuous if, for any k = 1,...,n<strong>and</strong> elements b 1 ,...,b k−1 ,b k+1 ,...,b n ∈ A, the functionA→Aa → f(b 1 ,...,b k−1 ,a,b k+1 ,...,b n )is continuous. Equivalently, one can say that for any net (a i ) i∈I such that(a i ) i∈I → a, wehave(f(b 1 ,...,b k−1 ,a i ,b k+1 ,...,b n )) i∈I → f(b 1 ,...,b k−1 ,a,b k+1 ,...,b n ).The fact that we allow a weaker form of continuity than it is costumaryon the algebraic operations forces us to impose a condition on the allowedequations.Definition 5.18. A polynomial over A in the variables x 1 ,...,x n is linear ifeach of the variables occurs at most once.Lemma 5.19. Let A=⟨A; F⟩ be a topological algebra with separately continuousoperations, <strong>and</strong> p a linear polynomial over A in the variables x 1 ,...,x n .Then the functioñp ∶A n → A(a 1 ,...,a n )→∣p∣ ν∶xi →a iis separately continuous.
86 A. Döring <strong>and</strong> R. Soares BarbosaProof. The proof goes by induction on (linear) polynomials (refer back todefinitions 5.10 <strong>and</strong> 5.11):● If p = x j ,theñp(a 1 ,...,a n )=∣x j ∣ ν∶xi →a i= ν(x j )=a j . (5.4)So ̃p = π j , which is clearly separately continuous (either constant or theidentity when arguments taken separately).● If p = f(p 1 ,...,p l ), assume as the induction hypothesis that ̃p 1 ,...,̃p lare separately continuous. Theñp = f ⋅⟨̃p 1 ,...,̃p l ⟩ . (5.5)Let us see that this is separately continuous in the first argument x 1(the other arguments can be treated analogously). Let b 2 ,...,b n ∈ A.Since p is linear, the variable x 1 occurs on at most one subpolynomialp k (with k = 1,...,l). Without loss of generality, say it occurs on p 1 .Then the functionst k = a → ̃p k (a, b 2 ,...,b n )for k = 2,...,l are constant (since x 1 does on occur in p k ), whereas thefunctiont 1 = a → ̃p 1 (a, b 2 ,...,b n )is continuous (by the induction hypothesis).Now, we consider the function ̃p with only the first argument varying.This is the function f ⋅⟨t 1 ,...,t l ⟩.Butt 2 ,...,t l are constant functions,which means that only the first argument of f varies. Since thisis given by a continuous function t 1 <strong>and</strong> f is continuous in the firstargument (because it is separately continuous), f ⋅⟨t 1 ,...,t l ⟩∶A → Ais continuous, meaning that ̃p is (separately) continuous in the first argument.For the other arguments, the proof is similar.◻We shall denote by cl(−) the (Kuratowski) closure operator associatedwith the topology of A.Lemma 5.20. For a topological algebra A=⟨A; F⟩ with separately continuousoperations:1. If B ∈ Sub A ,thencl(B)∈Sub A .2. Given a system of linear polynomial equations E over A, ifB ∈ Sub A /E,then cl(B)∈Sub A /E.Proof. (1) We need to show that cl(B) is closed under the operations in F.We consider only the case of a binary operation f ∈F. It will be apparentthat the general case follows from a simple inductive argument.Let a, b ∈ cl(B). Then there exist nets (a i ) i∈I <strong>and</strong> (b j ) j∈J consisting ofelements of B such that (a i ) i∈I → a <strong>and</strong> (b j ) j∈J → b. Byfixinganindexjat a time, we conclude, by separate continuity, that∀j ∈ J ∶(f(a i ,b j )) i∈I → f(a, b j ). (5.6)
Unsharp Values, Domains <strong>and</strong> Topoi 87Since all the elements of the net (f(a i ,b j )) i∈I are in B (for a i ,b j ∈ B <strong>and</strong>B is closed under f), all elements of the form f(a, b j ) are in cl(B), astheyare limits of nets of elements of B. Now, letting j vary again, by separatecontinuity we obtain(f(a, b j )) j∈J → f(a, b). (5.7)But (f(a, b j )) j∈J is a net of elements of cl(B), thusf(a, b) ∈cl(cl(B)) =cl(B). This completes the proof that cl(B) is closed under f.(2) The procedure is very similar <strong>and</strong> again we consider only a polynomialequation ⟨p, q⟩ in two variables x <strong>and</strong> y. Wewillprovethatcl(B)satisfies the equation whenever B does.Let a, b ∈ cl(B), with sets (a i ) i∈I <strong>and</strong> (b j ) j∈J as before. We use thefunction ̃p ∶ A 2 → A from lemma 5.19, which is given bỹp(k 1 ,k 2 )=∣p∣ [x1↦k 1,x 2↦k 2]. (5.8)Because of Lemma 5.19, we can follow the same procedure as in the first part(for the separately continuous function f) to conclude that∀j ∈ J ∶(̃p(a i ,b j )) i∈I → ̃p(a, b j ) ∧ (̃q(a i ,b j )) i∈I → ̃q(a, b j ). (5.9)But a i ,b j ∈ B <strong>and</strong> B satisfies the equations, so the nets (̃p(a i ,b j )) i∈I <strong>and</strong>(̃q(a i ,b j )) i∈I are the same. Since we have Hausdorff spaces by assumption, anet has at most one limit. This implies that∀j ∈ J ∶ ̃p(a, b j )=̃q(a, b j ). (5.10)As before, we take (again by separate continuity of ̃p <strong>and</strong> ̃q)(̃p(a, b j )) j∈J → ̃p(a, b) ∧ (̃q(a, b j )) j∈J → ̃q(a, b). (5.11)Because of 5.10 <strong>and</strong> 5.11, the nets are the same again, yielding̃p(a, b) =̃q(a, b). (5.12)This proves that cl(B) satifies ⟨p, q⟩.◻We now can state our result. As far as completeness (or convergence)properties are concerned, we have a similar situation as in the non-topologicalcase.Proposition 5.21. Let A=⟨A; F⟩ be a topological algebra with separately continuousoperations. Let E be a system of linear polynomial equations. Then:1. CSub A is a complete lattice.2. CSub A /E is a bounded-complete dcpo.Proof. If S is a set of closed subalgebras, then, by 5.20-1, there is a least closedsubalgebra containing S, givenbycl(⟨⋃ S⟩). This proves the first claim.Now, suppose that all B ∈ S satisfy E <strong>and</strong> that S is directed. By directedness,the supremum is given by cl(⟨⋃ S⟩) = cl(⋃ S). By Lemma 5.14, ⋃ Ssatisfies the equations E. Then, by 5.20-2, cl(⋃ S) also does.Similarly, suppose all B ∈ S satisfy E <strong>and</strong> that S is bounded above byC ∈ CSub A /E. Then the supremum of S in CSub A , namely cl(⟨⋃ S⟩), isa
88 A. Döring <strong>and</strong> R. Soares Barbosaclosed subalgebra smaller than C. Therefore, cl(⟨⋃ S⟩) must also satisfy E,hence it is a supremum for S in CSub A /E.◻V(N) as a dcpo. The results above apply easily to the situation we are interestedin, namely the poset of abelian von Neumann subalgebras of a nonabelianvon Neumann algebra N . We can describe N as the algebra 5⟨N ; 1, +, ⋅ c , ×, ∗⟩ (5.13)whose operations have arities 0, 2, 1, 2 <strong>and</strong> 1, respectively. The operationsare just the usual ∗-algebra operations. Note that ⋅ c (scalar multiplication byc ∈ C) is in fact an uncountable set of operations indexed by C. Note thatthe inclusion of 1 as an operation allows to restrict our attention to unitalsubalgebras with the same unit <strong>and</strong> to unital homomorphisms. Recall thata C ∗ -subalgebra of N is a ∗-subalgebra that is closed with respect to thenorm topology. A von Neumann subalgebra is a ∗-subalgebra that is closedwith respect to the weak operator topology. Since both these topologies areHausdorff <strong>and</strong> the operations are separately continuous with respect to both,the results from the previous sections apply.Proposition 5.22. Let N be a von Neumann algebra. Then● the set of unital ∗-subalgebras (respectively, C ∗ -subalgebras, <strong>and</strong> vonNeumann subalgebras) of N with the same unit element as N is a completelattice.● The set of abelian unital ∗-subalgebras (respectively, C ∗ -subalgebras, <strong>and</strong>von Neumann subalgebras) of N with the same unit element as N is abounded complete dcpo (or complete semilattice).The set of abelian von Neumann subalgebras of N with the same unitelement as N is simply V(N). Hence, we have shown that V(N) is a dcpo.Moreover, it is clear that V(N) is bounded complete if we include the trivialalgebra as the bottom element. Note that this holds for all types of vonNeumann algebras. Furthermore, our proof shows that analogous results holdfor the poset of abelian C ∗ -subalgebras of unital C ∗ -algebras (<strong>and</strong> otherkinds of topological algebras <strong>and</strong> equations, such as associative closed Jordansubalgebras, or closed subgroups of a topological group).As already remarked at the beginning of this subsection on topologicallyclosed subalgebras, V(N) is an algebraic domain in the case that N is afinite-dimensional algebra (finite direct sum of matrix algebras over C). Thiscan be seen directly from the fact that, in the finite-dimensional situation,any subalgebra is weakly (resp. norm) closed. Hence, the topological aspectsdo not play a rôle <strong>and</strong> proposition 5.14 applies. The result that V(N) isan algebraic domain if N is a finite-dimensional matrix algebra can also bededuced from the fact that V(N) has finite height in this case. This clearlymeans that any directed set has a maximal element <strong>and</strong>, consequently, the5 “Algebra” here has two different meanings. The first is the (traditional) meaning, of whichthe ∗-algebras, C ∗ -algebras <strong>and</strong> von Neumann algebras are special cases. When we say justalgebra, though, we mean it in the sense of universal algebra as in the previous two sections.
Unsharp Values, Domains <strong>and</strong> Topoi 89way-below relation coincides with the order relation, thus V(N) is triviallyan algebraic domain.One can also see directly that the context category V(N) isadcpoforan arbitrary von Neumann algebra N :letS ⊂V(N)be a directed subset,<strong>and</strong> letA ∶= ⟨ ⋃ V ⟩ (5.14)V ∈Sbe the algebra generated by the elements of S. Clearly, A is a self-adjointabelian algebra that contains ˆ1, but not a von Neumann algebra in general,since it need not be weakly closed. By von Neumann’s double commutanttheorem, the weak closure Ṽ ∶= Aw of A is given by the double commutantA ′′ of A. The double commutant construction preserves commutativity, soA ′′ = Ṽ is an abelian von Neumann subalgebra of N , namely the smallestvon Neumann algebra containing all the V ∈ S, soṼ = ⊔ ↑ S. (5.15)Hence, every directed subset S has a supremum in V(N).A functor from von Neumann algebras to dcpos. We have seen that to eachvon Neumann algebra N , we can associate a dcpo V(N). We will now showthat this assignment is functorial. Again, this result arises as a special caseof a more general proposition, which is quite easy to show.Definition 5.23. Let A <strong>and</strong> B be two algebras with the same signature Σ (thatis, they have the same set of operations). A function φ ∶ A → B is called ahomomorphism if it preserves every operation. That is, for each operation fof arity n,φ(f A (a 1 ,...,a n )) = f B (φ(a 1 ),...,φ(a n )). (5.16)Lemma 5.24. Let A <strong>and</strong> B be topological algebras <strong>and</strong> φ ∶ A → B a continuoushomomorphism that preserves closed subalgebras. Then, for any E, thefunctionφ → ∣ CSubA /E ∶ CSub A /E → CSub B /E (5.17)S → φ → Sis a Scott-continuous function.Proof. First, we check that the function is well-defined, that is, each elementof the domain determines one element in the codomain. Let S ∈ CSub A /E.Then S ′ ∶= φ → (S) is in CSub B by the assumption that φ preserves closedsubalgebras. The homomorphism condition implies that it also preserves thesatisfaction of equations E.As for the result itself, note that monotonicity is trivial. So we only needto show that this map preserves directed joins. We know that the directedjoin is given as ⊔ ↑ D = cl(⋃ D). Thus, we want to prove that for any directedset D:φ → (cl( ⋃ S)) ⊆ cl( ⋃ φ → (S)) .S∈DS∈D
90 A. Döring <strong>and</strong> R. Soares BarbosaThis follows easily since f → commutes with union of sets <strong>and</strong> the inequalityφ → cl(S)⊆cl(φ → S) is simply the continuity of φ.◻We can combine the previous lemma with Prop. 5.21 as follows:Proposition 5.25. Let C be a category such that● objects of C are topological Σ-algebras (for a fixed signature Σ) that fulfilthe conditions stated in Prop. 5.21,● morphisms of C are continuous homomorphisms preserving closed subalgebras.Then, for any system of linear polynomial equations E, the assignmentsA → CSub A /E (5.18)(φ ∶ A → B)→(φ → ∣ CSubA /E ∶ CSub A /E → CSub B /E) (5.19)define a functor F E ∶C→DCPO from C to the category of dcpos <strong>and</strong> Scottcontinuousfunctions.Proof. 5.21 <strong>and</strong> 5.24 imply that this is a well-defined map between the categories.Functoriality is immediate from the properties of images f → of afunction f: id → A = id PA <strong>and</strong> f → ○ g → =(f ○ g) → .◻This result can be applied to our situation. Let N be a von Neumannalgebra. Recall that a von Neumann subalgebra is a ∗-subalgebra closed withrespect to the σ-weak topology. 6 This topology is Hausdorff, <strong>and</strong> σ-weaklycontinuous (or normal) ∗-homomorphisms map von Neumann subalgebras tovon Neumann subalgebras. Hence, we getTheorem 5.26. The assignmentsV∶vNAlg → DCPON→V(N) (5.20)φ → φ → ∣ V(N) (5.21)define a functor from the category vNAlg of von Neumann algebras <strong>and</strong> σ-weakly continuous, unital algebra homomorphisms to the category DCPO ofdcpos <strong>and</strong> Scott-continuous functions.It is easy to see that the functor V is not full. It is currently an openquestion whether V is faithful.6. The context category V(N) in infinite dimensions – lack ofcontinuityThecaseofN=B(H)with infinite-dimensional H. We now show that foran infinite-dimensional Hilbert space H <strong>and</strong> the von Neumann algebra N=6 Equivalently, a von Neumann algebra is closed in the weak operator topology, as mentionedbefore.
Unsharp Values, Domains <strong>and</strong> Topoi 91B(H), theposetV(N) is not continuous, <strong>and</strong> hence not a domain. Thiscounterexample is due to Nadish de Silva, whom we thank.Let H be a separable, infinite-dimensional Hilbert space, <strong>and</strong> let (e i ) i∈Nbe an orthonormal basis. For each i ∈ N, let ˆP i be the projection onto the rayCe i . This is a countable set of pairwise-orthogonal projections.Let ˆP e ∶= ∑ ∞ i=1 ˆP 2i ,thatis, ˆP e is the sum of all the ˆP i with even indices.Let V e ∶= { ˆP e , ˆ1} ′′ = C ˆP e + Cˆ1 be the abelian von Neumann algebra generatedby ˆP e <strong>and</strong> ˆ1. Moreover, let V m be the (maximal) abelian von Neumannsubalgebra generated by all the ˆP i , i ∈ N. Clearly, we have V e ⊂ V m .Now let V j ∶= { ˆP 1 , ..., ˆP j , ˆ1} ′′ = C ˆP 1 + ... + C ˆP j + Cˆ1 be the abelian vonNeumann algebra generated by ˆ1 <strong>and</strong>thefirstj projections in the orthonormalbasis,wherej∈ N. The algebras V j form a directed set (actually a chain)(V j ) j∈N whose directed join is V m , obviously. But note that none of the algebrasV j contains V e , since the projection ˆP e is not contained in any of theV j . Hence, we have the situation that V e is contained in ⊔ ↑ (V j ) j∈N = V m , butV e ⊈ V j for any element V j of the directed set, so V e is not way below itself.Since V e is an atom in the poset V(N) (i.e. only the bottom element,the trivial algebra, is below V e in the poset), we have that ↡V e ={⊥}={Cˆ1},so ⊔ ↑ ↡V e = ⊔ ↑ {⊥} = ⊥ ≠ V e , which implies that V(N) is not continuous.Other types of von Neumann algebras. The proof that V(N ) = V(B(H)) isnot a continuous poset if H is infinite-dimensional can be generalised to othertypes of von Neumann algebras. (B(H) is a type I ∞ factor.) We will use thewell-known fact that an abelian von Neumann algebra V is of the formV ≃ l ∞ (X, μ) (6.1)for some measure space (X, μ). It is known that the following cases can occur:X ={1, ..., n}, foreachn ∈ N, orX = N, equipped with the counting measureμ c ;orX =[0, 1], equipped with the Lebesgue measure μ; or the combinationsX ={1, ..., n}⊔[0, 1] <strong>and</strong> X = N ⊔[0, 1].A maximal abelian subalgebra V m of B(H) generated by the projectionsonto the basis vectors of an orthonormal basis (e i ) i∈N of an infinitedimensional,separable Hilbert space H corresponds to the case of X = N, soV m ≃ l ∞ (N). The proof that in this case V(B(H)) is not continuous is basedon the fact that X = N can be decomposed into a countable family (S i ) i∈N ofmeasurable, pairwise disjoint subsets with ⋃ i S i = X such that μ(S i )>0foreach i. Of course, the sets S i can just be taken to be the singletons {i}, fori ∈ N, in this case. These are measurable <strong>and</strong> have measure greater than zerosince the counting measure on X = N is used.Yet, also for X =([0, 1],μ) we can find a countable family (S i ) i∈N ofmeasurable, pairwise disjoint subsets with ⋃ i S i = X such that μ(S i )>0for each i: for example, take S 1 ∶= [0, 1 2 ), S 2 ∶= [ 1 2 , 3 4 ), S 3 ∶= [ 3 4 , 7 8 ),etc.Each measurable subset S i corresponds to a projection ˆP i in the abelian vonNeumann algebra V ≃ l ∞ ([0, 1],μ), so we obtain a countable family ( ˆP i ) i∈N
92 A. Döring <strong>and</strong> R. Soares Barbosaof pairwise orthogonal projections with sum ˆ1 inV .Let ˆP e ∶= ∑ ∞ i=1 ˆP 2i ,<strong>and</strong>let V e ∶= { ˆP e , ˆ1} ′′ = C ˆP e + Cˆ1.Moreover, let V j ∶= { ˆP 1 , ..., ˆP j , ˆ1} ′′ be the abelian von Neumann algebragenerated by ˆ1 <strong>and</strong>thefirstj projections, where j ∈ N. Clearly, (V j ) j∈N isa directed set, <strong>and</strong> none of the algebras V j , j ∈ N, contains the algebra V e ,since ˆP e ∉ V j for all j. LetV ∶= ⊔ ↑ V j be the abelian von Neumann algebragenerated by all the V j .ThenwehaveV e ⊂ V ,soV e is not way below itself<strong>and</strong> ⊔ ↑ ↡V e ≠ V e .This implies that V(N) is not continuous if N contains an abeliansubalgebra of the form V ≃ l ∞ ([0, 1],μ). Clearly, analogous results hold ifN contains any abelian subalgebra of the form V = l ∞ ({1, ..., n}⊔[0, 1]) orV ≃ l ∞ (N ⊔[0, 1]) – in none of these cases is V(N) a domain.A von Neumann algebra N contains only abelian subalgebras of theform V ≃ l ∞ ({1, ..., n},μ c ) if <strong>and</strong> only if N is either (a) a finite-dimensionalmatrix algebra M n (C)⊗C with C an abelian von Neumann algebra (i.e., Nis a type I n -algebra) such that the centre C does not contain countably manypairwise orthogonal projections, 7 or (b) a finite direct sum of such matrixalgebras. Equivalently, N is a finite-dimensional von Neumann algebra, thatis, N is represented (faithfully) on a finite-dimensional Hilbert space. Wehave shown:Theorem 6.1. The context category V(N), that is, the partially ordered set ofabelian von Neumann subalgebras of a von Neumann algebra N which sharethe unit element with N , is a continuous poset <strong>and</strong> hence a domain if <strong>and</strong>only if N is finite-dimensional.7. Outlook: Space <strong>and</strong> space-time from interval domains?There are many further aspects of the link between the topos approach <strong>and</strong>domain theory to be explored. In this brief section, we indicate some veryearly ideas about an application to concepts of space <strong>and</strong> space-time. 8 Theideas partly go back to discussions with Chris Isham, but we take full responsibility.Let us take seriously the idea suggested by the topos approach thatin quantum theory it is not the real numbers R where physical quantitiestake their values, but rather the presheaf R ↔ . This applies in particular tothe physical quantity position. We know from ordinary quantum theory thatdepending on its state, a quantum particle will be localised more or lesswell. (Strictly speaking, there are no eigenstates of position, so a particle cannever be localised perfectly.) The idea is that we can now describe ‘unsharppositions’ as well as sharp ones by using the presheaf R ↔ <strong>and</strong> its globalelements. A quantum particle will not ‘see’ a point in the continuum R as its7 This of course means that C ≃ l ∞ ({1, ..., k},μ c), that is, C is isomorphic to a finitedimensionalalgebra of diagonal matrices with complex entries.8 The reader not interested in speculation may well skip this section!
Unsharp Values, Domains <strong>and</strong> Topoi 93position, but, depending on its state, it will ‘see’ some global element of R ↔as its (generalised) position.These heuristic arguments suggest the possibility of building a model ofspace based not on the usual continuum R, but on the quantity-value presheafR ↔ . The real numbers are embedded (topologically) in R ↔ as a tiny part,they correspond to global elements γ =(μ, ν) such that μ = ν = const a ,wherea ∈ R <strong>and</strong> const a is the constant function with value a.It is no problem to form R ↔3 or R ↔n within the topos, <strong>and</strong> also externallyit is clear what these presheaves look like. It is also possible to generalisenotions of distance <strong>and</strong> metric from R n to IR n , i.e., powers of the intervaldomain. This is an intermediate step to generalising metrics from R n to R ↔n .Yet, there are many open questions.The main difficulty is to find a suitable, physically sensible group thatwould act on R ↔n . Let us focus on space, modelled on R ↔3 ,fornow.Wepicture an element of R ↔3 V as a ‘box’: there are three intervals in three independentcoordinate directions. We see that in this interpretation, thereimplicitly is a spatial coordinate system given. Translations are unproblematic,they will map a box to another box. But rotations will map a box to arotated box that in general cannot be interpreted as a box with respect to thesame (spatial) coordinate system. This suggests that one may want to considera space of (box) shapes rather than the boxes themselves. These shapeswould then represent generalised, or ‘unsharp’, positions. Things get evenmore difficult if we want to describe space-time. There is no time operator,so we have to take some liberty when interpreting time intervals as ‘unsharpmoments in time’. More crucially, the Poincaré group acting on classical specialrelativistic space-time is much more complicated than the Galilei groupthat acts separately on space <strong>and</strong> time, so it will be harder to find a suitable‘space of shapes’ on which the Poincaré group, or some generalisation of it,will act.There is ongoing work by the authors to build a model of space <strong>and</strong>space-time based on the quantity-value presheaf R ↔ . It should be mentionedthat classical, globally hyperbolic space-times can be treated as domains, aswas shown by Martin <strong>and</strong> Panangaden [27]. These authors consider generalisedinterval domains over so-called bicontinuous posets. This very interestingwork serves as a motivation for our ambitious goal of defining a notion ofgeneralised space <strong>and</strong> space-time suitable for quantum theory <strong>and</strong> theories‘beyond quantum theory’, in the direction of quantum gravity <strong>and</strong> quantumcosmology. Such a model of space <strong>and</strong> space-time would not come from adiscretisation, but from an embedding of the usual continuum into a muchricher structure. It is very plausible that domain-theoretic techniques will stillapply, which could give a systematic way to define a classical limit of thesequantum space-times. For now, we merely observe that the topos approach,which so far applies to non-relativistic quantum theory, provides suggestivestructures that may lead to new models of space <strong>and</strong> space-time.
94 A. Döring <strong>and</strong> R. Soares Barbosa8. ConclusionWe developed some connections between the topos approach to quantum theory<strong>and</strong> domain theory. The topos approach provides new ‘spaces’ of physicalrelevance, given as objects in a presheaf topos. Since these new spaces are noteven sets, many of the usual mathematical techniques do not apply, <strong>and</strong> wehave to find suitable tools to analyse them. In this article, we focused on thequantity-value object R ↔ in which physical quantities take their values, <strong>and</strong>showed that domain theory provides tools for underst<strong>and</strong>ing the structure ofthis space of generalised, unsharp values. This also led to some preliminaryideas about models of space <strong>and</strong> space-time that would incorporate unsharppositions, extending the usual continuum picture of space-time.AcknowledgementsWe thank Chris Isham, Prakash Panangaden <strong>and</strong> Samson Abramsky for valuablediscussions <strong>and</strong> encouragement. Many thanks to Nadish de Silva forproviding the counterexample in Section 6, <strong>and</strong> to the anonymous refereefor a number of valuable suggestions <strong>and</strong> hints. RSB gratefully acknowledgessupport by the Marie Curie Initial Training Network in <strong>Mathematical</strong> Logic- MALOA - From MAthematical LOgic to Applications, PITN-GA-2009-238381, as well as previous support by Sant<strong>and</strong>er Abbey bank. We also thankFelix Finster, Olaf Müller, Marc Nardmann, Jürgen Tolksdorf <strong>and</strong> EberhardZeidler for organising the very enjoyable <strong>and</strong> interesting “<strong>Quantum</strong> <strong>Field</strong><strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>” conference in Regensburg in September 2010.References[1] S. Abramsky, A. Jung. Domain <strong>Theory</strong>. In H<strong>and</strong>book of Logic in ComputerScience, eds. S. Abramsky, D.M. Gabbay, T.S.E. Maibaum, Clarendon Press,1–168 (1994).[2] M. Caspers, C. Heunen, N.P. L<strong>and</strong>sman, B. Spitters. Intuitionistic quantumlogic of an n-level system. Found. Phys. 39, 731–759 (2009).[3] B. Coecke, K. Martin. A partial order on classical <strong>and</strong> quantum states. In NewStructures for Physics, ed. B. Coecke, Springer, Heidelberg (2011).[4] A. Döring. Kochen-Specker theorem for von Neumann algebras. Int. Jour.Theor. Phys. 44, 139–160 (2005).[5] A. Döring. Topos theory <strong>and</strong> ‘neo-realist’ quantum theory. In <strong>Quantum</strong> <strong>Field</strong><strong>Theory</strong>, Competitive Models, eds. B. Fauser, J. Tolksdorf, E. Zeidler, Birkhäuser(2009).[6] A. Döring. <strong>Quantum</strong> states <strong>and</strong> measures on the spectral presheaf. Adv. Sci.Lett. 2, special issue on “<strong>Quantum</strong> <strong>Gravity</strong>, Cosmology <strong>and</strong> Black Holes”, ed.M. Bojowald, 291–301 (2009).[7] A. Döring. Topos quantum logic <strong>and</strong> mixed states. In Proceedings of the 6thInternational Workshop on <strong>Quantum</strong> Physics <strong>and</strong> Logic (QPL 2009), Oxford,eds. B. Coecke, P. Panangaden, <strong>and</strong> P. Selinger. Electronic Notes in TheoreticalComputer Science 270(2) (2011).
Unsharp Values, Domains <strong>and</strong> Topoi 95[8] A. Döring. The physical interpretation of daseinisation. In Deep Beauty, ed.Hans Halvorson, 207–238, Cambridge University Press, New York (2011).[9] A. Döring, <strong>and</strong> C.J. Isham. A topos foundation for theories of physics: I. Formallanguages for physics. J. Math. Phys 49, 053515 (2008).[10] A. Döring, <strong>and</strong> C.J. Isham. A topos foundation for theories of physics: II.Daseinisation <strong>and</strong> the liberation of quantum theory. J. Math. Phys 49, 053516(2008).[11] A. Döring, <strong>and</strong> C.J. Isham. A topos foundation for theories of physics: III.<strong>Quantum</strong> theory <strong>and</strong> the representation of physical quantities with arrowsĂ ∶ Σ →PR. J. Math. Phys 49, 053517 (2008).[12] A. Döring, <strong>and</strong> C.J. Isham. A topos foundation for theories of physics: IV.Categories of systems. J. Math. Phys 49, 053518 (2008).[13] A. Döring, <strong>and</strong> C.J. Isham. ‘What is a thing?’: topos theory in the foundationsof physics. In New Structures for Physics, ed. B. Coecke, Springer (2011).[14] T. Erker, M. H. Escardó <strong>and</strong> K. Keimel. The way-below relation of functionspaces over semantic domains. Topology <strong>and</strong> its Applications 89, 61–74 (1998).[15] C. Flori. A topos formulation of consistent histories. Jour. Math. Phys 51053527 (2009).[16] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove, D.S. Scott.Continuous Lattices <strong>and</strong> Domains. Encyclopedia of Mathematics <strong>and</strong> its Applications93, Cambridge University Press (2003).[17] C. Heunen, N.P. L<strong>and</strong>sman, B. Spitters. A topos for algebraic quantum theory.Comm. Math. Phys. 291, 63–110 (2009).[18] C. Heunen, N.P. L<strong>and</strong>sman, B. Spitters. The Bohrification of operator algebras<strong>and</strong> quantum logic. Synthese, in press (2011).[19] C. Heunen, N.P. L<strong>and</strong>sman, B. Spitters. Bohrification. In Deep Beauty, ed.H. Halvorson, 271–313, Cambridge University Press, New York (2011).[20] C.J. Isham. Topos methods in the foundations of physics. In Deep Beauty, ed.Hans Halvorson, 187–205, Cambridge University Press, New York (2011).[21] C.J. Isham. Topos theory <strong>and</strong> consistent histories: The internal logic of the setof all consistent sets. Int. J. Theor. Phys., 36, 785–814 (1997).[22] C.J. Isham <strong>and</strong> J. Butterfield. A topos perspective on the Kochen-Speckertheorem: I. <strong>Quantum</strong> states as generalised valuations. Int. J. Theor. Phys. 37,2669–2733 (1998).[23] C.J. Isham <strong>and</strong> J. Butterfield. A topos perspective on the Kochen-Speckertheorem: II. <strong>Conceptual</strong> aspects, <strong>and</strong> classical analogues. Int. J. Theor. Phys.38, 827–859 (1999).[24] C.J. Isham, J. Hamilton <strong>and</strong> J. Butterfield. A topos perspective on the Kochen-Specker theorem: III. Von Neumann algebras as the base category. Int. J.Theor. Phys. 39, 1413-1436 (2000).[25] C.J. Isham <strong>and</strong> J. Butterfield. Some possible roles for topos theory in quantumtheory <strong>and</strong> quantum gravity. Found. Phys. 30, 1707–1735 (2000).[26] C.J. Isham <strong>and</strong> J. Butterfield. A topos perspective on the Kochen-Speckertheorem: IV. Interval valuations. Int. J. Theor. Phys 41, 613–639 (2002).[27] K. Martin, P. Panangaden. A domain of spacetime intervals in general relativity.Commun. Math. Phys. 267, 563-586 (2006).
96 A. Döring <strong>and</strong> R. Soares Barbosa[28] D.S. Scott. Outline of a mathematical theory of computation. In Proceedingsof 4th Annual Princeton Conference on Information Sciences <strong>and</strong> Systems,169–176 (1970).[29] D.S. Scott. Continuous lattices. In Toposes, Algebraic Geometry <strong>and</strong> Logic,Lecture Notes in Mathematics 274, Springer, 93–136 (1972).[30] D.S. Scott. Formal semantics of programming languages. In Lattice theory, datatypes <strong>and</strong> semantics, Englewood Cliffs, Prentice-Hall, 66–106 (1972).[31] S. Willard. General Topology. Addison-Wesley Series in Mathematics, AddisonWesley (1970).[32] L. Ying-Ming, L. Ji-Hua. Solutions to two problems of J.D. Lawson <strong>and</strong> M.Mislove. Topology <strong>and</strong> its Applications 69, 153–164 (1996).Andreas Döring <strong>and</strong> Rui Soares Barbosa<strong>Quantum</strong> GroupDepartment of Computer ScienceUniversity of OxfordWolfson BuildingParks RoadOxford OX1 3QDUKe-mail: <strong>and</strong>reas.doering@cs.ox.ac.ukrui.soaresbarbosa@wolfson.ox.ac.uk
Causal Boundary of Spacetimes:Revision <strong>and</strong> Applications toAdS/CFT CorrespondenceJosé Luis Flores, Jónatan Herrera <strong>and</strong> Miguel SánchezAbstract. The aim of this work is to explain the status <strong>and</strong> role of theso-called causal boundary of a spacetime in <strong>Mathematical</strong> Physics <strong>and</strong>Differential Geometry. This includes: (a) the consistency of its latestredefinition, (b) its role as an intrinsic conformally invariant boundaryin the AdS/CFT correspondence, (c) its relation with the Gromov <strong>and</strong>Cauchy boundaries for a Riemannian manifold, <strong>and</strong> (d) its further relationwith boundaries in Finsler Geometry.Mathematics Subject Classification (2010). Primary 83C75; Secondary53C50, 83E30, 83C35, 53C60.Keywords. Causal boundary, conformal boundary, AdS/CFT correspondence,plane wave, static spacetime, Gromov compactification, Cauchycompletion, Busemann function, stationary spacetime, Finsler metric,R<strong>and</strong>ers metric, pp-wave, arrival functional, Sturm-Liouville theory.1. IntroductionSince the seminal paper by Geroch, Kronheimer <strong>and</strong> Penrose in 1972 (see[17]), the definition <strong>and</strong> computation of a causal boundary (c-boundary, forshort) for reasonably well-behaved spacetimes have been a long-st<strong>and</strong>ing issuein <strong>Mathematical</strong> Relativity. This boundary would help to underst<strong>and</strong> betterthe global causal structure of the spacetime, <strong>and</strong> would represent a “concreteplace” where asymptotic conditions could be posed. Nevertheless, the lackof a satisfactory definition, as well as the difficulty of the computation ofelements associated to it, has favored the usage of the so-called conformalboundary. This boundary is not intrinsic to the spacetime, <strong>and</strong> it has severeThe authors are partially supported by the Spanish MICINN Grant MTM2001-60731 <strong>and</strong>Regional J. Andalucía Grant P09-FQM-4496, both with FEDER funds. Also, JH is supportedby Spanish MEC Grant AP2006-02237.F. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_ 6,© Springer Basel AG 201297
98 J. L. Flores, J. Herrera <strong>and</strong> M. Sánchezproblems of existence <strong>and</strong> uniqueness. However, sometimes it is easy to compute<strong>and</strong> it remains as a non-general construction which is useful in manysimple particular cases. Thus it is not strange that when developments inString <strong>Theory</strong> required a holographic boundary for the (bulk) spacetime, theconformal boundary was chosen. However, the limitations of this boundaryfor both fields, <strong>Mathematical</strong> Relativity <strong>and</strong> String <strong>Theory</strong>, have become apparent.Fortunately, there has been huge recent progress in the underst<strong>and</strong>ingof the c-boundary, which will be briefly reviewed now.In [11], [12] the authors have developed carefully a new notion of thec-boundary. This notion was suggested by one of the authors in [31], by selectingelements already developed by many other authors: GKP [17], Budic& Sachs [5], Rácz [30], Szabados [34], Harris [20], Marolf & Ross [25], Low[23], Flores [9], etc. In Sections 2, 3 <strong>and</strong> 4 we review briefly some intuitivemotivations for the c-boundary, our formal definition of this boundary <strong>and</strong>the arguments in favor of our choice, respectively. These arguments are of twodifferent types. First, we state some formal hypotheses on the minimum propertieswhich any admissible c-boundary must satisfy a priori. Then we arguethat the alternatives to our choice are very few. Second, the nice propertiessatisfied a posteriori by our chosen c-boundary are stressed. In particular,the consistency with the conformal boundary (in natural cases) is explained.This becomes essential, as many previous redefinitions of the c-boundaryhave been disregarded because the c-boundary did not coincide with the (obvious<strong>and</strong> natural) conformal boundary in some concrete examples. Moreover,the c-boundary allows to formalize rigorously assertions which are commonlyused for the conformal boundary in a loose way (see for example Corollary4.5). For a complete development of the material in this part, the readeris referred to [31] (historical development <strong>and</strong> AdS/CFT correspondence onplane waves) <strong>and</strong> [11] (redefinition <strong>and</strong> analysis of the c-boundary).In Sections 5, 6 <strong>and</strong> 7 the c-boundary is computed explicitly for threefamilies of spacetimes (where a natural conformal embedding which woulddefine the conformal boundary is not expected in general): static, stationary<strong>and</strong> pp-wave type spacetimes, respectively. However, our aim goes much further.In fact, the c-boundary of a static spacetime suggests a compactificationfor any (possibly incomplete) Riemannian manifold (as first studied in [10]),which can be compared with the classical Gromov compactification. This hasits own interest for Riemannian Geometry (see Remarks 5.1, 5.4), <strong>and</strong> can beextended to reversible Finsler Geometry. Moreover, the c-boundary for a stationaryspacetime suggests a compactification for any non-reversible Finslermanifold. We have developed not only this compactification but also Gromov’s<strong>and</strong> the Cauchy completion for a non-symmetric distance. The naturalrelations between these boundaries demonstrate the power of the c-boundaryapproach. Finally, the c-boundary for pp-wave type spacetimes suggests toassociate a boundary to certain Lagrangians, an idea which might be developedfurther. For the complete development of the material of this part, werefer to [12] (c-boundary for static <strong>and</strong> stationary spacetimes, <strong>and</strong> the relation
Causal Boundaries 99to the Gromov <strong>and</strong> Cauchy boundaries), [6] (connection between causality ofstationary spacetimes <strong>and</strong> Finsler Geometry) <strong>and</strong> [13] (boundary of pp-wavetype spacetimes).Summing up, it seems that now we have a consistent notion of c-boundary, based on “first principles”. From the practical viewpoint, thisboundary agrees with the conformal one in natural cases, it is closely connectedto other boundaries in Riemannian <strong>and</strong> Finslerian Geometry (Cauchy,Gromov), <strong>and</strong> specific tools for its computation have been developed. Moreover,it allows a better underst<strong>and</strong>ing of the AdS/CFT correspondence: thec-boundary should be used in this correspondence because its constructionis intrinsic, systematic <strong>and</strong> completely general; moreover, a concrete applicationfor plane waves has been obtained. In conclusion, the c-boundary isavailable as a tool for both, <strong>Mathematical</strong> Relativity <strong>and</strong> String <strong>Theory</strong>.2. Boundaries in <strong>Mathematical</strong> Relativity <strong>and</strong> String <strong>Theory</strong>2.1. Boundaries for spacetimesIn Differential Geometry, there are different boundaries which can be assignedto a space, depending on the problem which is being studied. For example,Cauchy’s <strong>and</strong> Gromov’s boundaries are two natural ones in the Riemanniansetting, which will be considered later.For a spacetime (i.e. a connected time-oriented Lorentzian (−, +,...,+)manifold), the most-used boundary in <strong>Mathematical</strong> Relativity is the socalled(Penrose) conformal boundary. In principle, the rough idea is simple.First one tries to find an open conformal embedding i : V ↩→ V 0 of the originalspacetime V in some (“aphysical”) spacetime V 0 with (say, compact) closurei(V ). Then the topological boundary ∂(i(V )) ⊂ V 0 is taken as the conformalboundary ∂ i V of the spacetime.However, such a procedure has obvious problems of both, existence (nogeneral procedure to find the embedding i is prescribed) <strong>and</strong> uniqueness(different choices of i may yield conformal boundaries with different properties,recall the examples in [11, Sect. 4.5]). The problem of uniqueness ledAshtekar <strong>and</strong> Hansen [1] to formulate intrinsic conditions which define asymptoticflatness (including the metric level, not only the conformal one).García-Parrado <strong>and</strong> Senovilla introduced isocausal extensions [15], becausetheir flexibility helps to ensure existence. The abstract boundary [33] by Scott<strong>and</strong> Szekeres is formulated to attach a unique point-set boundary to anyspacetime by using embeddings. But even though all these approaches areinteresting in several cases, the general question of existence <strong>and</strong> uniquenessfor the conformal boundary remains unsolved. The present status can besummarized in the recent results by Chrusciel [7, Ths. 4.5, 3.1], roughly: (a)one can ensure that a maximal extension exists assuming that a conformalextension i already exists <strong>and</strong> some technical properties between any pairof lightlike geodesics hold, <strong>and</strong> (b) uniqueness will hold under technical assumptions,which, a priori, are not satisfied in simple cases, for instance when
100 J. L. Flores, J. Herrera <strong>and</strong> M. Sánchezthe boundary changes from timelike to lightlike around some point. In spiteof these problems, the success of the conformal boundary comes from theexistence of very natural conformal embeddings in many physically relevantspacetimes. So, in many cases, one can define Penrose diagrams which allowto visualize in a simple way the conformal structure of the spacetime. As wewill see later, one remarkable property of the c-boundary is that it is expectedto coincide with the conformal boundary in such favorable cases.Finally, let us point out that Schmidt’s bundle boundary [32] <strong>and</strong> Geroch’sgeodesic boundary [16] are other boundaries (non-conformally invariant)applicable to any Lorentzian manifold. They are not used as widely in<strong>Mathematical</strong> Relativity as the conformal one, <strong>and</strong> it is known that in somesimple cases these boundaries may have undesirable properties (say, a pointin the boundary may be non-T 1 -separated from a point of the manifold, see[18]). But in any case, they are very appealing mathematical constructionswhich deserve to be studied further.2.2. Intrinsic boundary for AdS/CFT correspondenceThe AdS/CFT correspondence, or Maldacena duality, is a conjectured equivalencebetween a string theory on a bulk space (typically the product of anti-deSitter AdS n by a round sphere S m , or by another compact manifold) <strong>and</strong> aquantum field theory without gravity on the boundary of the initial space,which behaves as a hologram of lower dimension.The pertinent question here is which boundary must be chosen as theholographic one. Anti-de Sitter spacetime has a natural open conformal embeddingin Lorentz-Minkowski space which yields a simple (<strong>and</strong> non-compact)conformal boundary. Such a boundary was chosen in the AdS/CFT correspondenceas holographic boundary, but the situation changes when theholography on plane waves (put forward by Berenstein, Maldacena <strong>and</strong> Nastase[2]) is considered. In fact, Berenstein <strong>and</strong> Nastase [3] found a remarkable1-dimensional behavior for the boundary of the plane wave obtainedby Blau, Figueroa-O’Farrill, Hull <strong>and</strong> Papadopoulos [4] (a Penrose limit ofa lightlike geodesic in AdS 5 × S 5 which rotates on S 5 ). While studying towhat extent this behavior occurs in other cases, Marolf <strong>and</strong> Ross [24] realizedthat, among other limitations, the conformal boundary is not available fornon-conformally flat plane waves. Under some minimum hypotheses on thenotion of c-boundary, they proved that the c-boundary also is 1-dimensional,<strong>and</strong> this behavior holds for other plane waves as well. Therefore the usage ofthe c-boundary as the natural boundary for the AdS/CFT correspondencewas proposed.3. Formal definition of the c-boundary3.1. Intuitive ideasIn what follows V will be a strongly causal spacetime, i.e., the topology of Vis generated by the intersections between the chronological futures <strong>and</strong> pasts,
Causal Boundaries 101I + (ρ)ρPI − (γ)γFigure 1. In L 2 \{(t, x) :x ≥ 0}, the boundary point P isrepresented by both, I − (γ) <strong>and</strong>I + (ρ).I + (p) ∩ I − (q) forp, q ∈ V (we use st<strong>and</strong>ard notation as in [21, 28, 26]; inparticular, the chronological relation is denoted by ≪, i.e., p ≪ q iff thereexists a future-directed timelike curve from p to q). Strong causality will turnout to be essential to ensure the compatibility of the topologies on V <strong>and</strong> itsc-completion V , among other desirable properties.The purpose of the c-boundary ∂V is to attach a boundary endpointP ∈ ∂V to any inextensible future-directed (γ) or past-directed (ρ) timelikecurve. The basic idea is that the boundary points are represented by thechronological past I − (γ) or future I + (ρ) of the curve. It is also clear intuitivelythat some boundary points must be represented by both, some I − (γ)<strong>and</strong> some I + (ρ) (see Fig. 1). So, ∂V would be the disjoint union of threeparts: the future infinity ∂ +∞ V , reached by future-directed timelike curvesbut not by past-directed ones; the past infinity ∂ −∞ V , dual to the former;<strong>and</strong> the timelike boundary ∂ 0 V , containing the points which are reached byboth, future <strong>and</strong> past-directed timelike curves. The latter represents intrinsically“losses of global hyperbolicity” (the causal intersections J + (p) ∩ J − (q)are not always compact), i.e. naked singularities. From the viewpoint of hyperbolicpartial differential equations, boundary conditions would be posedon the timelike boundary, while initial conditions would be posed on someappropriate acausal hypersurface of the spacetime, or even at future or pastinfinity.Recall that the essential structure involved in the c-boundary constructionis the conformal structure (causality) of the spacetime. In principle, thec-boundary does not yield directly information on singularities, except if theyare conformally invariant; for example, this is the case for naked singularities.The name “singularity” may seem inappropriate, as in some cases the metricmay be extensible through it (as in Fig. 1); however, such an extension willnot be possible for all the representatives of the conformal class.Next, our aim will be to give precise definitions of the c-completion V :=V ∪ ∂V at all the natural levels; i.e. point set, chronological <strong>and</strong> topologicallevel.
102 J. L. Flores, J. Herrera <strong>and</strong> M. Sánchez3.2. Point set level <strong>and</strong> chronological levelAs a preliminary step, we define the future <strong>and</strong> past (pre)completions ˆV , ˇV ,resp., of V . A subset P ⊂ V is called a past set if I − (P )=P . In this case, P isdecomposable when P = P 1 ∪P 2 with P i past sets such that P 1 ≠ P ≠ P 2 ,<strong>and</strong>is an indecomposable past set (IP) otherwise. The future (pre)completion ˆV ofV is defined as the set of all IPs. One can prove that if P ∈ ˆV then either Pis a proper past set (PIP), i.e. P = I − (p) forsomep ∈ V ,orP is a terminalpast set (TIP), <strong>and</strong>thenP = I − (γ) for some inextensible future-directedtimelike curve γ. Thefuture (pre)boundary ˆ∂V of V is defined as the set ofall TIPs. Therefore, the spacetime V , which is identifiable with the set ofall PIPs, coincides with ˆV \ ˆ∂V . Analogously, one defines the indecomposablefuture sets (IF), which constitute the past (pre)completion ˇV . Each IF is eitherterminal (TIF) or proper (PIF). The set of TIFs is the past (pre)boundaryˇ∂V . The total precompletion is then V ♯ =(ˆV ∪ ˇV )/∼, whereI + (p) ∼ I − (p)∀p ∈ V . Note, however, that this identification is insufficient, as naturallysome TIPs <strong>and</strong> TIFs should represent the same boundary point (Fig. 1). Tosolve this problem, a new viewpoint on the boundary points is introduced.Recall that for a point p ∈ V one has I + (p) =↑I − (p) <strong>and</strong>I − (p) =↓I + (p), where the arrows ↑, ↓ denote the common future <strong>and</strong> past (e.g. ↑P =I + ({q ∈ V : p ≪ q ∀p ∈ P })). Let ˆV ∅ := ˆV ∪ {∅}, ˇV∅ := ˇV ∪ {∅}, <strong>and</strong>choose any (P, F) ∈ ( ˆV ∅ × ˇV ∅ )\{(∅, ∅)}. WesaythatP is S-related to F ifF is included <strong>and</strong> maximal in ↑P <strong>and</strong> P is included <strong>and</strong> maximal in ↓F .Bymaximal we mean that no other F ′ ∈ ˇV ∅ (resp. P ′ ∈ ˆV ∅ ) satisfies the statedproperty <strong>and</strong> includes strictly F (resp. P ); notice that such a maximal setmay be non-maximum (Fig. 3). In addition, if P (resp. F ) does not satisfythe previous condition for any F (resp. P ), we also say that P is S-relatedto ∅ (resp. ∅ is S-related to F ); notice that ∅ is never S-related to itself. Nowwe define the c-completion V of V as:V := {(P, F) ∈ ˆV ∅ × ˇV ∅ : P ∼ S F }. (3.1)In particular, for p ∈ V we have I − (p) ∼ S I + (p); so V is regarded as thesubset {(I − (p),I + (p)) : p ∈ V }⊂V .Recall that the c-completion has been defined only as a point set. However,it can be endowed with the following chronological relation (i.e. a binaryrelation which is transitive, anti-reflexive, without isolates <strong>and</strong> chronologicallyseparable, see [11, Def. 2.1]) which extends naturally the chronologicalrelation ≪ on V :(P, F) ≪ (P ′ ,F ′ ) ⇔ F ∩ P ′ ≠ ∅for all (P, F), (P ′ ,F ′ ) ∈ V .3.3. Topological levelIn order to define the topology on V , let us wonder how the convergence ofsequences in V can be characterized in terms of chronological futures <strong>and</strong>pasts. More precisely, it is not difficult to realize that the following operator
Causal Boundaries 103Figure 2. The sequence σ = {x n } converges to x. However,σ does not converge to x ′ as P ′ = I − (x) violates the secondcondition of the points in ˆL(σ).ˆL determines the limit x (or the possible limits if the Hausdorffness of V werenot assumed a priori) of any sequence σ = {x n } n ⊂ V as follows:{x ∈ ˆL(σ) y ≪ x ⇒ y ≪ xn ,⇔I − (x) ⊂ P ′ (∈ ˆV ), I − (x) ≠ P ′ ⇒∃z ∈ P ′ for large n: z ̸≪ x n(see Fig. 2). This operator suggests the natural notion of convergence for thefuture causal completion ˆV . The operator ˆL determines the possible limits ofany sequence σ = {P n } n ⊂ ˆV as:{P ∈ ˆL(σ) P ⊂ LI({Pn })⇔, (3.2)P is a maximal IP in LS({P n })where LI <strong>and</strong> LS denote the set-theoretic lim inf <strong>and</strong> lim sup, respectively.Now, one can check (see [11, Sect. 3.6]) that a topology is defined on ˆVas follows: C is closed iff ˆL(σ) ∈ C for any sequence σ ⊂ C. For this topology,P ∈ ˆL(σ) only if the sequence σ converges to P . A dual operator Ľ will definea topology on ˇV analogously.Finally, the topology on the completion V or chronological topology isintroduced by means of a new operator L on sequences which defines theclosed subsets <strong>and</strong> limits as above. Let σ = {(P n ,F n )}⊂V (⊂ ˆV ∅ × ˇV ∅ ),then:{when P ≠ ∅, P∈ ˆL({Pn })(P, F) ∈ L(σ) ⇔when F ≠ ∅, F∈ Ľ({F . (3.3)n})This topology is sequential, but it may be non-Hausdorff (Fig. 3).4. Plausibility of the boundary4.1. Admissible redefinitions of the c-completionDue to the long list of redefinitions of the c-boundary, the authors discussedextensively in [11] the plausible conditions to be fulfilled by such a definition
104 J. L. Flores, J. Herrera <strong>and</strong> M. SánchezFx n(P, F)(P ′ ,F)P P ′Figure 3. In L 2 \{(t, 0) : t ≤ 0}, P ∼ S F <strong>and</strong> P ′ ∼ S F .The sequence {(I − (x n ),I + (x n ))} converges to both, (P, F)<strong>and</strong> (P ′ ,F).in order to obtain a boundary which would be deemed satisfactory. We sketchsuch conditions <strong>and</strong> refer to the original article for more precise discussions.4.1.1. Admissibility as a point set. The minimum conditions are:• Ambient: the causal completion V must be a subset of ˆV ∅ × ˇV ∅ ,<strong>and</strong>Vmust be recovered from the pairs (I + (p),I − (p)), p ∈ V .This states just a general framework which allows many differentviewpoints. The possibility to consider a bigger ambient space (theproduct of the set of all the past sets by the set of all the future ones)was discussed, <strong>and</strong> disregarded, in [9].• Completeness: every TIP <strong>and</strong> TIF in V is the component of some pairin the boundary of the completion.This collects the main motivation of the c-boundary: every inextensibletimelike curve must determine a boundary point.• S-relation: if(P, F) ∈ V then P <strong>and</strong> F must be S-related.This condition generalizes naturally what happens in the spacetime.One could weaken it by relaxing the definition of S-relation whensome component is the empty set. About this, it is worth pointing outthat the S-relation appears naturally under a minimality assumption forthe c-boundary, among all the possible definitions satisfying that anyfuture or past-directed timelike curve has some endpoint [9]. If one imposedthe same condition for causal curves (not only timelike ones) then
Causal Boundaries 105more pairings would be admitted. However, under very simple hypotheses,the existence of endpoints for timelike curves ensures the existencefor causal curves too [11, Th. 3.35].Among all the completions which satisfy these hypotheses, our choice (i.e.take all the S-related pairs) yields the biggest boundary. This choice is madebecause there exists no unique minimal choice; nevertheless, our choice isunique.4.1.2. Admissibility as a chronological set. Let V be an admissible completionas a point set. A chronology ˜≪ on V is called:• an extended chronology ifp ∈ P ⇒ (I − (p),I + (p)) ˜≪ (P, F)q ∈ F ⇒ (P, F) ˜≪ (I − (q),I + (q))for all p, q ∈ V <strong>and</strong> (P, F) ∈ V ;• an admissible chronology if I ± ((P, F)) ⊂ V computed with ˜≪ satisfies:I − ((P, F)) ∩ V = P <strong>and</strong> I + ((P, F)) ∩ V = F ∀ (P, F) ∈ V.With these definitions, it is proved that our choice ≪ for the chronology ofV is the minimum extended chronology, as well as the unique admissible onecompatible with a minimally well-behaved topology.4.1.3. Admissibility as a topological space. In general the chronological futures<strong>and</strong> pasts of a spacetime generate the Alex<strong>and</strong>rov topology on the manifold,<strong>and</strong> this topology agrees with the manifold topology on strongly causalspacetimes. However, it is easy to realize that, for the completion, the analogoustopology, called the coarsely extended Alex<strong>and</strong>rov topology (CEAT), isnot enough (see [11, Figs. 1 (A), (B)]). Now, the conditions for the admissibilityof a topology τ on V are:• Compatibility with Alex<strong>and</strong>rov: τ is finer than CEAT. This only meansthat the chronological future <strong>and</strong> past of any (P, F) ∈ V must be open.• Compatibility of the limits with the empty set: if{(P n ,F n )}→(P, ∅)(resp. (∅,F)) <strong>and</strong> P ⊂ P ′ ⊂ LI({P n }) (resp. F ⊂ F ′ ⊂ LI({F n })) forsome (P ′ ,F ′ ) ∈ V ,then(P ′ ,F ′ )=(P, ∅) (resp. (P ′ ,F ′ )=(∅,F)). Thisproperty is natural; cf. also the extensive discussion in [11].• Minimality: among all the topologies satisfying the previous two conditions,ordered by the relation “is finer than”, τ is a minimal element.This requirement avoids the possibility to introduce arbitrarily new opensubsets—for example, the discrete topology would not fulfill it.Our choice of the chronological topology in (3.3) is justified because it isadmissible at least when L is of first order (i.e., if {(P n ,F n )}→(P, F) then(P, F) ∈ L(P n ,F n ), which happens in all the known cases), <strong>and</strong> is then uniquein very general circumstances [11, Th. 3.22]:Theorem 4.1. The chronological topology is the unique admissible sequentialtopology when L is of first order.
106 J. L. Flores, J. Herrera <strong>and</strong> M. Sánchez4.1.4. Properties a posteriori. As a justification a posteriori of our choice ofc-completion, recall that none of the previous choices in the literature satisfiedthe following simple properties simultaneously.Theorem 4.2. Under our definition of c-completion V , one has:(a) (consistence with the original motivation:) each future (resp. past) timelikecurve γ converges to some pair (P, F) with P = I − (γ) (resp.F = I + (γ)),(b) the c-boundary ∂V isaclosedsubsetofthecompletionV ,(c) V is T 1 ,(d) V is a sequential space,(e) I ± ((P, F)) (computed with ≪) isopeninV .In the remainder of the present article, we will study other propertieswhich also justify a posteriori our definition.4.2. Relation to the conformal boundaryConsider an envelopment or open conformal embedding i : V ↩→ V 0 ,asexplainedin Section 2.1. Let us see under which conditions the conformal completionV i := i(V ) <strong>and</strong> boundary ∂ i V can be identified with the causal ones.4.2.1. Focusing the problem. First we have to define a chronological relationin V i . There are two natural options, which we will denote p ≪ i q <strong>and</strong>p ≪ S i q, forp, q ∈ V i . In the first option, ≪ i , one assumes that there existssome continuous curve γ :[a, b] → V i with γ(a) =p, γ(b) =q <strong>and</strong>suchthatγ| (a,b) is future-directed smooth timelike <strong>and</strong> contained in i(V ). In the secondoption, ≪ S i , one assumes additionally that γ is also smooth <strong>and</strong> timelike attheendpointsasacurveinV 0 (recall that ∂ i V may be non-smooth). Thesecond option is stronger than (<strong>and</strong> non-equivalent to) the first one, <strong>and</strong> itis not intrinsic to V i . Therefore we will choose the first one, ≪ i .Notice that in order to relate the conformal <strong>and</strong> causal boundaries, onehas to focus only on the accessible (conformal) boundary ∂i ∗ V , that is, on thosepoints of ∂ i V which are endpoints of some timelike curve or, more precisely,which are ≪ i -chronologically related with some point in i(V ). Accordingly,one focuses on the natural accessible completion V ∗ i . The following is easy toprove:Proposition 4.3. If V i is an embedded manifold with C 1 boundary then allthe points of ∂ i V areaccessible(i.e.∂ ∗ i V = ∂ iV ).However, a point like the spacelike infinity i 0 in the st<strong>and</strong>ard conformalembedding of Lorentz-Minkowski space into the Einstein Static UniverseL n ↩→ ESU n is not accessible. Obviously, such a point cannot correspond toany point of the c-boundary.Analogously, we will require the envelopment i to be chronologicallycomplete, that is: any inextensible (future- or past-directed) timelike curvein V has an endpoint in the conformal boundary. Obviously, otherwise thecorresponding point in the causal boundary would not correspond to any
Causal Boundaries 107point of the conformal one. If, for example, V ∗ i is compact, this property isfulfilled. It is also fulfilled by L n ↩→ ESU n in spite of the fact that V ∗ i is notcompact in this case (as i 0 has been removed).Summing up, if ∂ i V ∗ is chronologically complete, then the natural projectionsˆπ : ˆ∂V → ∂i ∗ V, P = I − (γ) ↦→ the limit of γˇπ : ˇ∂V → ∂i ∗ V, F = I + (ρ) ↦→ the limit of ρare well-defined. We say that the causal <strong>and</strong> (accessible) conformal completionsare identifiable, <strong>and</strong>writeV ≡ V ∗ i , when the map{π : ∂V → ∂i ∗ ˆπ(P ) if P ≠ ∅V, π((P, F)) =ˇπ(F ) if F ≠ ∅satisfies the following conditions: (i) it is well-defined (i.e., (P, F) ∈ ∂V <strong>and</strong>P ≠ ∅ ̸= F imply ˆπ(P )=ˇπ(F )); (ii) π is bijective; <strong>and</strong> (iii) its naturalextension to the completions π : V → V ∗ i is both, a homeomorphism <strong>and</strong> achronological isomorphism.4.2.2. Sufficient conditions for the identification V ≡ V ∗ i . Even though thereare simple examples where both boundaries differ, there are also simple conditionswhich ensure that they can be identified. They are studied systematicallyin [11] <strong>and</strong>, essentially, are fulfilled when the points of the conformalboundary are regularly accessible. We will not study this notion here, we willonly state some consequences.As explained above, given an envelopment there are two natural notionsof chronological relation, ≪ i <strong>and</strong> ≪ S i . Analogously, there are two naturalnotions of causal relation, ≤ i <strong>and</strong> ≤ S i ,definedasfollows.Wesayp ≤ i qif either p = q or there exists some continuous curve γ :[a, b] → V i withγ(a) =p, γ(b) =q such that γ is future-directed <strong>and</strong> continuously causalwhen regarded as a curve in V 0 (this means that t 1
108 J. L. Flores, J. Herrera <strong>and</strong> M. Sánchezx nx 2x 1(0, 0)Figure 4. M C <strong>and</strong> M G are equal as point sets, <strong>and</strong> theyinclude (0, 0) as a boundary point. However, their topologiesdiffer: M C is not locally compact, since sequences like {x n }violate the compactness of the balls centered at (0, 0).Corollary 4.5. Let V be a spacetime which admits a chronologically completeenvelopment such that V i is a C 1 manifold with boundary. Then V is globallyhyperbolic iff ∂ i V does not have any timelike point.5. Static spacetimes <strong>and</strong> boundaries on a Riemannian manifold5.1. Classical Cauchy <strong>and</strong> Gromov boundaries on a Riemannian manifoldAny (connected, positive definite) Riemannian manifold (M,g), regarded asa metric space with distance d, admits the classical Cauchy completion M C(say, by using classes of Cauchy sequences). Recall that such a completion isnot a compactification <strong>and</strong>, even more, that ∂ C M may be non-locally compact(see, for example, Fig. 4).Gromov developed a compactification M G for any complete Riemannianmanifold. The idea is as follows. Let L 1 (M,d) bethespaceofalltheLipschitz functions on (M,g) with Lipschitz constant equal to 1, <strong>and</strong> endowit with the topology of pointwise convergence. Consider the equivalence relationin L 1 (M,d) which relates any two functions which only differ by anadditive constant, <strong>and</strong> let L 1 (M,d)/R be the corresponding quotient topologicalspace. Each point x ∈ M determines the function d x ∈L 1 (M,d) definedas d x (y) =d(x, y) for all y ∈ M. Theclass[d x ] ∈L 1 (M,d)/R characterizesunivocally x. Moreover, the mapM↩→L 1 (M,d)/R, x ↦→ [−d x ]is a topological embedding, so we can regard M as a subset of L 1 (M,d)/R.Then Gromov’s completion M G is just the closure of M in L 1 (M,d)/R.
Causal Boundaries 109(0, 1)x nx 2x 1(0, 0)Figure 5. ∂ C M includes only (0, 0) <strong>and</strong> (0, 1), but ∂ G Mincludes the “segment” connecting both points. Observe thatno point in that segment (but (0, 0) <strong>and</strong> (0, 1)) is an endpointof a curve in M.Gromov’s completion can be extended also to any (not necessarily complete)Riemannian manifold, <strong>and</strong> one has again a natural inclusion of theCauchy completion M C into M G . However, this inclusion is only continuousin general (Fig. 4). In fact, the inclusion is an embedding iff M C is locallycompact.Remark 5.1. M G satisfies good topological properties, as being second countable,Hausdorff <strong>and</strong> compact. However, the following possibilities may occur:(1) Even though all the points in Gromov’s boundary ∂ G M are limits ofsequences in the manifold M, some of them may not be the limit of anycurve (see Fig. 5).(2) Define the Cauchy-Gromov boundary ∂ CG M (⊂ ∂ G M) as the set ofboundary points which are limits of some bounded sequence σ ⊂ M(i.e., σ is included in some ball), <strong>and</strong> the proper Gromov boundary as∂ G M = ∂ G M \ ∂ CG M. Clearly, the Cauchy boundary ∂ C M is includedin ∂ CG M,<strong>and</strong>this inclusion may be strict (Fig. 5)—apart from the factthat it may be not an embedding.Summing up, we have:Theorem 5.2. Let (M,g) be a (possibly incomplete) Riemannian manifold.Then Gromov’s completion M G is a compact, Hausdorff <strong>and</strong> second countablespace, <strong>and</strong> M is included in M G as an open dense subset. Moreover, Gromov’sboundary ∂ G M is the disjoint union of the Cauchy-Gromov boundary∂ CG M <strong>and</strong> the proper Gromov boundary ∂ G M. The former includes (perhapsstrictly) the Cauchy boundary ∂ C M, <strong>and</strong> the inclusion M C ↩→ M G
110 J. L. Flores, J. Herrera <strong>and</strong> M. Sánchez(i) is continuous;(ii) is an embedding iff M C is locally compact.Finally, we emphasize that Gromov’s compactification coincides withEberlein <strong>and</strong> O’Neill’s compactification [8] for Hadamard manifolds (i.e. simplyconnected complete Riemannian manifolds of non-positive curvature).The latter can be described in terms of Busemann functions associated torays. More generally, given any curve c :[α, Ω) → M with |ċ| ≤1inanyRiemannian manifold, we define the Busemann function associated to c as:b c (x) = lim (t − d(x, c(t))) ∀x ∈ M. (5.1)t→ΩOne can check that this limit always exists <strong>and</strong>, if it is ∞ at some point,then it is constantly equal to ∞, b c ≡∞.LetB(M) be the set of all finitevaluedBusemann functions. For a Hadamard manifold, B(M) can be obtainedby taking the curves c as rays, i.e. unit distance-minimizing completehalf-geodesics (with [α, Ω) = [0, ∞)). The points of Eberlein <strong>and</strong> O’Neill’sboundary can be described as the quotient B(M)/R, wheref,f ′ in B(M)are identified iff f − f ′ is a constant. Such a quotient is endowed with thecone topology, defined by using angles between rays.5.2. Static spacetimes induce a new (Riemannian) Busemann boundaryNext we consider the following product spacetime:V =(R × M,g L = −dt 2 + π ∗ g), (5.2)where g is a Riemannian metric on M with distance d, <strong>and</strong>π : R × M → Mthe natural projection. Recall that any st<strong>and</strong>ard static spacetime is conformalto such a product <strong>and</strong> thus can be expressed as in (5.2) for the study of thec-boundary.The chronological relation ≪ is easily characterized as follows:(t 0 ,x 0 ) ≪ (t 1 ,x 1 ) ⇔ d(x 0 ,x 1 )
Causal Boundaries 111to Busemann functions b c with converging c (thus Ω < ∞), <strong>and</strong> the TIPswith non-converging c.Up to now, we have regarded B(M) ∪{f ≡∞}as a point set. It iseasy to check that all the functions in B(M) are 1-Lipschitz, but we arenot going to regard B(M) as a topological subspace of L 1 (M,d). On thecontrary, as ˆV was endowed with a topology (recall (3.2)), we can definea topology on B(M) ∪{f ≡∞}so that the identification (5.5) becomesa homeomorphism (of course, this topology can be also defined directly onB(M) ∪{f ≡∞}, with no mention of the spacetime, see 1 [12, Sect. 5.2.2]).Now, we can consider also the quotient topological space M B = B(M)/Robtained by identifying Busemann functions up to a constant. Again M canbe regarded as a topological subspace of M B .So,M B is called the Busemanncompletion of M <strong>and</strong> ∂ B M := M B \M the Busemann boundary.Theorem 5.3. Let M B be the Busemann completion of a Riemannian manifold(M,g). Then:(1) M B is sequentially compact, <strong>and</strong> M is naturally embedded as an opendense subset in M B .(2) All the points in ∂ B M can be reached as limits of curves in M.(3) M C is naturally included in M B , <strong>and</strong> the inclusion is continuous (theCauchy topology on M C is finer).(4) M B is the disjoint union of M, the Cauchy boundary ∂ C M <strong>and</strong> theasymptotic Busemann boundary ∂ B M,where:– M C = M ∪ ∂ C M corresponds to the subset of B(M) constructedfrom curves c with Ω < ∞ (identified with PIPs or TIPs in ˆVwhichareS-relatedtosomeTIF).– ∂ B M corresponds to the subset of B(M) constructed from curvesc with Ω=∞ (unpaired TIPs different from P = M).(5) M B is T 1 , <strong>and</strong> any non-T 2 -related points must lie in ∂ B M.(6) M B ↩→ M G in a natural way, <strong>and</strong> the inverse of the inclusion is continuous(the topology on M G is finer).(7) M B is homeomorphic to M G iff M B is Hausdorff. In particular, M Bcoincides with Eberlein <strong>and</strong> O’Neill’s compactification for Hadamardmanifolds.Remark 5.4. The possible non-Hausdorff character of ∂ B M, which alerts onthe discrepancy between M B <strong>and</strong> M G (by the assertion (7)), also points outa possible non-nice behavior of M G . Recall that neither property (2) nor ananalog to property (4) hold for M G in general (the latter, as well as theCauchy-Gromov boundary, may include strictly ∂ C M), see Remark 5.1.In fact, the discrepancy comes from the fact that, in some sense, M Bis a compactification of M by (finite or asymptotic) directions, <strong>and</strong> M G is amore analytical one.1 Therefore the Busemann boundary will be extensible to much more general spaces. Inparticular, it can be extended directly to reversible Finsler metrics. However, the Finslercase (including the non-reversible one) will be considered later.
112 J. L. Flores, J. Herrera <strong>and</strong> M. Sánchez5.3. Structure of the c-boundary for static spacetimesRecall that, in a static spacetime, the map (t, x) ↦→ (−t, x) isanisometry.So, the behavior of the past causal completion ˇV is completely analogous tothe one studied for the future causal completion, <strong>and</strong> an identification as in(5.5) holds for ˇV . Moreover, the S-relation, which defines the pairings in V(see above (3.1)), can be computed easily. In fact, the unique pair (P, F) withP ≠ ∅ ≠ F appears when both, P <strong>and</strong> F , regarded as elements of B(M),project on points of the Cauchy boundary ∂ C M ⊂ M B .Remark 5.5. We say that a causal completion V is simple as a point set if anynon-empty component of each pair (P, F) ∈ V determines univocally the pair(i.e. (P, F), (P, F ′ ) ∈ V with P ≠ ∅ implies F = F ′ ,<strong>and</strong>(P, F), (P ′ ,F) ∈ Vwith F ≠ ∅ implies P = P ′ ). In this case, <strong>and</strong> when L is of first order, wesay that V is simple if, for any sequence {(P n ,F n )}⊂V , the convergenceof the first component {P n } or the second one {F n } to a (non-empty) IPor IF implies the convergence of the sequence itself. Then the boundary ofany static spacetime is always simple as a point set. If the Cauchy boundary∂ C M is locally compact, then one can prove that it is also simple.Summing up (Fig. 6; see also [12, Cor. 6.28] for a more precise statement<strong>and</strong> definitions):Theorem 5.6. Let V = R × M be a st<strong>and</strong>ard static spacetime. If M C islocally compact <strong>and</strong> M B Hausdorff, then the c-boundary ∂V coincides withthe quotient topological space ( ˆ∂V ∪ ˇ∂V )/∼ S ,where ˆ∂V <strong>and</strong> ˇ∂V have thestructures of cones with base ∂ B M <strong>and</strong> apexes i + , i − , respectively.As a consequence, each point of ∂ B M\∂ C M yields two lines in ∂V(which are horismotic in a natural sense), one starting at i − <strong>and</strong> the otherone at i + , <strong>and</strong> each point in ∂ C M yields a single timelike line from i − to i + .6. Stationary spacetimes <strong>and</strong> boundaries of a Finsler manifold6.1. Cauchy <strong>and</strong> Gromov boundaries of a Finsler manifold6.1.1. Finsler elements. Recall that, essentially, a Finsler metric F on a manifoldM assigns smoothly a positively homogeneous tangent space norm to eachp ∈ M, where positive homogeneity means that the equality F (λv) =|λ|F (v)for v ∈ TM, λ ∈ R (of the usual norms) is only ensured now when λ>0. So,givensuchanF , one can define the reverse Finsler metric: F rev (v) :=F (−v).Any Finsler metric induces a map d : M × M → R, whered(x, y) is the infimumof the lengths l(c) of the curves c :[a, b] → M which start at x <strong>and</strong> endat y (by definition, l(c) = ∫ bF (ċ(s))ds). Such a d is a generalized distance,athat is, it satisfies all the axioms of a distance except symmetry (i.e., d is aquasidistance) <strong>and</strong>, additionally, d satisfies the following condition: a sequence{x n }⊂M satisfies d(x, x n ) → 0iffd(x n ,x) → 0. So, centered at any x 0 ∈ M,one can define the forward <strong>and</strong> backward closed balls of radius r ≥ 0as:B + (x 0 ,r)={x ∈ M : d(x 0 ,x)
Causal Boundaries 113respectively. Both types of balls generate the topology of M. Recall that thebackward balls for d are the forward ones for the distance d rev , defined asd rev (x, y) =d(y, x) for all x, y. One can define also the symmetrized distanced s =(d+d rev )/2, which is a (true) distance even though it cannot be obtainedas a length space (i.e. as the infimum of lengths of connecting curves).6.1.2. Cauchy completions. For any generalized distance d one can defineCauchy sequences <strong>and</strong> completions. Namely, {x n }⊂M is (forward) Cauchyif for all ɛ>0 there exists n 0 ∈ N such that d(x n ,x m )
114 J. L. Flores, J. Herrera <strong>and</strong> M. Sánchez6.2. Stationary spacetimes induce a new (Finslerian) Busemann boundaryConsider the spacetimeV =(R × M,g L = −dt 2 + π ∗ ω ⊗ dt + dt ⊗ π ∗ ω + π ∗ g), (6.1)where ω is a 1-form <strong>and</strong> g is a Riemannian metric, both on M, <strong>and</strong>π :R × M → M the natural projection. As in the static case, the conformalinvariance of the c-boundary allows to consider expression (6.1) as the one ofany st<strong>and</strong>ard stationary spacetime, without loss of generality.The elements in (6.1) allow to construct the following Finsler metrics(of R<strong>and</strong>ers type) on M:F ± (v) = √ g(v, v)+ω(v) 2 ± ω(v) (6.2)(recall that F − is the reverse metric of F + ). It is not difficult to check that acurve γ(t) =(t, c(t)) (resp. γ(t) =(−t, c(t))), t ∈ [α, Ω) is timelike <strong>and</strong> future(resp. past) directed iff F ± (ċ) < 1. So the characterization of the chronologicalrelation in (5.3) <strong>and</strong> the IPs in (5.4) hold now with d replaced by thegeneralized distance d + associated to F + (<strong>and</strong> with the arguments of thedistances written in the same order as before, which is now essential). Thusthe IPs are expressed in terms of Finslerian Busemann functions (defined asin (5.1) but with d + ), <strong>and</strong> the future causal completion ˆV is identifiable tothe set B + (M) ∪{f ≡∞}of all the F + -Busemann functions as in (5.5).Moreover, this set is topologized to make it homeomorphic to ˆV ,<strong>and</strong>thequotient topological space M + B= B+ (M)/R is called the (forward) Busemanncompletion of (M,F + ). On the other h<strong>and</strong>, the construction can becarried out for any Finsler manifold (M,F) (whereF is not necessarily aR<strong>and</strong>ers metric as in (6.2)), just by defining the topology directly in terms ofthe Finslerian elements, with no reference to spacetimes (see [12, Sect. 5.2.2]for details). Summing up, one finds:Theorem 6.3. Let (M,F) be a Finsler manifold. The (forward) Busemanncompletion M + Bsatisfies all the properties stated in Theorem 5.3 except property(3), which must be replaced by:(3’) M + C is naturally included as a subset in M + B, <strong>and</strong> the inclusion is continuousif the natural topology generated in M + C by the d Q-backward ballsis finer than the one generated by the forward balls (see Remark 6.1).In particular, this happens if d Q is a generalized distance.Remark 6.4. Obviously, given a Finsler manifold (M,F), one can define alsoa Busemann completion M − Bby using the reverse metric. For the R<strong>and</strong>ersmetrics F ± in (6.2), the differences between both completions reflect asymmetriesbetween the future <strong>and</strong> past causal boundaries, which did not appearin the static case. Moreover, the S-relation (defined for spacetimes) suggestsa general relation between some points in M + C with M − C (<strong>and</strong> M + B with M − B )[12, Sects. 3.4, 6.3].
Causal Boundaries 1156.3. Structure of the c-boundary for stationary spacetimesWith all the previously obtained elements at h<strong>and</strong>, one can give a precisedescription of the c-boundary for any stationary spacetime. However, a notablecomplication appears with respect to the static case. In that case, theS-relation was trivial, <strong>and</strong> the c-boundary was always simple as a point set.In the stationary case, however, the S-relation may be highly non-trivial. Infact, it is worth emphasizing that all the classical difficulties for the definitionof the S-relation <strong>and</strong>, then, for obtaining a consistent c-boundary, canbe found in st<strong>and</strong>ard stationary spacetimes. So we give here the c-boundaryof a stationary spacetime under some additional natural hypotheses, whichimply that the c-boundary is simple, according to Remark 5.5 (see Fig. 6).Theorem 6.5. Let V be a st<strong>and</strong>ard stationary spacetime (as in (6.1)), <strong>and</strong>assume that the associated Finsler manifolds (M,F ± ) satisfy: (a) the quasidistancesd ± Q are generalized distances, (b) the Cauchy completion M C s for thesymmetrized distance is locally compact, <strong>and</strong> (c) the Busemann completionsM ± Bare Hausdorff.Then ∂V coincides with the quotient topological space ( ˆ∂V ∪ ˇ∂V )/∼ S ,where ˆ∂V <strong>and</strong> ˇ∂V have the structures of cones with bases ∂ + B M, ∂− B M <strong>and</strong>apexes i + , i − , respectively.As a consequence, the description of the c-boundary for the static casein Theorem 5.6 holds with the following differences: (a) ∂ B M\∂ C M must bereplaced by ∂ + B M\∂s C M (resp. ∂− B M\∂s CM) for the horismotic lines startingat i + (resp. i − ), <strong>and</strong> (b) the timelike lines correspond to points in ∂C s M(instead of ∂ C M).In the general (non-simple) case, the points in ∂ + C M\∂s CM may yieldlocally horismotic curves starting at i + , which may be eventually identifiedwith the corresponding ones starting at i − for ∂ − C M\∂s CM. We refer to [12]for a full description.7. Boundary of pp-wave type spacetimesThe c-boundary of pp-wave type spacetimes was studied systematically in [13]<strong>and</strong> was explained further in the context of the AdS/CFT correspondence in[31]. Here we would just like to point out that the computation of this c-boundary is related to the critical curves associated to a Lagrangian J ona Riemannian manifold (M,g). So, in the line of previous cases, one couldtry to assign a boundary to this situation, even dropping the reference to thespacetime.Consider the pp-wave type spacetimes:V =(M 0 × R 2 ,g L = π ∗ 0g 0 − F (x, u) du 2 − 2 du dv),where (M 0 ,g 0 ) is a Riemannian manifold, π 0 : V → M 0 is the natural projection,(v, u) are the natural coordinates of R 2 <strong>and</strong> F : M 0 ×R → R is a smoothfunction. These spacetimes include all the pp-waves (i.e. (M 0 ,g 0 )=R n0 ).
116 J. L. Flores, J. Herrera <strong>and</strong> M. Sánchezi +Future cone∂ + B M \ ∂s C M(P, F) :P, F from ∂ s C M∂ − B M \ ∂s C Mi −Past coneFigure 6. c-boundary for a st<strong>and</strong>ard stationary spacetimeunder the assumptions of Theorem 6.5. In the static case, onehas the simplifications ∂ ± B M ≡ ∂ BM <strong>and</strong> ∂ + C M = ∂− C M =∂C s M ≡ ∂ CM.Among pp-waves, the plane waves are characterized by F being quadratic inx ∈ R n 0.In order to compute the TIPs <strong>and</strong> TIFs, one must take into accountthat the chronological relation z 0 ≪ z 1 ,forz i =(x i ,v i ,u i ) ∈ V , <strong>and</strong> thusI + (z 0 )<strong>and</strong>I − (z 0 ), are controlled by the infimum of the “arrival functional”:∫ |Δu|Ju Δu0: C→R, Ju Δu0(y) = 1 (|ẏ(s)| 2 − F (y(s),u ν (s)))ds,2 0where C is the space of curves in M 0 joining x 0 with x 1 ,Δu = u 1 − u 0 ,u ν (s) =u 0 + ν(Δu)s, <strong>and</strong>ν =+1forI + (z 0 )(ν = −1 forI − (z 0 )). Thecareful study of this situation yields a precise computation of the boundary.A goal in this computation is to characterize when the TIPs “collapse” to a1-dimensional future boundary, in agreement with the Berenstein & Nastase<strong>and</strong> Marolf & Ross results. That was achieved by studying a critical Sturm-Liouville problem. This depends on abstract technical conditions for Ju Δu0,which turn out to depend on certain asymptotic properties of F (see Table1).We would like to emphasize that Ju Δu0is a Lagrangian which depends onthe “classical time” u. This opens the possibility of further study, as remarkedabove.
Causal Boundaries 117Qualitative F Boundary ∂V ExamplesF superquad.−F at most quad.no boundarypp-waves yieldingsine-Gordon string<strong>and</strong> related onesλ-asymp. quad.λ>1/21-dimensional,lightlikeplane waveswith some eigenvalueμ 1 ≥ λ 2 /(1 + u 2 )for |u| largeλ-asymp. quad.λ ≤ 1/2criticalpp-wave withF (x, u) =λ 2 x 2 /(1 + u) 2(for u>0)subquadraticno identif.in ˆ∂V, ˇ∂Vexpectedhigher dim.(1) L n <strong>and</strong> statictype Mp-waves(2) plane waves with−F quadraticTable 1. Rough properties of the c-boundary for a pp-wavetype spacetime depending on the qualitative behavior of F .References[1] A. Ashtekar, R. O. Hansen, A unified treatment of null <strong>and</strong> spatial infinity ingeneral relativity. I. Universal structure, asymptotic symmetries, <strong>and</strong> conservedquantities at spatial infinity, J. Math. Phys. 19 (1978), no. 7, 1542–1566.[2] D. Berenstein, J. M. Maldacena <strong>and</strong> H. Nastase, Strings in flat space <strong>and</strong> ppwaves from N = 4 super Yang Mills, J. High Energy Phys. 0204 (2002) 013, 30pp.[3] D. Berenstein <strong>and</strong> H. Nastase, On lightcone string field theory from super Yang-Mills <strong>and</strong> holography. Available at arXiv:hep-th/0205048.[4] M. Blau, J. Figueroa-O’Farrill, C. Hull, G. Papadopoulos, Penrose limits <strong>and</strong>maximal supersymmetry, Class. Quant. Grav. 19 (2002) L87–L95.[5] R. Budic, R. K. Sachs, Causal boundaries for general relativistic spacetimes,J. Math. Phys. 15 (1974) 1302–1309.
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Causal Boundaries 119[26] E. Minguzzi, M. Sánchez, The causal hierarchy of spacetimes, in: Recent developmentsin pseudo-Riemannian geometry (2008) 359–418. ESI Lect. in Math.Phys., European <strong>Mathematical</strong> Society Publishing House. Available at grqc/0609119.[27] O. Müller, M. Sánchez, Lorentzian manifolds isometrically embeddable in L N ,Trans. Amer. Math. Soc. 363 (2011) 5367–5379.[28] B. O’Neill, Semi-Riemannian geometry. With applications to relativity, AcademicPress, 1983.[29] S. Scott, P. Szekeres, The abstract boundary—a new approach to singularitiesof manifolds, J. Geom. Phys. 13 (1994), no. 3, 223–253.[30] I. Rácz, Causal boundary of space-times, Phys. Rev. D 36 (1987) 1673–1675.Causal boundary for stably causal space-times, Gen. Relat. Grav. 20 (1988)893–904.[31] M. Sánchez, Causal boundaries <strong>and</strong> holography on wave type spacetimes, NonlinearAnal., 71 (2009) e1744–e1764.[32] B. G. Schmidt, A new definition of singular points in general relativity. Gen.Relat. Grav. 1 (1970/71), no. 3, 269–280.[33] S. Scott, P. Szekeres, The abstract boundary–a new approach to singularitiesof manifolds, J. Geom. Phys. 13 (1994) 223–253.[34] L. B. Szabados, Causal boundary for strongly causal spaces, Class. Quant.Grav. 5 (1988) 121–34. Causal boundary for strongly causal spacetimes: II,Class. Quant. Grav. 6 (1989) 77–91.José Luis Flores <strong>and</strong> Jónatan HerreraDepartamento de Álgebra, Geometría y TopologíaFac. Ciencias, Campus Teatinos s/n,Universidad de MálagaE 29071 MálagaSpaine-mail: floresj@agt.cie.uma.esjherrera@uma.esMiguel SánchezDepartamento de Geometría y TopologíaFac. Ciencias, Campus de Fuentenueva s/n,Universidad de GranadaE 18071 GranadaSpaine-mail: sanchezm@ugr.es
Some <strong>Mathematical</strong> Aspects of the HawkingEffect for Rotating Black HolesDietrich HäfnerAbstract. The aim of this work is to give a mathematically rigorous descriptionof the Hawking effect for fermions in the setting of the collapseof a rotating charged star.Mathematics Subject Classification (2010). 35P25, 35Q75, 58J45, 83C47,83C57.Keywords. General relativity, Kerr-Newman metric, quantum field theory,Hawking effect, Dirac equation, scattering theory.1. IntroductionIt was in 1975 that S. W. Hawking published his famous paper about thecreation of particles by black holes (see [10]). Later this effect was analyzedby other authors in more detail (see e.g. [13]), <strong>and</strong> we can say that the effectwas well-understood from a physical point of view at the end of the 1970s.From a mathematical point of view, however, fundamental questions linkedto the Hawking radiation, such as scattering theory for field equations onblack hole space-times, were not addressed at that time.Scattering theory for field equations on the Schwarzschild metric hasbeen studied from a mathematical point of view since the 1980s, see e.g.[7]. In 1999 A. Bachelot [2] gave a mathematically rigorous description ofthe Hawking effect in the spherically symmetric case. The methods used byDimock, Kay <strong>and</strong> Bachelot rely in an essential way on the spherical symmetryof the problem <strong>and</strong> can’t be generalized to the rotating case.The aim of the present work is to give a mathematically precise descriptionof the Hawking effect for spin-1/2 fields in the setting of the collapse ofa rotating charged star, see [9] for a detailed exposition. We show that anobserver who is located far away from the black hole <strong>and</strong> at rest with respectto the Boyer-Lindquist coordinates observes the emergence of a thermal statewhen his proper time t goes to infinity. In the proof we use the results of [8]as well as their generalizations to the Kerr-Newman case in [4].F. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_ 7,© Springer Basel AG 2012121
122 D. HäfnerLet us give an idea of the theorem describing the effect. Let r ∗ be theRegge-Wheeler coordinate. We suppose that the boundary of the star is describedby r ∗ = z(t, θ). The space-time is then given byM col = ⋃ tΣ colt , Σ colt = { (t, r ∗ ,ω) ∈ R t × R r∗ × S 2 ; r ∗ ≥ z(t, θ) } .The typical asymptotic behavior of z(t, θ) is(A(θ) > 0, κ + > 0):z(t, θ) =−t − A(θ)e −2κ +t + B(θ)+O(e −4κ +t ), t →∞.Here κ + is the surface gravity of the outer horizon. LetH t = L 2( (Σ colt , dVol); C 4) .The Dirac equation can be written as∂ t Ψ=i /D t Ψ + boundary condition. (1.1)We will put an MIT boundary condition on the surface of the star. Thiscondition makes the boundary totally reflecting, we refer to [9, Section 4.5]for details. The evolution of the Dirac field is then described by an isometricpropagator U(s, t) :H s →H t . The Dirac equation on the whole exteriorKerr-Newman space-time M BH will be written as∂ t Ψ=i /DΨ.Here /D is a selfadjoint operator on H = L 2 ((R r∗ × S 2 ,dr ∗ dω); C 4 ). There existsan asymptotic velocity operator P ± such that for all continuous functionsJ with lim |x|→∞ J(x) =0wehave(J(P ± ) = s-limt→±∞ e−it /D r∗)J e it /D .tLet U col (M col ) (resp. U BH (M BH )) be the algebras of observables outside thecollapsing body (resp. on the space-time describing the eternal black hole)generated by Ψ ∗ col (Φ 1)Ψ col (Φ 2 ) (resp. Ψ ∗ BH (Φ 1)Ψ BH (Φ 2 )). Here Ψ col (Φ) (resp.Ψ BH (Φ)) are the quantum spin fields on M col (resp. M BH ). Let ω col be avacuum state on U col (M col ); ω vac a vacuum state on U BH (M BH )<strong>and</strong>ω η,σHawbe a KMS-state on U BH (M BH ) with inverse temperature σ>0<strong>and</strong>chemicalpotential μ = e ση (see Section 5 for details). For a function Φ ∈ C0 ∞ (M BH )we define:Φ T (t, r ∗ ,ω)=Φ(t − T,r ∗ ,ω).The theorem about the Hawking effect is the following:Theorem 1.1 (Hawking effect). LetΦ j ∈ ( C0 ∞ (M col ) ) 4, j =1, 2.Then we havelim ω (col Ψ∗col (Φ T 1 )Ψ col (Φ T 2 ) ) = ω η,σ (Haw Ψ∗BH (1 R +(P − )Φ 1 )Ψ BH (1 R +(P − )Φ 2 ) )T →∞(+ ω vac Ψ∗BH (1 R −(P − )Φ 1 )Ψ BH (1 R −(P − )Φ 2 ) ) ,(1.2)
Hawking Effect for Rotating Black Holes 123T Haw =1/σ = κ + /2π, μ = e ση , η = qQr +r+ 2 + a + aD ϕ2 r+ 2 + a . 2Here q is the charge of the field, Q the charge of the black hole, a√is the angular momentum per unit mass of the black hole, r + = M +M2− (a 2 + Q 2 ) defines the outer event horizon, <strong>and</strong> κ + is the surfacegravity of this horizon. The interpretation of (1.2) is the following. We startwith a vacuum state which we evolve in the proper time of an observer atrest with respect to the Boyer-Lindquist coordinates. The limit as the propertime of this observer goes to infinity is a thermal state coming from the eventhorizon in formation <strong>and</strong> a vacuum state coming from infinity as expressedon the R.H.S. of (1.2). The Hawking effect comes from an infinite Dopplereffect <strong>and</strong> the mixing of positive <strong>and</strong> negative frequencies. To explain thisa little bit more, we describe the analytic problem behind the effect. Letf(r ∗ ,ω) ∈ C0∞ (R × S 2 ). The key result about the Hawking effect is:∥lim ∥1 [0,∞) ( /D 0 )U(0,T)f ∥ 2 = 〈 10 R +(P − )f, μe σ /D (1 + μe σ /D ) −1 1 R +(P − )f 〉T →∞+ ∥ ∥1 [0,∞) ( /D)1 R −(P − )f ∥ ∥ 2 ,(1.3)where μ, η, σ are as in the above theorem. Here 〈., .〉 <strong>and</strong> ‖.‖ (resp. ‖.‖ 0 )are the st<strong>and</strong>ard inner product <strong>and</strong> norm on H (resp. H 0 ). Equation (1.3)implies (1.2).The term on the L.H.S. comes from the vacuum state we consider. Wehave to project onto the positive frequency solutions (see Section 5 for details).Note that in (1.3) we consider the time-reversed evolution. This comesfrom the quantization procedure. When time becomes large, the solution hitsthe surface of the star at a point closer <strong>and</strong> closer to the future event horizon.Figure 1 shows the situation for an asymptotic comparison dynamics,which satisfies Huygens’ principle. For this asymptotic comparison dynamicsthe support of the solution concentrates more <strong>and</strong> more when time becomeslarge, which means that the frequency increases. The consequence of thechange in frequency is that the system does not stay in the vacuum state.2. The analytic problemLet us consider a model where the eternal black hole is described by a staticspace-time (although the Kerr-Newman space-time is not even stationary,the problem will be essentially reduced to this kind of situation). Then theproblem can be described as follows. Consider a Riemannian manifold Σ 0 withone asymptotically euclidean end <strong>and</strong> a boundary. The boundary will movewhen t becomes large asymptotically with the speed of light. The manifoldat time t is denoted Σ t . The ”limit” manifold Σ is a manifold with two ends,one asymptotically euclidean <strong>and</strong> the other asymptotically hyperbolic (seeFigure 2). The problem consists in evaluating the limit∥lim ∥1 [0,∞) ( /D 0 )U(0,T)f ∥ 0,T →∞
124 D. HäfnerFigure 1. The collapse of the starwhere U(0,T) is the isometric propagator for the Dirac equation on the manifoldwith moving boundary <strong>and</strong> suitable boundary conditions <strong>and</strong> /D 0 is theDirac Hamiltonian at time t = 0. It is worth noting that the underlyingscattering theory is not the scattering theory for the problem with movingboundary but the scattering theory on the ”limit” manifold. It is shown in[9] that the result does not depend on the chiral angle in the MIT boundarycondition. Note also that the boundary viewed in ⋃ t {t}×Σ t is only weaklytimelike, a problem that has been rarely considered (but see [1]).One of the problems for the description of the Hawking effect is toderive a reasonable model for the collapse of the star. We will suppose thatthe metric outside the collapsing star is always given by the Kerr-Newmanmetric. Whereas this is a genuine assumption in the rotational case, in thespherically symmetric case Birkhoff’s theorem assures that the metric outsidethe star is the Reissner-Nordström metric. We will suppose that a point on thesurface of the star will move along a curve which behaves asymptotically likea timelike geodesic with L = Q = Ẽ =0,whereL is the angular momentum,Ẽ the rotational energy <strong>and</strong> Q the Carter constant. The choice of geodesicsis justified by the fact that the collapse creates the space-time, i.e., angular
Hawking Effect for Rotating Black Holes 125momenta <strong>and</strong> rotational energy should be zero with respect to the space-time.We will need an additional asymptotic condition on the collapse. It turns outthat there is a natural coordinate system (t, ˆr, ω) associated to the collapse.In this coordinate system the surface of the star is described by ˆr =ẑ(t, θ).We need to assume the existence of a constant C such that|ẑ(t, θ)+t + C| →0, t →∞. (2.1)It can be checked that this asymptotic condition is fulfilled if we use theabove geodesics for some appropriate initial condition. We think that it ismore natural to impose a (symmetric) asymptotic condition than an initialcondition. If we would allow in (2.1) a function C(θ) rather than a constant,the problem would become more difficult. Indeed one of the problems fortreating the Hawking radiation in the rotational case is the high frequenciesof the solution. In contrast with the spherically symmetric case, the differencebetween the Dirac operator <strong>and</strong> an operator with constant coefficients is nearthe horizon always a differential operator of order one 1 . This explains that inthe high-energy regime we are interested in, the Dirac operator is not closeto a constant-coefficient operator. Our method for proving (1.3) is to usescattering arguments to reduce the problem to a problem with a constantcoefficientoperator, for which we can compute the radiation explicitly. If wedo not impose a condition of type (2.1), then in all coordinate systems thesolution has high frequencies, in the radial as well as in the angular directions.With condition (2.1) these high frequencies only occur in the radial direction.Our asymptotic comparison dynamics will differ from the real dynamics onlyby derivatives in angular directions <strong>and</strong> by potentials.Let us now give some ideas of the proof of (1.3). We want to reduce theproblem to the evaluation of a limit that can be explicitly computed. To doso, we use the asymptotic completeness results obtained in [8] <strong>and</strong> [4]. Thereexists a constant-coefficient operator /D ← such that the following limits exist:W ← ± := s-limt→±∞ e−it /D e it /D ←1 R ∓(P ←),±Ω ± ← := s-limt→±∞ e−it /D ←e it /D 1 R ∓(P ± ).Here P ←± is the asymptotic velocity operator associated to the dynamicse it /D ← . Then the R.H.S. of (1.3) equals:∥∥1 [0,∞) ( /D)1 R −(P − )f ∥ 2 + 〈 Ω − ←f, μe σ /D ←(1 + μe σ /D ←) −1 Ω − ←f 〉 .The aim is to show that the incoming part is:∥lim ∥1 [0,∞) (D ←,0 )U ← (0,T)Ω − ←f ∥ 2 = 〈 Ω − 0 ←f, μe σ /D ←(1 + μe σ /D ←) −1 Ω − ←f 〉 ,T →∞where the equality can be shown by explicit calculation. Here /D ←,t <strong>and</strong>U ← (s, t) are the asymptotic operator with boundary condition <strong>and</strong> the associatedpropagator. The outgoing part is easy to treat.1 In the spherically symmetric case we can diagonalize the operator. After diagonalizationthe difference is just a potential.
126 D. HäfnerFigure 2. The manifold at time t =0Σ 0 <strong>and</strong> the limitmanifold Σ.Note that (1.3) is of course independent of the choice of the coordinatesystem <strong>and</strong> the tetrad, i.e., both sides of (1.3) are independent of thesechoices.Theproofsofalltheresultsstatedinthisworkcanbefoundin[9].Thework is organized as follows:• In Section 3 we present the model of the collapsing star. We first analyzegeodesics in the Kerr-Newman space-time <strong>and</strong> explain how theCarter constant can be understood in terms of the Hamiltonian flow.We construct a coordinate system which is well adapted to the collapse.• In Section 4 we describe classical Dirac fields. The form of the Diracequation with an adequate choice of the Newman-Penrose tetrad isgiven. Scattering results are stated.• Dirac quantum fields are discussed in Section 5. The theorem about theHawking effect is formulated <strong>and</strong> discussed in Subsection 5.2.• In Section 6 we give the main ideas of the proof.3. The model of the collapsing starThe purpose of this section is to describe the model of the collapsing star. Wewill suppose that the metric outside the star is given by the Kerr-Newmanmetric. Geodesics are discussed in Subsection 3.2. We give a description of theCarter constant in terms of the associated Hamiltonian flow. A new position
Hawking Effect for Rotating Black Holes 127variable is introduced. In Subsection 3.3 we give the asymptotic behavior ofthe boundary of the star using this new position variable. We require that apoint on the surface behaves asymptotically like incoming timelike geodesicswith L = Q = Ẽ = 0, which are studied in Subsection 3.3.1.3.1. The Kerr-Newman metricWe give a brief description of the Kerr-Newman metric, which describes aneternal rotating charged black hole. In Boyer-Lindquist coordinates, a Kerr-Newman black hole is described by a smooth 4-dimensional Lorentzian manifoldM BH = R t × R r × Sω, 2 whose space-time metric g <strong>and</strong> electromagneticvector potential Φ a are given by:g =()1+ Q2 − 2Mrρ 2 dt 2 + 2a sin2 θ (2Mr − Q 2 )dt dϕρ 2− ρ2Δ dr2 − ρ 2 dθ 2 − σ2ρ 2 sin2 θ dϕ 2 ,ρ 2 = r 2 + a 2 cos 2 θ,(3.1)Δ=r 2 − 2Mr + a 2 + Q 2 ,σ 2 =(r 2 + a 2 ) 2 − a 2 Δsin 2 θ,Φ a dx a = − Qrρ (dt − a 2 sin2 θ dϕ).Here M is the mass of the black hole, a its angular momentum per unitmass, <strong>and</strong> Q the charge of the black hole. If Q =0,g reduces to the Kerrmetric, <strong>and</strong> if Q = a = 0 we recover the Schwarzschild metric. The expression(3.1) of the Kerr metric has two types of singularities. While the set of points{ρ 2 =0} (the equatorial ring {r =0, θ = π/2} of the {r =0} sphere) isa true curvature singularity, the spheres where Δ vanishes, called horizons,are mere coordinate singularities. We will consider in this work subextremalKerr-Newman space-times, that is, we suppose Q 2 + a 2
128 D. Häfner<strong>and</strong> the angular momentum L := −〈γ ′ ,∂ ϕ 〉. There exists a fourth constantof motion, the so-called Carter constant K (see e.g. [3]). We will also use theCarter constant Q = K−(L−aE) 2 , which has a somewhat more geometricalmeaning, but gives in general more complicated formulas. LetP := (r 2 + a 2 )E − aL, D := L − aE sin 2 θ. (3.3)Let ✷ g be the d’Alembertian associated to the Kerr-Newman metric. We willconsider the Hamiltonian flow of the principal symbol of 1 2 ✷ g <strong>and</strong>thenusethefact that a geodesic can be understood as the projection of the Hamiltonianflow on M BH . The principal symbol of 1 2 ✷ g is:LetP := 12ρ 2 ( σ2C p :=Δ τ 2 − 2a(Q2 − 2Mr)Δq ϕ τ − Δ − a2 sin 2 θΔsin 2 qϕ 2 − Δ|ξ| 2 − qθ2 θ{(t, r, θ, ϕ; τ,ξ,q θ ,q ϕ ) ; P (t, r, θ, ϕ; τ,ξ,q θ ,q ϕ )= 1 2 p }.).(3.4)Here (τ,ξ,q θ ,q ϕ ) is dual to (t, r, θ, ϕ). We have the following:(Theorem 3.1. Let x 0 =(t 0 ,r 0 ,ϕ 0 ,θ 0 ,τ 0 ,ξ 0 ,q θ0 ,q ϕ0 ) ∈C p ,<strong>and</strong>letx(s) =t(s),r(s), θ(s), ϕ(s); τ(s), ξ(s), qθ (s), q ϕ (s) ) be the associated Hamiltonianflow line. Then we have the following constants of motion:p =2P, E = τ, L = −q ϕ ,K = qθ 2 +D2sin 2 θ + pa2 cos 2 θ = P2Δ − (3.5)Δ|ξ|2 − pr 2 ,where D, P are defined in (3.3).The case L = Q = 0 is of particular interest. Let γ be a null geodesicwith energy E>0, Carter constant Q = 0, angular momentum L = 0 <strong>and</strong>given signs of ξ 0 <strong>and</strong> q θ0 . We can associate a Hamiltonian flow line using (3.5)to define the initial data τ 0 ,ξ 0 ,q θ0 ,q ϕ0 given t 0 ,r 0 ,θ 0 ,ϕ 0 . From (3.5) we inferthat ξ, q θ do not change their signs. Note that γ is either in the equatorialplane or it does not cross it. Under the above conditions ξ (resp. q θ )canbeunderstood as a function of r (resp. θ) alone. In this case let k <strong>and</strong> l suchthatdk(r)= ξ(r)dr E , l′ (θ) = q θ(θ), ˆr := k(r)+l(θ). (3.6)EIt is easy to check that (t, ˆr, ω) is a coordinate system on block I.Lemma 3.2. We have:∂ˆr= −1 along γ, (3.7)∂twhere t is the Boyer-Lindquist time.We will suppose from now on that our construction is based on incomingnull geodesics. From the above lemma follows that for given sign of q θ0 thesurfaces C c,± = {(t, r ∗ ,θ,ϕ); ±t =ˆr(r ∗ ,θ)+c} are characteristic.
Hawking Effect for Rotating Black Holes 129Remark 3.3. The variable ˆr is a Bondi-Sachs type coordinate. This coordinatesystem is discussed in some detail in [12]. As in [12], we will call the nullgeodesics with L = Q =0simple null geodesics (SNGs).3.3. The model of the collapsing starLet S 0 be the surface of the star at time t = 0. We suppose that elementsx 0 ∈ S 0 will move along curves which behave asymptotically like certainincoming timelike geodesics γ p . All these geodesics should have the sameenergy E, angular momentum L, Carter constant K (resp. Q = K−(L−aE) 2 )<strong>and</strong> “mass” p := 〈γ p,γ ′ p〉. ′ We will suppose:(A) The angular momentum L vanishes: L =0.(B) The rotational energy vanishes: Ẽ = a 2 (E 2 − p) =0.(C) The total angular momentum about the axis of symmetry vanishes:Q =0.The conditions (A)–(C) are imposed by the fact that the collapse itself createsthe space-time, so that momenta <strong>and</strong> rotational energy should be zero withrespect to the space-time.3.3.1. Timelike geodesics with L = Q = Ẽ =0. Next, we will study theabove family of geodesics. The starting point of the geodesic is denoted(0,r 0 ,θ 0 ,ϕ 0 ). Given a point in the space-time, the conditions (A)–(C) definea unique cotangent vector provided one adds the condition that the correspondingtangent vector is incoming. The choice of p is irrelevant because itjust corresponds to a normalization of the proper time.Lemma 3.4. Let γ p be a geodesic as above. Along this geodesic we have:∂θ∂t =0, ∂ϕ∂t = a(2Mr − Q2 ), (3.8)σ 2where t is the Boyer-Lindquist time.The function ∂ϕis usually called the local angular velocityσ 2of the space-time. Our next aim is to adapt our coordinate system to thecollapse of the star. The most natural way of doing this is to choose anincoming null geodesic γ with L = Q = 0 <strong>and</strong> then use the Bondi-Sachstype coordinate as in the previous subsection. In addition we want that k(r ∗ )behaves like r ∗ when r ∗ →−∞. We therefore put:∫ (√)r∗ak(r ∗ )=r ∗ + 1 −2 Δ(s)−∞ (r(s) 2 + a 2 ) − 1 ds, l(θ) =a sin θ. (3.9)2Thechoiceofthesignofl ′ is not important, the opposite sign would havebeen possible. Recall that cos θ does not change its sign along a null geodesicwith L = Q = 0. We put ˆr = k(r ∗ )+l(θ), <strong>and</strong> by Lemma 3.2 we have∂ˆr∂t= −1 alongγ.In order to describe the model of the collapsing star we have to evaluate∂ˆr∂t along γ p. Note that θ(t) =θ 0 = const along γ p .∂t = a(2Mr−Q2 )
130 D. HäfnerLemma 3.5. There exist smooth functions Â(θ, r 0) > 0, ˆB(θ, r0 ) such thatalong γ p we have uniformly in θ, r 0 ∈ [r 1 ,r 2 ] ⊂ (r + , ∞):where κ + =ˆr = −t − Â(θ, r 0)e −2κ +t + ˆB(θ, r 0 )+O(e −4κ +t ), t →∞, (3.10)r +−r −2(r 2 + +a2 )is the surface gravity of the outer horizon.3.3.2. Assumptions. We will suppose that the surface at time t = 0 is given inthe (t, ˆr, θ, ϕ) coordinate system by S 0 = { (ˆr 0 (θ 0 ),θ 0 ,ϕ 0 ); (θ 0 ,ϕ 0 ) ∈ S 2} ,where ˆr 0 (θ 0 ) is a smooth function. As ˆr 0 does not depend on ϕ 0 , we willsuppose that ẑ(t, θ 0 ,ϕ 0 ) will be independent of ϕ 0 :ẑ(t, θ 0 ,ϕ 0 )=ẑ(t, θ 0 )=ẑ(t, θ) as this is the case for ˆr(t) along timelike geodesics with L = Q =0.Wesuppose that ẑ(t, θ) satisfies the asymptotics (3.10) with ˆB(θ, r 0 ) independentof θ, see [9] for the precise assumptions. Thus the surface of the star is givenby:S = { (t, ẑ(t, θ),ω); t ∈ R, ω∈ S 2} . (3.11)The space-time of the collapsing star is given by:M col = { (t, ˆr, θ, ϕ) ; ˆr ≥ ẑ(t, θ) } .We will also note:Σ colt = { (ˆr, θ, ϕ) ; ˆr ≥ ẑ(t, θ) } , thus M col = ⋃ tΣ colt .The asymptotic form (3.10) with ˆB(θ, r 0 ) can be achieved by incoming timelikegeodesics with L = Q = Ẽ = 0, see [9, Lemma 3.5].4. Classical Dirac fieldsIn this section we will state some results on classical Dirac fields <strong>and</strong> explainin particular how to overcome the difficulty linked to the high-frequency problem.The key point is the appropriate choice of a Newman-Penrose tetrad.Let (M,g) be a general globally hyperbolic space-time. Using the Newman-Penrose formalism, the Dirac equation can be expressed as a system of partialdifferential equations with respect to a coordinate basis. This formalism isbased on the choice of a null tetrad, i.e. a set of four vector fields l a , n a , m a<strong>and</strong> ¯m a , the first two being real <strong>and</strong> future oriented, ¯m a being the complexconjugate of m a , such that all four vector fields are null <strong>and</strong> m a is orthogonalto l a <strong>and</strong> n a ,thatistosay,l a l a = n a n a = m a m a = l a m a = n a m a =0.Thetetrad is said to be normalized if in addition l a n a =1,m a ¯m a = −1. Thevectors l a <strong>and</strong> n a usually describe ”dynamic” or scattering directions, i.e.directions along which light rays may escape towards infinity (or more generallyasymptotic regions corresponding to scattering channels). The vector m atends to have, at least spatially, bounded integral curves; typically m a <strong>and</strong>¯m a generate rotations. To this Newman-Penrose tetrad is associated a spinframe. The Dirac equation is then a system of partial differential equations
Hawking Effect for Rotating Black Holes 131for the components of the spinor in this spin frame. For the Weyl equation(m = 0) we obtain:{n a ∂ a φ 0 − m a ∂ a φ 1 +(μ − γ)φ 0 +(τ − β)φ 1 =0,l a ∂ a φ 1 − ¯m a ∂ a φ 1 +(α − π)φ 0 +(ɛ − ˜ρ)φ 1 =0.The μ, γ etc. are the so-called spin coefficients, for exampleμ = − ¯m a δn a , δ = m a ∇ a .For the formulas of the spin coefficients <strong>and</strong> details about the Newman-Penrose formalism see e.g. [11].Our first result is that there exists a tetrad well-adapted to the high-frequencyproblem. Let H = L 2 ((Rˆr ×S 2 , dˆr dω); C 4 ), Γ 1 = Diag (1, −1, −1, 1).Proposition 4.1. There exists a Newman-Penrose tetrad such that the Diracequation in the Kerr-Newman space-time can be written as∂ t ψ = iHψ; H =Γ 1 Dˆr + P ω + W,where W is a real potential <strong>and</strong> P ω is a differential operator of order one withderivatives only in the angular directions. The operator H is selfadjoint withdomain D(H) ={v ∈H; Hv ∈H}.Remark 4.2.(i) l a , n a are chosen to be generators of the simple null geodesics.(ii) Note that the local velocity in ˆr direction is ±1:v =[ˆr, H] =Γ 1 .This comes from the fact that ∂ˆr = ±1 along simple null geodesics (±∂tdepending on whether the geodesic is incoming or outgoing).(iii) Whereas the above tetrad is well-adapted to the high-frequency analysis,it is not the good choice for the proof of asymptotic completeness results.See [8] for an adequate choice.(iv) ψ are the components of the spinor which is multiplied by some density.LetH ← =Γ 1 aDˆr −r+ 2 + a D 2 ϕ − qQr +r+ 2 + a , 2H + = { v =(v 1 ,v 2 ,v 3 ,v 4 ) ∈H; v 1 = v 4 =0 } ,H − = { v =(v 1 ,v 2 ,v 3 ,v 4 ) ∈H; v 2 = v 3 =0 } .The operator H ← is selfadjoint on H with domainD(H ← )= { v ∈H; H ← v ∈H } .Remark 4.3. The above operator is our comparison dynamics. Note thatthe difference between the full dynamics <strong>and</strong> the comparison dynamics isa differential operator with derivatives only in the angular directions. Thehigh frequencies will only be present in the ˆr directions; this solves the highfrequencyproblem.
132 D. HäfnerProposition 4.4. There exist selfadjoint operators P ± such that for all g ∈C(R) with lim |x|→∞ g(x) =0, we have:( ) ˆrg(P ± ) = s-limt→±∞ e−itH g e itH . (4.1)tLet P H ∓ be the projections from H to H ∓ .Theorem 4.5. ThewaveoperatorsW ← ± = s-limt→±∞ e−itH e itH ←P H ∓,Ω ± ← = s-limt→±∞ e−itH ←e itH 1 R ∓(P ± )exist.Using the above tetrad, the Dirac equation with MIT boundary conditionon the surface of the star (chiral angle ν) can be written in the followingform:⎫∂ t Ψ=iHΨ, ẑ(t, θ) < ˆr,( ∑ˆμ∈{t,ˆr,θ,ϕ} Nˆμˆγ ˆμ) Ψ ( t, ẑ(t, θ),ω ) = −ie −iνγ5 Ψ ( t, ẑ(t, θ),ω ) ⎬,⎭ (4.2)Ψ(t = s, .) =Ψ s (.).Here Nˆμ are the coordinates of the conormal, ˆγ ˆμ are some appropriate Diracmatrices <strong>and</strong> γ 5 = Diag (1, 1, −1, −1). LetH t = L 2(({ (ˆr, ω) ∈ R × S 2 ;ˆr ≥ ẑ(t, θ) } , dˆr dω ) ; C 4) .Proposition 4.6. The equation (4.2) can be solved by a unitary propagatorU(t, s) :H s →H t .5. Dirac quantum fieldsWe adopt the approach of Dirac quantum fields in the spirit of [5] <strong>and</strong> [6]. Thisapproach is explained in Section 5.1. In Section 5.2 we present the theoremabout the Hawking effect.5.1. Quantization in a globally hyperbolic space-timeFollowing J. Dimock [6] we construct the local algebra of observables in thespace-time outside the collapsing star. This construction does not dependon the choice of the representation of the CARs, or on the spin structure ofthe Dirac field, or on the choice of the hypersurface. In particular we canconsider the Fermi-Dirac-Fock representation <strong>and</strong> the following foliation ofour space-time (see Subsection 3.3):M col = ⋃ Σ colt , Σ colt = { (t, ˆr, θ, ϕ) ; ˆr ≥ ẑ(t, θ) } .t∈RWe construct the Dirac field Ψ 0 <strong>and</strong> the C ∗ -algebra U(H 0 ) in the usual way.We define the operator:S col : Φ ∈ ( C0 ∞ (M col ) ) ∫4↦→ S col Φ:= U(0,t)Φ(t)dt ∈H 0 , (5.1)R
Hawking Effect for Rotating Black Holes 133where U(0,t) is the propagator defined in Proposition 4.6. The quantum spinfield is defined by:Ψ col :Φ∈ ( C0 ∞ (M col ) ) 4↦→ Ψ col (Φ) := Ψ 0 (S col Φ) ∈L(F(H 0 )).Here F(H 0 ) is the Dirac-Fermi-Fock space associated to H 0 . For an arbitraryset O ⊂ M col , we introduce U col (O), the C ∗ -algebra generated byψcol ∗ (Φ 1)Ψ col (Φ 2 ), supp Φ j ⊂O, j =1, 2. Eventually, we have:U col (M col )=⋃U col (O).O⊂M colThen we define the fundamental state on U col (M col ) as follows:(ω col Ψ∗col (Φ 1 )Ψ col (Φ 2 ) ) (:= ω vac Ψ∗0 (S col Φ 1 )Ψ 0 (S col Φ 2 ) )= 〈 〉1 [0,∞) (H 0 )S col Φ 1 ,S col Φ 2 .Let us now consider the future black hole. The algebra U BH (M BH )<strong>and</strong>thevacuum state ω vac are constructed as before working now with the group e itHrather than the evolution system U(t, s). We also define the thermal Hawkingstate (S is analogous to S col , Ψ BH to Ψ col , <strong>and</strong> Ψ to Ψ 0 ):(Ψ∗BH (Φ 1 )Ψ BH (Φ 2 ) ) = 〈 μe σH (1 + μe σH ) −1 〉SΦ 1 ,SΦ 2ω η,σHaw=: ω η,σ (KMS Ψ ∗ (SΦ 1 )Ψ(SΦ 2 ) )withT Haw = σ −1 , μ = e ση , σ > 0,where T Haw is the Hawking temperature <strong>and</strong> μ is the chemical potential.5.2. The Hawking effectIn this subsection we formulate the main result of this work.Let Φ ∈ (C0 ∞ (M col )) 4 . We putΦ T (t, ˆr, ω) =Φ(t − T, ˆr, ω). (5.2)Theorem 5.1 (Hawking effect). LetΦ j ∈ (C0 ∞ (M col )) 4 , j =1, 2.Then we havelim ω (col Ψ∗col (Φ T 1 )Ψ col (Φ T 2 ) ) = ω η,σ (Haw Ψ∗BH (1 R +(P − )Φ 1 )Ψ BH (1 R +(P − )Φ 2 ) )T →∞(+ ω vac Ψ∗BH (1 R −(P − )Φ 1 )Ψ BH (1 R −(P − )Φ 2 ) ) ,(5.3)T Haw =1/σ = κ + /2π, μ = e ση , η = qQr +r+ 2 + + aD ϕa2 r+ 2 + .a2In the above theorem P ± is the asymptotic velocity introduced in Section4. The projections 1 R ±(P ± ) separate outgoing <strong>and</strong> incoming solutions.Remark 5.2. The result is independent of the choices of coordinate system,tetrad <strong>and</strong> chiral angle in the boundary condition.H
134 D. Häfner6. Strategy of the proofThe radiation can be explicitly computed for the asymptotic dynamics nearthe horizon. For f =(0,f 2 ,f 3 , 0) <strong>and</strong> T large, the time-reversed solution ofthe mixed problem for the asymptotic dynamics is well approximated by theso called geometric optics approximation:F T (ˆr, ω) :=1√−κ+ˆr (f 3, 0, 0, −f 2 )(T + 1κ +ln(−ˆr) − 1κ +ln Â(θ), ω ).For this approximation the radiation can be computed explicitly:Lemma 6.1. We have:∥lim ∥1 [0,∞) (H ← )F ∥ T 2 =T →∞〈f, e 2πκ +H ←(1+e 2πκ +H ←) −1f〉.The strategy of the proof is now the following:i) We decouple the problem at infinity from the problem near the horizonby cut-off functions. The problem at infinity is easy to treat.ii) We consider U(t, T )f on a characteristic hypersurface Λ. The resultingcharacteristic data is denoted g T . We will approximate Ω − ←f by a function(Ω − ←f) R with compact support <strong>and</strong> higher regularity in the angularderivatives. Let U ← (s, t) be the isometric propagator associated to theasymptotic Hamiltonian H ← with MIT boundary conditions. We alsoconsider U ← (t, T )(Ω − ←f) R on Λ. The resulting characteristic data is denotedg←,R T . The situation for the asymptotic comparison dynamics isshown in Figure 1.iii) We solve a characteristic Cauchy problem for the Dirac equation withdata g←,R T . The solution at time zero can be written in a region nearthe boundary asG(g←,R) T =U(0,T/2+c 0 )Φ(T/2+c 0 ), (6.1)where Φ is the solution of a characteristic Cauchy problem in the wholespace (without the star). The solutions of the characteristic problems forthe asymptotic Hamiltonian are written in a similar way <strong>and</strong> denotedG ← (g←,R T )<strong>and</strong>Φ ←, respectively.iv) Using the asymptotic completeness result we show that g T − g←,R T → 0when T,R →∞. By continuous dependence on the characteristic datawe see that:G(g T ) − G(g←,R) T → 0, T,R →∞.v) We writeG(g←,R)−G T ← (g←,R) T =U(0,T/2+c 0 ) ( Φ(T/2+c 0 ) − Φ ← (T/2+c 0 ) )+ ( U(0,T/2+c 0 )−U ← (0,T/2+c 0 ) ) Φ ← (T/2+c 0 ).The first term becomes small near the boundary when T becomes large.We then note that for all ɛ>0 there exists t ɛ > 0 such that∥ ( U(t ɛ ,T/2+c 0 ) − U ← (t ɛ ,T/2+c 0 ) ) Φ ← (T/2+c 0 ) ∥ ∥
Hawking Effect for Rotating Black Holes 135uniformly in T when T is large. We fix the angular momentum D ϕ =n. The function U ← (t ɛ ,T/2+c 0 )Φ ← (T/2+c 0 ) will be replaced by ageometric optics approximation Ft T ɛwhich has the following properties:supp Ft T ɛ⊂ ( )− t ɛ −|O(e −κ +T )|, −t ɛ , (6.2)Ft T ɛ⇀ 0, T →∞, (6.3)∀λ >0 Op ( χ(〈ξ〉 ≤λ〈q〉) ) Ft T ɛ→ 0, T →∞. (6.4)Here ξ <strong>and</strong> q are the dual coordinates to ˆr <strong>and</strong> θ, respectively. Op(a) isthe pseudo-differential operator associated to the symbol a (Weyl calculus).The notation χ(〈ξ〉 ≤λ〈q〉) means that the symbol is supportedin 〈ξ〉 ≤λ〈q〉.vi) We show that for λ sufficiently large possible singularities ofOp ( χ(〈ξ〉 ≥λ〈q〉) ) Ft T ɛare transported by the group e −itɛH in such a way that they always stayaway from the surface of the star.vii) The points i) tov) imply:∥lim ∥1 [0,∞) (H 0 ) j − U(0,T)f ∥ 2 = lim∥∥∥10 [0,∞) (H 0 ) U(0,t ɛ ) Ft T 2ɛT →∞T →∞where j − is a smooth cut-off which equals 1 near the boundary <strong>and</strong> 0at infinity. Let φ δ be a cut-off outside the surface of the star at time 0.If φ δ = 1 sufficiently close to the surface of the star at time 0, we see bythe previous point that(1 − φ δ )e −itɛH Ft T ɛ→ 0, T →∞. (6.5)Using (6.5) we show that (modulo a small error term):(U(0,tɛ ) − φ δ e ) −it ɛHFt T ɛ→ 0, T →∞.Therefore it remains to consider:∥∥lim ∥1 [0,∞) (H 0 )φ δ e −itɛH Ft T ∥0ɛ.T →∞viii) We show that we can replace 1 [0,∞) (H 0 )by1 [0,∞) (H). This will essentiallyallow us to commute the energy cut-off <strong>and</strong> the group. We thenshow that we can replace the energy cut-off by 1 [0,∞) (H ← ). We end upwith:∥∥lim ∥1 [0,∞) (H ← ) e −it ɛH ←Ft T ɛ, (6.6)T →∞which we have already computed.0 ,References[1] A. Bachelot, Scattering of scalar fields by spherical gravitational collapse, J.Math. Pures Appl. (9) 76, 155–210 (1997).[2] A. Bachelot, The Hawking effect, Ann. Inst. H. Poincaré Phys. Théor. 70, 41–99(1999).
136 D. Häfner[3] B. Carter, Black hole equilibrium states, Black holes/Les astres occlus (Écoled’Été Phys. Théor., Les Houches, 1972), 57–214, Gordon <strong>and</strong> Breach, NewYork, 1973.[4] T. Daudé, Sur la théorie de la diffusion pour des champs de Dirac dansdivers espaces-temps de la relativité générale, PhD thesis Université Bordeaux1 (2004).[5] J. Dimock, Algebras of local observables on a manifold, Comm.Math.Phys.77, 219–228 (1980).[6] J. Dimock, Dirac quantum fields on a manifold, Trans.Amer.Math.Soc.269,133–147 (1982).[7]J.Dimock,B.S.Kay,Classical <strong>and</strong> quantum scattering theory for linear scalarfields on the Schwarzschild metric. I, Ann.Physics175, 366–426 (1987).[8] D. Häfner <strong>and</strong> J.-P. Nicolas, Scattering of massless Dirac fields by a Kerr blackhole, Rev. Math. Phys. 16, 29–123 (2004).[9] D. Häfner, Creation of fermions by rotating charged black holes, Mémoires dela SMF 117 (2009), 158 pp.[10] S. W. Hawking, Particle creation by black holes, Comm.Math.Phys.43, 199–220 (1975).[11] R. Penrose, W. Rindler, Spinors <strong>and</strong> space-time, Vol. I, Cambridge monographson mathematical physics, Cambridge University Press 1984.[12] S. J. Fletcher <strong>and</strong> A. W. C. Lun, The Kerr spacetime in generalized Bondi-Sachs coordinates, Classical <strong>Quantum</strong> <strong>Gravity</strong> 20, 4153–4167 (2003).[13] R. Wald, On particle creation by black holes, Comm.Math.Phys.45, 9–34(1974).Dietrich HäfnerUniversité deGrenoble1Institut Fourier-UMR CNRS 5582100, rue des MathsBP 7438402 Saint Martin d’Hères CedexFrancee-mail: Dietrich.Hafner@ujf-grenoble.fr
Observables in theGeneral Boundary FormulationRobert OecklAbstract. We develop a notion of quantum observable for the generalboundary formulation of quantum theory. This notion is adapted tospacetime regions rather than to hypersurfaces <strong>and</strong> naturally fits into thetopological-quantum-field-theory-like axiomatic structure of the generalboundary formulation. We also provide a proposal for a generalized conceptof expectation value adapted to this type of observable. We showhow the st<strong>and</strong>ard notion of quantum observable arises as a special casetogether with the usual expectation values. We proceed to introducevarious quantization schemes to obtain such quantum observables includingpath integral quantization (yielding the time-ordered product),Berezin-Toeplitz (antinormal-ordered) quantization <strong>and</strong> normal-orderedquantization, <strong>and</strong> discuss some of their properties.Mathematics Subject Classification (2010). Primary 81P15; Secondary81P16, 81T70, 53D50, 81S40, 81R30.Keywords. <strong>Quantum</strong> field theory, general boundary formulation, observable,expectation value, quantization, coherent state.1. Motivation: Commutation relations <strong>and</strong> quantum fieldtheoryIn st<strong>and</strong>ard quantum theory one is used to think of observables as encoded inoperators on the Hilbert space of states. The algebra formed by these is thenseen as encoding fundamental structure of the quantum theory. Moreover,this algebra often constitutes the primary target of quantization schemesthat aim to produce a quantum theory from a classical theory. Commutationrelations in this algebra then provide a key ingredient of correspondence principlesbetween a classical theory <strong>and</strong> its quantization. We shall argue in thefollowing that while this point of view is natural in a non-relativistic setting,F. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_ 8,© Springer Basel AG 2012137
138 R. Oecklit is less compelling in a special-relativistic setting <strong>and</strong> becomes questionablein a general-relativistic setting. 1In non-relativistic quantum mechanics (certain) operators correspondto measurements that can be applied at any given time, meaning that themeasurement is performed at that time. Let us say we consider the measurementof two quantities, one associated with the operator A <strong>and</strong> anotherassociated with the operator B. Inparticular,wecanthenalsosaywhichoperator is associated with the consecutive measurement of both quantities.If we first measure A <strong>and</strong> then B the operator is the product BA, <strong>and</strong>ifwe first measure B <strong>and</strong> then A the operator is the product AB. 2 Hence,the operator product has the operational meaning of describing the temporalcomposition of measurements. One of the key features of quantum mechanicsis of course the fact that in general AB ≠ BA, i.e., a different temporalordering of measurements leads to a different outcome.The treatment of operators representing observables is different in quantumfield theory. Here, such operators are labeled with the time at which theyare applied. For example, we write φ(t, x)forafieldoperatorattimet. Hence,if we want to combine the measurement processes associated with operatorsφ(t, x) <strong>and</strong>φ(t ′ ,x ′ ) say, there is only one operationally meaningful way to doso. The operator associated with the combined process is the time-orderedproduct of the two operators, Tφ(t, x)φ(t ′ ,x ′ ). Of course, this time-orderedproduct is commutative since the information about the temporal ordering ofthe processes associated with the operators is already contained in their labels.Nevertheless, in traditional treatments of quantum field theory one firstconstructs a non-commutative algebra of field operators starting with equaltimecommutation relations. Since the concept of equal-time hypersurface isnot Poincaré-invariant, one then goes on to generalize these commutationrelations to field operators at different times. In particular, one finds that fortwo localized operators A(t, x) <strong>and</strong>B(t ′ ,x ′ ), the commutator obeys[A(t, x),B(t ′ ,x ′ )] = 0 if (t, x) <strong>and</strong>(t ′ ,x ′ ) are spacelike separated, (1)which is indeed a Poincaré-invariant condition. The time-ordered product isusually treated as a concept that is derived from the non-commutative operatorproduct. From this point of view, condition (1) serves to make sure thatit is well-defined <strong>and</strong> does not depend on the inertial frame. Nevertheless,it is the former <strong>and</strong> not the latter that has a direct operational meaning.Indeed, essentially all the predictive power of quantum field theory derivesfrom the amplitudes <strong>and</strong> the S-matrix which are defined entirely in terms1 By “general-relativistic setting” we shall underst<strong>and</strong> here a context where the metric ofspacetime is a dynamical object, but which is not necessarily limited to Einstein’s theoryof General Relativity.2 The notion of composition of measurements considered here is one where the output valuegenerated by the composite measurement can be understood as the product of outputvalues of the individual measurements, rather than one where one would obtain separateoutput values for both measurements.
Observables in the General Boundary Formulation 139of time-ordered products. On the other h<strong>and</strong>, the non-commutative operatorproduct can be recovered from the time-ordered product. Equal-timecommutation relations can be obtained as the limit[A(t, x),B(t, x ′ )] = lim TA(t + ɛ, x)B(t − ɛ,ɛ→+0 x′ ) − TA(t − ɛ, x)B(t + ɛ, x ′ ).The property (1) can then be seen as arising from the transformation propertiesof this limit <strong>and</strong> its non-equal time generalization.We conclude that in a special-relativistic setting, there are good reasonsto regard the time-ordered product of observables as more fundamental thanthe non-commutative operator product. This suggests to try to formulatethe theory of observables in terms of the former rather than the latter. In a(quantum) general-relativistic setting with no predefined background metrica condition such as (1) makes no longer sense, making the postulation of anon-commutative algebra structure for observables even more questionable.In this paper we shall consider a proposal for a concept of quantumobservable that takes these concerns into account. The wider framework inwhich we embed this is the general boundary formulation of quantum theory(GBF) [1]. We start in Section 2 with a short review of the relevant ingredientsof the GBF. In Section 3 we introduce a concept of quantum observable inan axiomatic way, provide a suitably general notion of expectation value<strong>and</strong> show how st<strong>and</strong>ard concepts of quantum observable <strong>and</strong> expectationvalues arise as special cases. In Section 4 we consider different quantizationprescriptions of classical observables that produce such quantum observables,mainly in a field-theoretic context.2. Short review of the general boundary formulation2.1. Core axiomsThe basic data of a general-boundary quantum field theory consists of twotypes: geometric objects that encode a basic structure of spacetime <strong>and</strong> algebraicobjects that encode notions of quantum states <strong>and</strong> amplitudes. Thealgebraic objects are assigned to the geometric objects in such a way thatthe core axioms of the general boundary formulation are satisfied. These maybe viewed as a special variant of the axioms of a topological quantum fieldtheory [2]. They have been elaborated, with increasing level of precision, in[3, 1, 4, 5]. In order for this article to be reasonably self-contained, we repeatbelow the version from [5].The geometric objects are of two kinds:Regions. These are (certain) oriented manifolds of dimension d (the spacetimedimension), usually with boundary.Hypersurfaces. These are (certain) oriented manifolds of dimension d − 1,here assumed without boundary. 33 The setting may be generalized to allow for hypersurfaces with boundaries along the linesof [4]. However, as the required modifications are of little relevance in the context of thepresent paper, we restrict to the simpler setting.
140 R. OecklDepending on the theory to be modeled, the manifolds may carry additionalstructure such as that of a Lorentzian metric in the case of quantum fieldtheory. For more details see the references mentioned above. The core axiomsmay be stated as follows:(T1) Associated to each hypersurface Σ is a complex separable Hilbert spaceH Σ , called the state space of Σ. We denote its inner product by 〈·, ·〉 Σ .(T1b) Associated to each hypersurface Σ is a conjugate-linear isometry ι Σ :H Σ →H¯Σ. This map is an involution in the sense that ι¯Σ ◦ ι Σ is theidentity on H Σ .(T2) Suppose the hypersurface Σ decomposes into a disjoint union of hypersurfacesΣ = Σ 1 ∪···∪Σ n . Then, there is an isometric isomorphism ofHilbert spaces τ Σ1 ,...,Σ n ;Σ : H Σ1 ˆ⊗··· ˆ⊗H Σn →H Σ . The composition ofthe maps τ associated with two consecutive decompositions is identicalto the map τ associated to the resulting decomposition.(T2b) The involution ι is compatible with the above decomposition. That is,τ ¯Σ1 ,...,¯Σ n ;¯Σ ◦ (ι Σ1 ˆ⊗··· ˆ⊗ι Σn )=ι Σ ◦ τ Σ1 ,...,Σ n ;Σ.(T4) Associated with each region M is a linear map from a dense subspaceH∂M ◦ of the state space H ∂M of its boundary ∂M (which carries theinduced orientation) to the complex numbers, ρ M : H∂M ◦ → C. Thisiscalled the amplitude map.(T3x) Let Σ be a hypersurface. The boundary ∂ ˆΣ of the associated emptyregion ˆΣ decomposes into the disjoint union ∂ ˆΣ =¯Σ ∪ Σ ′ ,whereΣ ′denotes a second copy of Σ. Then, τ¯Σ,Σ ′ ;∂ ˆΣ (H ¯Σ⊗H Σ ′) ⊆H ◦ .Moreover,∂ ˆΣρˆΣ◦ τ ¯Σ,Σ ′ ;∂ ˆΣ restricts to a bilinear pairing (·, ·) Σ : H¯Σ ×H Σ ′ → C suchthat 〈·, ·〉 Σ =(ι Σ (·), ·) Σ .(T5a) Let M 1 <strong>and</strong> M 2 be regions <strong>and</strong> M := M 1 ∪ M 2 be their disjoint union.Then ∂M = ∂M 1 ∪∂M 2 is also a disjoint union <strong>and</strong> τ ∂M1 ,∂M 2 ;∂M(H∂M ◦ 1⊗H∂M ◦ 2) ⊆H∂M ◦ . Then, for all ψ 1 ∈H∂M ◦ 1<strong>and</strong> ψ 2 ∈H∂M ◦ 2,ρ M ◦ τ ∂M1 ,∂M 2 ;∂M(ψ 1 ⊗ ψ 2 )=ρ M1 (ψ 1 )ρ M2 (ψ 2 ). (2)(T5b) Let M be a region with its boundary decomposing as a disjoint union∂M =Σ 1 ∪ Σ ∪ Σ ′ ,whereΣ ′ is a copy of Σ. Let M 1 denote the gluingof M with itself along Σ, Σ ′ <strong>and</strong> suppose that M 1 is a region. Note that∂M 1 =Σ 1 . Then, τ Σ1 ,Σ,Σ ′ ;∂M (ψ ⊗ ξ ⊗ ι Σ(ξ)) ∈H∂M ◦ for all ψ ∈H◦ ∂M 1<strong>and</strong> ξ ∈H Σ . Moreover, for any orthonormal basis {ξ i } i∈I of H Σ ,wehave for all ψ ∈H∂M ◦ 1:ρ M1 (ψ) · c(M;Σ, Σ ′ )= ∑ ρ M ◦ τ Σ1,Σ,Σ ′ ;∂M (ψ ⊗ ξ i ⊗ ι Σ (ξ i )), (3)i∈Iwhere c(M;Σ, Σ ′ ) ∈ C\{0} is called the gluing-anomaly factor <strong>and</strong> dependsonly on the geometric data.As in [5] we omit in the following the explicit mention of the maps τ.
Observables in the General Boundary Formulation 1412.2. Amplitudes <strong>and</strong> probabilitiesIn st<strong>and</strong>ard quantum theory transition amplitudes can be used to encodemeasurements. The setup, in its simplest form, involves an initial state ψ<strong>and</strong> a final state η. The initial state encodes a preparation of or knowledgeabout the measurement, while the final state encodes a question about orobservation of the system. The modulus square |〈η, Uψ〉| 2 ,whereU is thetime-evolution operator between initial <strong>and</strong> final time, is then the probabilityfor the answer to the question to be affirmative. (We assume states to benormalized.) This is a conditional probability P (η|ψ), namely the probabilityto observe η given that ψ was prepared.In the GBF this type of measurement setup generalizes considerably. 4Given a spacetime region M, apreparation of or knowledge about the measurementis encoded through a closed subspace S of the boundary Hilbertspace H ∂M . Similarly, the question or observation is encoded in anotherclosed subspace A of H ∂M .Theconditional probability for observing A giventhat S is prepared (or known to be the case) is given by the following formula[1, 6]:P (A|S) = ‖ρ M ◦ P S ◦ P A ‖ 2‖ρ M ◦ P S ‖ 2 . (4)Here P S <strong>and</strong> P A are the orthogonal projectors onto the subspaces S <strong>and</strong> Arespectively. ρ M ◦ P S is the linear map H ∂M → C given by the compositionof the amplitude map ρ M with the projector P S . A requirement for (4) tomake sense is that this composed map is continuous but does not vanish.(The amplitude map ρ M is generically not continuous.) That is, S must beneither “too large” nor “too small”. Physically this means that S must onthe one h<strong>and</strong> be sufficiently restrictive while on the other h<strong>and</strong> not imposingan impossibility. The continuity of ρ M ◦ P S meansthatitisanelementinthe dual Hilbert space H∂M ∗ . The norm in H∗ ∂Mis denoted in formula (4)by ‖·‖. With an analogous explanation for the numerator the mathematicalmeaning of (4) is thus clear.In [1], where this probability interpretation of the GBF was originallyproposed, the additional assumption A⊆Swas made, <strong>and</strong> with good reason.Physically speaking, this condition enforces that we only ask questions ina way that takes into account fully what we already know. Since it is ofrelevance in the following, we remark that formula (4) might be rewritten inthis case as follows:P (A|S) = 〈ρ M ◦ P S ,ρ M ◦ P A 〉. (5)‖ρ M ◦ P S ‖ 2Here the inner product 〈·, ·〉 is the inner product of the dual Hilbert spaceH∂M ∗ . Indeed, whenever P S <strong>and</strong> P A commute, (5) coincides with (4).4 Even in st<strong>and</strong>ard quantum theory, generalizations are possible which involve subspaces ofthe Hilbert space instead of states. A broader analysis of this situation shows that formula(4) is a much milder generalization of st<strong>and</strong>ard probability rules than might seem at firstsight, see [1].
142 R. Oeckl2.3. Recovery of st<strong>and</strong>ard transition amplitudes <strong>and</strong> probabilitiesWe briefly recall in the following how st<strong>and</strong>ard transition amplitudes arerecovered from amplitude functions. Similarly, we recall how st<strong>and</strong>ard transitionprobabilities arise as special cases of the formula (4). Say t 1 is someinitial time <strong>and</strong> t 2 >t 1 some final time, <strong>and</strong> we consider the spacetime regionM =[t 1 ,t 2 ] × R 3 in Minkowski space. ∂M is the disjoint union Σ 1 ∪ ¯Σ 2 ofhypersurfaces of constant t = t 1 <strong>and</strong> t = t 2 respectively. We have chosenthe orientation of Σ 2 here to be opposite to that induced by M, but equal(under time-translation) to that of Σ 1 . Due to axioms (T2) <strong>and</strong> (T1b), wecan identify the Hilbert space H ∂M with the tensor product H Σ1 ˆ⊗HΣ ∗ 2.Theamplitude map ρ M associated with M can thus be viewed as a linear mapH Σ1 ˆ⊗H Σ ∗ 2→ C.In the st<strong>and</strong>ard formalism, we have on the other h<strong>and</strong> a single Hilbertspace H of states <strong>and</strong> a unitary time-evolution map U(t 1 ,t 2 ):H → H. Torelate the two settings we should think of H, H Σ1 <strong>and</strong> H Σ2 as really identical(due to time-translation being an isometry). Then, for any ψ, η ∈ H, theamplitude map ρ M <strong>and</strong> the operator U are related asρ M (ψ ⊗ ι(η)) = 〈η, U(t 1 ,t 2 )ψ〉. (6)Consider now a measurement in the same spacetime region, where aninitial state ψ is prepared at time t 1 <strong>and</strong> a final state η is tested at time t 2 .The st<strong>and</strong>ard formalism tells us that the probability for this is (assumingnormalized states):P (η|ψ) =|〈η, U(t 1 ,t 2 )ψ〉| 2 . (7)In the GBF, the preparation of ψ <strong>and</strong> observation of η are encoded in thefollowing subspaces of H Σ1 ˆ⊗HΣ ∗ 2:S = {ψ ⊗ ξ : ξ ∈H Σ2 } <strong>and</strong> A = {λψ ⊗ ι(η) :λ ∈ C}. (8)Using (6) one can easily show that then P (A|S) =P (η|ψ), i.e., the expressions(4) <strong>and</strong> (7) coincide. This remains true if, alternatively, we define Adisregarding the knowledge encoded in S, i.e., asA = {ξ ⊗ ι(η) :ξ ∈H Σ1 }. (9)3. A conceptual framework for observables3.1. AxiomaticsTaking account of the fact that realistic measurements are extended both inspace as well as in time, it appears sensible to locate also the mathematicalobjects that represent observables in spacetime regions. This is familiarfor example from algebraic quantum field theory, while being in contrast toidealizing measurements as happening at instants of time as in the st<strong>and</strong>ardformulation of quantum theory.<strong>Mathematical</strong>ly, we model an observable associated with a given spacetimeregion M as a replacement of the corresponding amplitude map ρ M .That is, an observable in M is a linear map H∂M ◦ → C, whereH◦ ∂Mis the
Observables in the General Boundary Formulation 143dense subspace of H ∂M appearing in core axiom (T4). Not any such mapneeds to be an observable, though. Which map exactly qualifies as an observablemay generally depend on the theory under consideration.(O1) Associated to each spacetime region M is a real vector space O M oflinear maps H∂M ◦ → C, called observable maps. Inparticular,ρ M ∈O M .The most important operation that can be performed with observablesis that of composition. This composition is performed exactly in the sameway as prescribed for amplitude maps in core axioms (T5a) <strong>and</strong> (T5b). Thisleads to an additional condition on the spaces O M of observables, namelythat they be closed under composition.(O2a) Let M 1 <strong>and</strong> M 2 be regions as in (T5a), <strong>and</strong> let O 1 ∈O M1 , O 2 ∈O M2 .Then there is O 3 ∈O M1 ∪M 2such that for all ψ 1 ∈H∂M ◦ 1<strong>and</strong> ψ 2 ∈H∂M ◦ 2,O 3 (ψ 1 ⊗ ψ 2 )=ρ M1 (ψ 1 )ρ M2 (ψ 2 ). (10)(O2b) Let M be a region with its boundary decomposing as a disjoint union∂M =Σ 1 ∪ Σ ∪ Σ ′ <strong>and</strong> M 1 given as in (T5b) <strong>and</strong> O ∈O M . Then, thereexists O 1 ∈O M1 such that for any orthonormal basis {ξ i } i∈I of H Σ <strong>and</strong>for all ψ ∈H∂M ◦ 1,O 1 (ψ) · c(M;Σ, Σ ′ )= ∑ i∈IO(ψ ⊗ ξ i ⊗ ι Σ (ξ i )). (11)We generally refer to the gluing operations of observables of the types describedin (O2a) <strong>and</strong> (O2b) as well as their iterations <strong>and</strong> combinations ascompositions of observables. Physically, the composition is meant to representthe combination of measurements. Combination is here to be understood asin classical physics, when the product of observables is taken.3.2. Expectation valuesAs in the st<strong>and</strong>ard formulation of quantum theory, the expectation value ofan observable depends on a preparation of or knowledge about a system. Asrecalled in Section 2.2, this is encoded for a region M in a closed subspaceS of the boundary Hilbert space H ∂M . Given an observable O ∈O M <strong>and</strong> aclosed subspace S⊆H ∂M , the expectation value of O with respect to S isdefined as〈O〉 S := 〈ρ M ◦ P S ,O〉‖ρ M ◦ P S ‖ 2 . (12)We use notation here from Section 2.2. Also, as there we need ρ M ◦ P S to becontinuous <strong>and</strong> different from zero for the expectation value to make sense.We proceed to make some remarks about the motivation for postulatingthe expression (12). Clearly, the expectation value must be linear in theobservable. Another important requirement is that we would like probabilitiesin the sense of Section 2.2 to arise as a special case of expectation values.Indeed, given a closed subspace A of H ∂M <strong>and</strong> setting O = ρ M ◦ P A , we seethat expression (12) reproduces exactly expression (5). At least in the case
144 R. Oecklwhere the condition A⊆Sis met, this coincides with expression (4) <strong>and</strong>represents the conditional probability to observe A given S.3.3. Recovery of st<strong>and</strong>ard observables <strong>and</strong> expectation valuesOf course it is essential that the present proposal for implementing observablesin the GBF can reproduce observables <strong>and</strong> their expectation values asoccurring in the st<strong>and</strong>ard formulation of quantum theory. There observablesare associated to instants of time, i.e., equal-time hypersurfaces. To modelthese we use “infinitesimally thin” regions, also called empty regions, whichgeometrically speaking are really hypersurfaces, but are treated as regions.Concretely, consider the equal-time hypersurface at time t in Minkowskispace, i.e., Σ = {t}×R 3 . We denote the empty region defined by the hypersurfaceΣ as ˆΣ. The relation between an observable map O ∈O M <strong>and</strong> thecorresponding operator Õ is then analogous to the relation between the amplitudemap <strong>and</strong> the time-evolution operator as expressed in equation (6). Bydefinition, ∂ ˆΣ is equal to the disjoint union Σ ∪ ¯Σ, so that H ∂ ˆΣ= H Σ ˆ⊗HΣ ∗ .The Hilbert space H Σ is identified with the conventional Hilbert space H,<strong>and</strong> for ψ, η ∈ H we requireO(ψ ⊗ ι(η)) = 〈η, Õψ〉 Σ. (13)Note that we can glue two copies of ˆΣ together, yielding again a copy of ˆΣ.The induced composition of observable maps then translates via (13) preciselyto the composition of the corresponding operators. In this way we recover theusual operator product for observables of the st<strong>and</strong>ard formulation.Consider now a normalized state ψ ∈ H = H Σ encoding a preparation.This translates in the GBF language to the subspace S = {ψ ⊗ ξ : ξ ∈HΣ ∗ }of H ∂ ˆΣas reviewed in Section 2.3. The amplitude map ρˆΣcan be identifiedwith the inner product of H = H Σ due to core axiom (T3x). Thus,ρˆΣ◦ P S (ξ ⊗ ι(η)) = 〈η, P ψ ξ〉 Σ ,whereP ψ is the orthogonal projector in H Σonto the subspace spanned by ψ. This makes it straightforward to evaluatethe denominator of (12). Let {ξ i } i∈N be an orthonormal basis of H Σ ,which,moreover, we choose for convenience such that ξ 1 = ψ. Then‖ρ M ◦ P S ‖ 2 =∞∑|ρ M ◦ P S (ξ i ⊗ ι(ξ j ))| 2 =i,j=1For the numerator of (12) we observe〈ρ M ◦ P S ,O〉 =∞∑|〈ξ j ,P ψ ξ i 〉 Σ | 2 =1. (14)i,j=1∞∑ρ M ◦ P S (ξ i ⊗ ι(ξ j )) O(ξ i ⊗ ι(ξ j ))i,j=1=∞∑〈P ψ ξ i ,ξ j 〉 Σ 〈ξ j , Õξ i〉 Σ = 〈ψ, Õψ〉 Σ. (15)i,j=1Hence, the GBF formula (12) recovers in this case the conventional expectationvalue of Õ with respect to the state ψ.
4. QuantizationObservables in the General Boundary Formulation 145We turn in this section to the problem of the quantization of classical observables.On the one h<strong>and</strong>, we consider the question of how specific quantizationschemes that produce Hilbert spaces <strong>and</strong> amplitude functions satisfying thecore axioms can be extended to produce observables. On the other h<strong>and</strong>,we discuss general features of quantization schemes for observables <strong>and</strong> therelation to conventional schemes.4.1. Schrödinger-Feynman quantizationCombining the Schrödinger representation with the Feynman path integralyields a quantization scheme that produces Hilbert spaces for hypersurfaces<strong>and</strong> amplitude maps for regions in a way that “obviously” satisfies the coreaxioms [3, 7, 8]. We shall see that it is quite straightforward to include observablesinto this scheme. Moreover, the resulting quantization can be seen to bein complete agreement with the results of st<strong>and</strong>ard approaches to quantumfield theory.We recall that in this scheme states on a hypersurface Σ arise as wavefunctions on the space space K Σ of field configurations on Σ. These form aHilbert space H Σ of square-integrable functions with respect to a (fictitious)translation-invariant measure μ Σ :∫〈ψ ′ ,ψ〉 Σ := ψ ′ (ϕ)ψ(ϕ)dμ Σ (ϕ). (16)K ΣThe amplitude map for a region M arises as the Feynman path integral,∫ρ M (ψ) := ψ (φ| Σ ) e iS M (φ) dμ M (φ), (17)K Mwhere S M is the action evaluated in M, <strong>and</strong>K M is the space of field configurationsin M.The Feynman path integral is of course famous for resisting a rigorousdefinition, <strong>and</strong> it is a highly non-trivial task to make sense of expressions(17) or even (16) in general. Nevertheless, much of text-book quantum fieldtheory relies on the Feynman path integral <strong>and</strong> can be carried over to thepresent context relatively easily for equal-time hypersurfaces in Minkowskispace <strong>and</strong> regions bounded by such. Moreover, for other special types of regions<strong>and</strong> hypersurfaces this quantization program has also been successfullycarried through for linear or perturbative quantum field theories. Notably,this includes timelike hypersurfaces [7, 8] <strong>and</strong> has led to a widening of theconcept of an asymptotic S-matrix [9, 10].We proceed to incorporate observables into the quantization scheme.To this end, a classical observable F in a region M is modeled as a real- (orcomplex-) valued function on K M . According to Section 3.1 the quantizationof F , which we denote here by ρ F M , must be a linear map H◦ ∂M→ C. Wedefine it as∫ρ F M (ψ) := ψ (φ| Σ ) F (φ) e iS M (φ) dμ M (φ). (18)K M
146 R. OecklBefore we proceed to interpret this formula in terms of text-book quantumfield theory language, we emphasize a key property of this quantization prescription.Suppose we have disjoint but adjacent spacetime regions M 1 <strong>and</strong>M 2 supporting classical observables F 1 : K M1 → R <strong>and</strong> F 2 : K M2 → R,respectively. Applying first the operation of (O2a) <strong>and</strong> then that of (O2b),we can compose the corresponding quantum observables ρ F 1M 1<strong>and</strong> ρ F 2M 2to anew observable, which we shall denote ρ F 1M 1⋄ρ F 2M 2, supported on the spacetimeregion M := M 1 ∪M 2 . On the other h<strong>and</strong>, the classical observables F 1 <strong>and</strong> F 2can be extended trivially to the spacetime region M <strong>and</strong> there be multipliedto a classical observable F 1 · F 2 : K M → R. The composition property of theFeynman path integral now implies the identityρ F 1·F 2M= ρ F 1M 1⋄ ρ F 2M 2. (19)That is, there is a direct correspondence between the product of classicalobservables <strong>and</strong> the spacetime composition of quantum observables. Thiscomposition correspondence, as we shall call it, is not to be confused withwhat is usually meant with the term “correspondence principle” such as arelation between the commutator of operators <strong>and</strong> the Poisson bracket ofclassical observables that these are representing. Indeed, at a careless glancethese concepts might even seem to be in contradiction.Consider now in Minkowski space a region M =[t 1 ,t 2 ] × R 3 ,wheret 1
Observables in the General Boundary Formulation 147to coherent states. We consider the simple setting of a massive free scalarfield theory in Minkowski space with equal-time hypersurfaces. Recall ([10],equation (26)) that a coherent state in the Schrödinger representation at timet can be written as(∫ d 3 x d 3 )kψ t,η (ϕ) :=C t,η expη(k) e −i(Et−kx) ϕ(x) ψ(2π) 30 (ϕ), (22)where η is a complex function on momentum space encoding a solution ofthe Klein-Gordon equation. ψ 0 is the vacuum wave function <strong>and</strong> C t,η is anormalization constant. Consider as above an initial time t 1 ,afinaltimet 2 >t 1 <strong>and</strong> the region M := [t 1 ,t 2 ]×R 3 in Minkowski space. Let F : K M → Crepresent a classical observable. Evaluating the quantized observable map ρ F Mon an initial coherent state encoded by η 1 <strong>and</strong> a final coherent state encodedby η 2 yields:( )ψt1,η 1⊗ ψ t2,η 2ρ F M= C t1 ,η 1C t2 ,η 2ψ 0 (ϕ 1 ) ψ 0 (ϕ 2 )∫K M(∫ d 3 x d 3 k() )· expη(2π) 3 1 (k) e −i(Et1−kx) ϕ 1 (x)+η 2 (k) e i(Et2−kx) ϕ 2 (x)· F (φ) e iS M (φ) dμ M (φ)= ρ M(ψt1 ,η 1⊗ ψ t2 ,η 2) ∫ K Mψ 0 (ϕ 1 ) ψ 0 (ϕ 2 ) F (φ +ˆη) e iS M (φ) dμ M (φ).(23)Here ϕ i denote the restrictions of the configuration φ to time t i . To obtainthe second equality we have shifted the integration variable φ by∫d 3 k)ˆη(t, x) :=(η(2π) 3 1 (k)e −i(Et−kx) + η 2 (k)e i(Et−kx) (24)2E<strong>and</strong> used the conventions of [10]. Note that ˆη is a complexified classical solutionin M determined by η 1 <strong>and</strong> η 2 .WehavesupposedthatF naturallyextends to a function on the complexified configuration space KM C .Viewingthe function φ → F (φ +ˆη) as a new observable F ˆη , the remaining integralin (23) can be interpreted in terms of (18), <strong>and</strong> we obtain the factorizationidentityρ F ( ) ( )M ψt1 ,η 1⊗ ψ t2 ,η 2 = ρM ψt1 ,η 1⊗ ψ t2 ,η 2 ρF ˆηM (ψ 0 ⊗ ψ 0 ). (25)That is, the quantum observable map evaluated on a pair of coherent statesfactorizes into the plain amplitude for the same pair of states <strong>and</strong> the quantumobservable map for a shifted observable evaluated on the vacuum. Notethat the second term on the right-h<strong>and</strong> side here is a vacuum expectationvalue.It turns out that factorization identities analogous to (25) are genericrather than special to the types of hypersurfaces <strong>and</strong> regions considered here.We will come back to this issue in the next section, where also the role of
148 R. Oecklthe complex classical solution ˆη will become clearer from the point of viewof holomorphic quantization. For the moment let us consider the particularlysimple case where F is a linear observable. In this case F ˆη (φ) =F (φ)+F (ˆη),<strong>and</strong> the second term on the right-h<strong>and</strong> side of (25) decomposes into a sum oftwo terms:ρ F ˆηM (ψ 0 ⊗ ψ 0 )=ρ F M (ψ 0 ⊗ ψ 0 )+F (ˆη)ρ M (ψ 0 ⊗ ψ 0 ). (26)The first term on the right-h<strong>and</strong> side is a one-point function which vanishes inthe present case of a linear field theory. (F is antisymmetric under exchangeof φ <strong>and</strong> −φ, while the other expressions in (18) are symmetric.) The secondfactor in the second term is the amplitude of the vacuum <strong>and</strong> hence equal tounity. Thus, in the case of a linear observable, (25) simplifies to( ) ( )ψt1 ,η 1⊗ ψ t2 ,η 2 = F (ˆη)ρM ψt1 ,η 1⊗ ψ t2 ,η 2 . (27)ρ F M4.2. Holomorphic quantizationA more rigorous quantization scheme that produces a GBF from a classicalfield theory is the holomorphic quantization scheme introduced in [5]. It isbased on ideas from geometric quantization, <strong>and</strong> its Hilbert spaces are versionsof “Fock representations”. An advantage of this scheme is that takingan axiomatically described classical field theory as input, it produces a GBFas output that can be rigorously proved to satisfy the core axioms of Section2.1. A shortcoming so far is that only the case of linear field theory hasbeen worked out.Concretely, the classical field theory is to be provided in the form ofa real vector space L Σ of (germs of) solutions near each hypersurface Σ.Moreover, for each region M thereistobegivenasubspaceL ˜Mof thespace L ∂M of solutions on the boundary of M. This space L ˜Mhas the interpretationof being the space of solutions in the interior of M (restricted tothe boundary). Also, the spaces L Σ carry non-degenerate symplectic structuresω Σ as well as complex structures J Σ . Moreover, for each hypersurfaceΣ, the symplectic <strong>and</strong> complex structures combine to a complete real innerproduct g Σ (·, ·) =2ω Σ (·,J Σ·) <strong>and</strong> to a complete complex inner product{·, ·} Σ = g Σ (·, ·)+2iω Σ (·, ·). Another important condition is that the subspaceL ˜M⊆ L ∂M is Lagrangian with respect to the symplectic structure ω ∂M .The Hilbert space H Σ associated with a hypersurface Σ is the space ofholomorphic square-integrable functions on ˆL Σ with respect to a Gaussianmeasure ν Σ . 6 That is, the inner product in H Σ is∫〈ψ ′ ,ψ〉 Σ := ψ(φ)ψ ′ (φ)dν Σ (φ). (28)ˆL Σ6 The space ˆL Σ is a certain extension of the space L Σ , namely the algebraic dual of itstopological dual. Nevertheless, due to Theorem 3.18 of [5] it is justified to think of wavefunctions ψ as functions merely on L Σ rather than on ˆL Σ , <strong>and</strong> to essentially ignore thedistinction between L Σ <strong>and</strong> ˆL Σ .
Observables in the General Boundary Formulation 149Heuristically, the measure ν Σ can be understood as(dν Σ (φ) ≈ exp − 1 )2 g ∂M(φ, φ) dμ Σ (φ), (29)where μ Σ is a fictitious translation-invariant measure on ˆL Σ . The space H Σ isessentially the Fock space constructed from L Σ viewed as a 1-particle spacewith the inner product {·, ·} Σ .The amplitude map ρ M : H ∂M → C associated with a region M is givenby the integral formula∫ρ M (ψ) := ψ(φ)dν ˜M(φ). (30)ˆL ˜MThe integration here is over the space ˆL ˜M⊆ ˆL ∂M of solutions in M with themeasure ν ˜M, which heuristically can be understood as(dν ˜M(φ) ≈ exp − 1 )4 g ∂M(φ, φ) dμ ˜M(φ), (31)where again μ ˜Mis a fictitious translation-invariant measure on ˆL ˜M.Particularly useful in the holomorphic quantization scheme turn outto be the coherent states that are associated to classical solutions near thecorresponding hypersurface. On a hypersurface Σ the coherent state K ξ ∈H Σassociated to ξ ∈ L Σ is given by the wave function) 1K ξ (φ) :=exp(2 {ξ,φ} Σ . (32)The natural vacuum, which we denote by 1, is the constant wave function ofunit value. Note that 1 = K 0 .4.2.1. Creation <strong>and</strong> annihilation operators. One-particle states on a hypersurfaceΣ are represented by non-zero continuous complex-linear maps p :L Σ → C, where complex-linearity here implies p(Jξ)=ip(ξ). By the RieszRepresentation Theorem such maps are thus in one-to-one correspondencewith non-zero elements of L Σ . Concretely, for a non-zero element ξ ∈ L Σ thecorresponding one-particle state is represented by the wave function p ξ ∈H Σgiven byp ξ (φ) = √ 1 {ξ,φ} Σ . (33)2The normalization is chosen here such that ‖p ξ ‖ = ‖ξ‖. Physically distinctone-particle states thus correspond to the distinct rays in L Σ , viewed asa complex Hilbert space. An n-particle state is represented by a (possiblyinfinite) linear combination of the product of n wave functions of this type.The creation operator a † ξ for a particle state corresponding to ξ ∈ L Σ is givenby multiplication,(a † ξ ψ)(φ) =p ξ(φ)ψ(φ) = 1 √2{ξ,φ} Σ ψ(φ). (34)
150 R. OecklThe corresponding annihilation operator is the adjoint. Using the reproducingproperty of the coherent states K φ ∈H Σ , we can write it as(a ξ ψ)(φ) =〈K φ ,a ξ ψ〉 Σ = 〈a † ξ K φ,ψ〉 Σ . (35)Note in particular, that the action of an annihilation operator on a coherentstate is by multiplication,a ξ K φ = 1 √2{φ, ξ} Σ K φ . (36)For ξ,η ∈ L Σ the commutation relations are, as usual,[a ξ ,a † η]={η, ξ} Σ , [a ξ ,a η ]=0, [a † ξ ,a† η]=0. (37)4.2.2. Berezin-Toeplitz quantization. A natural way to include observablesinto this quantization scheme seems to be the following. We model a classicalobservable F on a spacetime region M as a map L ˜M→ C (or L ˜M→ R) <strong>and</strong>define the associated quantized observable map via∫ρ ◭F M◮ (ψ) := ψ(φ)F (φ)dν ˜M(φ). (38)ˆL ˜MTo bring this into a more familiar form, we consider, as in Section 3.3, thespecial case of an empty region ˆΣ, given geometrically by a hypersurface Σ.Then, for ψ 1 ,ψ 1 ∈H Σ encoding “initial” <strong>and</strong> “final” state, we have∫ρ ◭F ˆΣ◮ (ψ 1 ⊗ ι(ψ 2 )) = ψ 1 (φ)ψ 2 (φ)F (φ)dν Σ (φ). (39)ˆL ΣWe can interpret this formula as follows: The wave function ψ 1 is multipliedby the function F . The resulting function is an element of the Hilbertspace L 2 (ˆL Σ ,ν Σ ) (supposing F to be essentially bounded), but not of thesubspace H Σ of holomorphic functions. We thus project back onto this subspace<strong>and</strong> finally take the inner product with the state ψ 2 . This is preciselyaccomplished by the integral. We may recognize this as a version of Berezin-Toeplitz quantization, where in the language of Berezin [11] the function Fis the contravariant symbol of the operator ˜F that is related to ρ ◭F ˆΣ◮ byformula (13). That is,ρ ◭F ˆΣ◮ (ψ 1 ⊗ ι(ψ 2 )) = 〈ψ 2 , ˜Fψ 1 〉 Σ . (40)In the following we shall refer to the prescription encoded in (38) simply asBerezin-Toeplitz quantization.Note that any complex-valued continuous real-linear observable F :L Σ → C can be decomposed into its holomorphic (complex-linear) <strong>and</strong> antiholomorphic(complex-conjugate-linear) partF (φ) =F + (φ)+F − (φ), where F ± (φ) = 1 2(F (φ) ∓ iF (JΣ φ) ) . (41)If we consider real-valued observables only, we can parametrize them by elementsof L Σ due to the Riesz Representation Theorem. (In the complexvaluedcase the parametrization is by elements of L C Σ , the complexification of
Observables in the General Boundary Formulation 151L Σ , instead.) If we associate to ξ ∈ L Σ the real-linear observable F ξ given bythenF ξ (φ) := √ 2 g Σ (ξ,φ), (42)F + ξ (φ) = 1 √2{ξ,φ} Σ , F − ξ (φ) = 1 √2{φ, ξ} Σ . (43)Using the results of Section 4.2.1, we see that F ± ξquantized according to theprescription (39) <strong>and</strong> expressed in terms of operators ˜F ± ξas in (40) yields˜F + ξ = a† ξ , − ˜Fξ= a ξ , <strong>and</strong> for the sum, ˜Fξ = a † ξ + a ξ. (44)Consider now n real-linear observables F 1 ,...,F n : L Σ → C.Weshallbeinterested in the antinormal-ordered product of the corresponding operators˜F 1 ,..., ˜F n , which we denote by ◭ ˜F 1 ··· ˜F n ◮. To evaluate matrix elementsof this antinormal-ordered product, we decompose the observables F i accordingto (41) into holomorphic <strong>and</strong> anti-holomorphic parts, correspondingto creation operators <strong>and</strong> annihilation operators, respectively. The creationoperators ˜F + ithen act on wave functions by multiplication with F + iaccordingto (34). Converting the annihilation operators into creation operators bymoving them to the left-h<strong>and</strong> side of the inner product, we see that thesecorrespondingly contribute factors F − iin the inner product (28). We obtain:〈〈ψ2 , ◭ ˜F 1 ··· ˜F〉n∏〉n ◮ ψ 1 = ψΣ 2 , ◭ ( ˜F + i + ˜F − i )◮ ψ 1i=1Σ∫ ( n)∏= ψ 2 (φ) (F + i (φ)+F − i (φ)) ψ 1 (φ)dν Σ (φ)ˆL Σ i=1∫= ψ 2 (φ)F 1 (φ) ···F n (φ)ψ 1 (φ)dν Σ (φ).ˆL ΣSetting F := F 1 ···F n , this coincides precisely with the right-h<strong>and</strong> side of(39). Thus in the case of a hypersurface (empty region) the Berezin-Toeplitzquantization precisely implements antinormal ordering.Remarkably, the Berezin-Toeplitz quantization shares with the Schrödinger-Feynmanquantization the factorization property exhibited in equation(25). In fact, it is in the present context of holomorphic quantization thatthis property attains a strikingly simple form. In order to state it rigorously,we need a bit of technical language. For a map F : L ˜M→ C <strong>and</strong> an elementξ ∈ L ˜M,wedenotebyF ξ : L ˜M→ C the translated map φ ↦→ F (φ + ξ).We say that F : L ˜M→ C is analytic iff for each pair φ, ξ ∈ L ˜Mthe mapz ↦→ F (φ+zξ) is real analytic. We denote the induced extension L C˜M → C alsoby F ,whereL C˜M is the complexification of L ˜M.WesaythatF : L ˜M→ C isanalytic <strong>and</strong> sufficiently integrable iff for any η ∈ L C˜M the map F η is integrablein (ˆL ˜M,ν ˜M). We recall (Lemma 4.1 of [5]) that elements ξ of L ∂M decomposeuniquely as ξ = ξ R + J Σ ξ I ,whereξ R ,ξ I areelementsofL ˜M.
152 R. OecklProposition 4.1 (Coherent factorization property). Let F : L M → C be analytic<strong>and</strong> sufficiently integrable. Then for any ξ ∈ L ∂M , we haveρ ◭F M◮ (K ξ)=ρ M (K ξ ) ρ ◭F ˆξ ◮M (1), (45)where ˆξ ∈ L C˜M is given by ˆξ = ξ R − iξ I .Proof. Recall that for φ ∈ ˆL ˜Mwe can rewrite the wave function of the coherentstate K ξ as follows:1K ξ (φ) =exp(2 g ∂M(ξ R ,φ) − i )2 g ∂M(ξ I ,φ) . (46)We restrict first to the special case ξ ∈ L ˜M, i.e., ξ I = 0. Translating theintegr<strong>and</strong> by ξ (using Proposition 3.11 of [5]), we find:∫)1F (φ)exp(ˆL ˜M2 g ∂M(ξ,φ) dν(φ)∫1= F (φ + ξ)exp(ˆL ˜M2 g ∂M(ξ,φ + ξ) − 1 )4 g ∂M(2φ + ξ,ξ) dν(φ)) ∫1=exp(4 g ∂M(ξ,ξ) F (φ + ξ)dν(φ)ˆL ˜M= ρ M (K ξ ) ρ ◭F ξ ◮M (1) .In order to work out the general case, we follow the strategy outlined inthe proof of Proposition 4.2 of [5]: We replace the i in (46) by a complexparameter <strong>and</strong> note that all relevant expressions are holomorphic in thisparameter. This must also hold for the result of the integration performedabove. But performing the integration is straightforward when this parameteris real, since we can then combine both terms in the exponential in (46). Onthe other h<strong>and</strong>, a holomorphic function is completely determined by its valueson the real line. This leads to the stated result.□It is clear at this point that equation (25) is just a special case of (theanalog for ρ F M of) equation (45). Indeed, it turns out that with a suitablechoice of complex structure (see [5]) the complexified classical solution ˆηgiven by (24) decomposes precisely as ˆη = η R − iη I . 7 From here onwards weshall say that a quantization scheme satisfying equation (45) has the coherentfactorization property.The coherent factorization property may also be interpreted as suggestingan intrinsic definition of a real observable in the quantum setting. It isclear that quantum observable maps must take values in the complex numbers<strong>and</strong> not merely in the real numbers since for example the amplitudemap is a special kind of quantum observable map. 8 On the other h<strong>and</strong>, wehave in axiom (O1) deliberately only required that the observable maps in a7 We differ here slightly from the conventions in [5] to obtain exact agreement.8 Proposition 4.2 of [5] implies that amplitude maps generically take complex <strong>and</strong> notmerely real values.
Observables in the General Boundary Formulation 153region M form a real vector space O M ,toallowforarestrictionto“real”observables,analogous to hermitian operators in the st<strong>and</strong>ard formulation <strong>and</strong>to real-valued maps in the classical theory. Of course, given a quantizationprescription such as (18) or (38), we can simply restrict the quantization toreal classical observables. However, equation (45) suggests a more intrinsicdefinition in case of availability of coherent states. Namely, we could say thata quantum observable map is real iff its evaluation on a coherent state K ξassociated to any element ξ in the subspace L ˜M⊆ L ∂M yields a real multipleof the amplitude map evaluated on the same coherent state. Note that thischaracterization is closed under real-linear combinations. Also, if a quantizationscheme satisfies the coherent factorization property, this characterizationcoincides with the condition for the classical observable to be real-valued, asis easily deduced using the completeness of the coherent states.Let us briefly return to the special case of linear observables. Supposethat F : L ˜M→ R is linear (hence analytic) <strong>and</strong> sufficiently integrable. Weevaluate the Berezin-Toeplitz quantum observable map ρ ◭F M◮ on the coherentstate K ξ associated to ξ ∈ L ∂M . As above we define ˆξ ∈ L C˜M as ˆξ = ξ R − iξ I .Using the coherent factorization property (45) as well as linearity of F ,weobtainρ ◭F M◮ (K ξ)=ρ M (K ξ )ρ ◭F ˆξ ◮M (1) =ρ M (K ξ )()ρ ◭F M ◮ (1)+F (ˆξ)ρ M (1) . (47)The first term in brackets vanishes by inspection of (38) due to anti-symmetryof F under exchange of φ <strong>and</strong> −φ, while ρ M (1) = 1. Thus, analogously to(27), we obtainρ ◭F ◮M (K ξ)=F (ˆξ)ρ M (K ξ ). (48)Supposing that the amplitudes of coherent states coincide between theSchrödinger-Feynman scheme <strong>and</strong> the holomorphic scheme (that is, if thecomplex structure of the holomorphic scheme <strong>and</strong> the vacuum of the Schrödinger-Feynmanscheme are mutually adapted), also the quantization of linearobservables according to (18) coincides with that of (38). Nevertheless,the quantization of non-linear observables is necessarily different. For one,classical observables in the Schrödinger-Feynman scheme are defined on configurationspaces rather than on spaces of solutions. Indeed, the quantizationof observables that coincide when viewed merely as functions on solutions differsin general. However, it is also clear that the Berezin-Toeplitz quantizationcannot satisfy the composition correspondence property (19) that is satisfiedby the Schrödinger-Feynman scheme. Indeed, consider adjacent regions M 1<strong>and</strong> M 2 that can be glued to a joint region M. Then the classical observablesin the disjoint region induce classical observables in the glued region, but notthe other way round. While the former are functions on L ˜M1×L ˜M2, the latterare functions on the subspace L ˜M⊆ L ˜M1× L ˜M2. In spite of the summationinvolved in axiom (O2b), one can use this to cook up a contradiction to thecomposition correspondence property (19). It is easy to see how this problemis avoided in the Schrödinger-Feynman scheme: There, classical observablesin a region M are functions on the space of field configurations K M ,whichis
154 R. Oecklmuch larger than the space of classical solutions L M <strong>and</strong> permits the “decoupling”of observables in adjacent regions. Indeed, the present considerationsindicate that in order for a quantization scheme to satisfy the compositioncorrespondence property, this kind of modification of the definition of whatconstitutes a classical observable is a necessity.The composition correspondence property suggests also a different routeto quantization of observables: We may consider a quantization scheme onlyfor linear observables <strong>and</strong> impose the composition correspondence propertyto define a quantization scheme for more general observables. In the Berezin-Toeplitz case this would lead to a scheme equivalent to the path integral(18). However, recall that the composition of quantum observable maps isonly possible between disjoint regions. On the other h<strong>and</strong>, well-known difficulties(related to renormalization) arise also for the path integral (18) whenconsidering field observables at coincident points.4.2.3. Normal-ordered quantization. Consider a hypersurface Σ <strong>and</strong> linearobservables F 1 ,...,F n : L Σ → C in the associated empty region ˆΣ. Considernow the normal-ordered product : ˜F 1 ··· ˜F n :<strong>and</strong>itsmatrixelements.Thesematrix elements turn out to be particularly simple for coherent states. Toevaluate these we decompose the maps F 1 ,...,F n into holomorphic (creation)<strong>and</strong> anti-holomorphic (annihilation) parts according to (41). The annihilationoperators act on coherent states simply by multiplication according to (36).The creation operators on the other h<strong>and</strong> can be converted to annihilationoperators by moving them to the left-h<strong>and</strong> side of the inner product. We find:〈Kη , :F 1 ···F n : K ξ〉Σ = n ∏i=1(F+i (η)+F − i (ξ)) 〈K η ,K ξ 〉 Σ . (49)While this expression looks quite simple, it can be further simplified by takingseriously the point of view that ˆΣ is an (empty) region. Hence, K ξ ⊗ι(K η ) is really the coherent state K (ξ,η) ∈H ∂ ˆΣassociated to the solution(ξ,η) ∈ L ∂ ˆΣ.As above we may decompose (ξ,η) = (ξ,η) R + J ∂ ˆΣ(ξ,η) I ,where (ξ,η) R , (ξ,η) I ∈ L˜ˆΣ.Identifying L˜ˆΣwith L Σ (<strong>and</strong> taking into accountJ ∂ ˆΣ=(J Σ , −J Σ )), we have(ξ,η) R = 1 2 (ξ + η), (ξ,η)I = − 1 2 (J Σξ − J Σ η). (50)But observe:F + i (η)+F − i (ξ) =1 (Fi (η + ξ) − iF i (J Σ (η − ξ)) )2(= F i (ξ,η)R ) (− iF i (ξ,η)I ) (= F i (ξ,η) R − i(ξ,η) I) , (51)where in the last step we have extended the domain of F i from L ˜ˆΣto itscomplexification L C˜ˆΣ.Defining a quantum observable map encoding the normal-ordered productρ :F 1···F n :(ψˆΣ1 ⊗ ι(ψ 2 )) := 〈 ψ 2 , : ˜F 1 ··· ˜F n : ψ 1〉Σ , (52)
Observables in the General Boundary Formulation 155the identity (49) becomes thusn∏ρ :F 1···F n(:(KˆΣ(ξ,η) )= F i (ξ,η) R − i(ξ,η) I) ρ ˆΣ(K (ξ,η) ). (53)i=1Note that in the above expression the fact that we consider an empty regionrather than a generic region is no longer essential. Rather, we may consider aregion M <strong>and</strong> replace (ξ,η) by some solution φ ∈ L ˜M.Also there is no longer anecessity to write the observable explicitly as a product of linear observables.A generic observable F : L ˜M→ C that has the analyticity property will do.We obtain:ρ M :F : (K φ ):=F ( ˆφ)ρ M (K φ ), (54)where ˆφ := φ R −iφ I . We may take this now as the definition of a quantizationprescription that we shall call normal-ordered quantization. Itcoincideswith— <strong>and</strong> provides an extremely concise expression for — the usual conceptof normal ordering in the case when M is the empty region associated to ahypersurface.Interestingly, expression (54) also coincides with expression (48) <strong>and</strong>with expression (27). However, the latter two expressions were only validin the case where F is linear. So, unsurprisingly, we obtain agreement ofnormal-ordered quantization with Berezin-Toeplitz quantization <strong>and</strong> withSchrödinger-Feynman quantization in the case of linear observables, whilethey differ in general. Remarkably, however, normal-ordered quantizationshares with these other quantization prescriptions the coherent factorizationproperty (45). To see this, note that (using (54)):ρ :F ˆφ :M (1) =F ˆφ(0)ρ M (1) =F ( ˆφ). (55)4.2.4. Geometric quantization. Since the holomorphic quantization schemedraws on certain ingredients of geometric quantization, it is natural to alsoconsider what geometric quantization has to say about the quantizationof observables [12]. For hypersurfaces Σ (empty regions ˆΣ) the geometricquantization of a classical observable F : L Σ → C is given by an operatorˇF : H Σ →H Σ . If the observable F preserves the polarization (which is thecase for example for linear observables), then ˇF is given by the formulaˇFψ = −idψ(F) − θ(F)ψ + Fψ. (56)Here F denotes the Hamiltonian vector field generated by the function F , θis the symplectic potential given here by θ η (Φ) = − i 2 {η, Φ η} Σ ,<strong>and</strong>dψ is theexterior derivative of ψ.Consider a real-linear observable F : L Σ → C. Without explainingthe details we remark that for the holomorphic part F + (recall (41)) weobtain dψ(F + ) = 0 as well as θ(F + ) = 0. On the other h<strong>and</strong>, for the antiholomorphicpart F − we have θ(F − )=F − . This simplifies (56) toˇFψ = −idψ(F − )+F + ψ. (57)
156 R. OecklSetting F equal to F ξ given by (42) for ξ ∈ L Σ results in F + ψ = a † ξ ψ <strong>and</strong>−idψ(F − )=a ξ ψ. That, is the operator ˇF coincides with the operator ˜Fobtained by quantizing F with any of the other prescriptions discussed. Onthe other h<strong>and</strong>, the quantization of non-linear observables will in generaldiffer from any of the other prescriptions. Indeed, it is at present not evenclear whether or how the prescription (56) can be generalized to non-emptyregions.AcknowledgmentsI would like to thank the organizers of the Regensburg Conference 2010:<strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong> for inviting me to present aspects of thiswork as well as for providing a stimulating <strong>and</strong> well-organized conference.This work was supported in part by CONACyT grant 49093.References[1] R. Oeckl, General boundary quantum field theory: Foundations <strong>and</strong> probabilityinterpretation, Adv. Theor. Math. Phys. 12 (2008), 319–352, hep-th/0509122.[2] M. Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ.Math. (1989), no. 68, 175–186.[3] R. Oeckl, A “general boundary” formulation for quantum mechanics <strong>and</strong> quantumgravity, Phys. Lett. B 575 (2003), 318–324, hep-th/0306025.[4] R. Oeckl, Two-dimensional quantum Yang-Mills theory with corners, J. Phys.A 41 (2008), 135401, hep-th/0608218.[5] R. Oeckl, Holomorphic quantization of linear field theory in the general boundaryformulation, preprint arXiv:1009.5615.[6] R. Oeckl, Probabilities in the general boundary formulation, J.Phys.:Conf.Ser.67 (2007), 012049, hep-th/0612076.[7] R. Oeckl, States on timelike hypersurfaces in quantum field theory, Phys. Lett.B 622 (2005), 172–177, hep-th/0505267.[8] R. Oeckl, General boundary quantum field theory: Timelike hypersurfaces inKlein-Gordon theory, Phys. Rev. D73(2006), 065017, hep-th/0509123.[9] D. Colosi <strong>and</strong> R. Oeckl, S-matrix at spatial infinity, Phys. Lett. B 665 (2008),310–313, arXiv:0710.5203.[10] D. Colosi <strong>and</strong> R. Oeckl, Spatially asymptotic S-matrix from general boundaryformulation, Phys. Rev. D78(2008), 025020, arXiv:0802.2274.[11] F. A. Berezin, Covariant <strong>and</strong> contravariant symbols of operators, Math. USSRIzvestija 6 (1972), 1117–1151.[12] N. M. J. Woodhouse, Geometric quantization, 2nd ed., Oxford University Press,Oxford, 1991.Robert OecklInstituto de MatemáticasUniversidad Nacional Autónoma de México, Campus MoreliaC.P. 58190, Morelia, MichoacánMexicoe-mail: robert@matmor.unam.mx
Causal Fermion Systems:A <strong>Quantum</strong> Space-Time Emerging From anAction PrincipleFelix Finster, Andreas Grotz <strong>and</strong> Daniela SchiefenederAbstract. Causal fermion systems are introduced as a general mathematicalframework for formulating relativistic quantum theory. By specializing,we recover earlier notions like fermion systems in discretespace-time, the fermionic projector <strong>and</strong> causal variational principles.We review how an effect of spontaneous structure formation gives riseto a topology <strong>and</strong> a causal structure in space-time. Moreover, we outlinehow to construct a spin connection <strong>and</strong> curvature, leading to a proposalfor a “quantum geometry” in the Lorentzian setting. We review recentnumerical <strong>and</strong> analytical results on the support of minimizers of causalvariational principles which reveal a “quantization effect” resulting in adiscreteness of space-time. A brief survey is given on the correspondenceto quantum field theory <strong>and</strong> gauge theories.Mathematics Subject Classification (2010). Primary 51P05, 81T20; Secondary49Q20, 83C45, 47B50, 47B07.Keywords. <strong>Quantum</strong> geometry, causal fermion systems, fermionic projectorapproach.1. The general framework of causal fermion systemsCausal fermion systems provide a general mathematical framework for theformulation of relativistic quantum theory. They arise by generalizing <strong>and</strong>harmonizing earlier notions like the “fermionic projector,” “fermion systemsin discrete space-time” <strong>and</strong> “causal variational principles.” After a brief motivationof the basic objects (Section 1.1), we introduce the general framework,trying to work out the mathematical essence from an abstract point of view(Sections 1.2 <strong>and</strong> 1.3). By specializing, we then recover the earlier notions(Section 1.4). Our presentation is intended as a mathematical introduction,which can clearly be supplemented by the more physical introductions in thesurvey articles [9, 12, 13].Supported in part by the Deutsche Forschungsgemeinschaft.F. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_ 9,© Springer Basel AG 2012157
158 F. Finster, A. Grotz <strong>and</strong> D. Schiefeneder1.1. Motivation of the basic objectsIn order to put the general objects into a simple <strong>and</strong> concrete context,we begin with the free Dirac equation in Minkowski space. Thus we let(M,〈., .〉) be Minkowski space (with the signature convention (+−−−)) <strong>and</strong>dμ the st<strong>and</strong>ard volume measure (thus dμ = d 4 x in a reference frame x =(x 0 ,...,x 3 )). We consider a subspace I of the solution space of the Diracequation (iγ j ∂ j − m)ψ =0(I may be finite- or infinite-dimensional). On Iwe introduce a scalar product 〈.|.〉 H . The most natural choice is to take thescalar product associated to the probability integral,∫〈ψ|φ〉 H =2π (ψγ 0 φ)(t, ⃗x) d⃗x (1)t=const(where ψ = ψ † γ 0 is the usual adjoint spinor; note that due to current conservation,the value of the integral is independent of t), but other choices arealso possible. In order not to distract from the main ideas, in this motivationwe disregard technical issues by implicitly assuming that the scalar product〈.|.〉 H is well-defined on I <strong>and</strong> by ignoring the fact that mappings on I maybe defined only on a dense subspace (for details on how to make the followingconsideration rigorous see [14, Section 4]). Forming the completion of I, weobtain a Hilbert space (H, 〈.|.〉 H ).Next, for any x ∈ M we introduce the sesquilinear formb : H × H → C : (ψ, φ) ↦→ −(ψφ)(x) . (2)As the inner product ψφ on the Dirac spinors is indefinite of signature (2, 2),the sesquilinear form b has signature (p, q) with p, q ≤ 2. Thus we mayuniquely represent it asb(ψ, φ) =〈ψ|Fφ〉 H (3)with a self-adjoint operator F ∈ L(H) of finite rank, which (counting withmultiplicities) has at most two positive <strong>and</strong> at most two negative eigenvalues.Introducing this operator for every x ∈ M, we obtain a mappingF : M → F , (4)where F ⊂ L(H) denotes the set of all self-adjoint operators of finite rankwith at most two positive <strong>and</strong> at most two negative eigenvalues, equippedwith the topology induced by the Banach space L(H).It is convenient to simplify this setting in the following way. In mostphysical applications, the mapping F will be injective with a closed image.Then we can identify M with the subset F (M) ⊂ F. Likewise, we can identifythe measure μ with the push-forward measure ρ = F ∗ μ on F (M) (defined byρ(Ω) = μ(F −1 (Ω))). The measure ρ is defined even on all of F, <strong>and</strong> the imageof F coincides with the support of ρ. Thus setting M = supp ρ, we can reconstructspace-time from ρ. This construction allows us to describe the physicalsystem by a single object: the measure ρ on F. Moreover, we can extend ournotion of space-time simply by allowing ρ to be a more general measure (i.e.a measure which can no longer be realized as the push-forward ρ = F ∗ μ ofthe volume measure in Minkowski space by a continuous function F ).
Causal Fermion Systems 159We have the situation in mind when I is composed of all the occupiedfermionic states of a physical system, including the states of the Dirac sea(for a physical discussion see [13]). In this situation, the causal structure isencoded in the spectrum of the operator product F (x)·F (y). In the remainderof this section, we explain how this works for the vacuum. Thus let us assumethat I is the subspace of all negative-energy solutions of the Dirac equation.We first compute F .Lemma 1.1. Let ψ, φ be two smooth negative-energy solutions of the free Diracequation. We set( )F (y) φ (x) =P (x, y) φ(y) ,where P (x, y) is the distribution∫P (x, y) =d 4 k(2π) (k/ + m) 4 δ(k2 − m 2 )Θ(−k 0 ) e −ik(x−y) . (5)Then the equation〈ψ|F (y) φ〉 H = −(ψφ)(y)holds, where all integrals are to be understood in the distributional sense.Proof. We can clearly assume that ψ is a plane-wave solution, which forconvenience we write asψ(x) =(q/ + m) χe −iqx , (6)where q =(q 0 ,⃗q) with⃗q ∈ R 3 <strong>and</strong> q 0 = − √ |⃗q| 2 + m 2 is a momentum on thelower mass shell <strong>and</strong> χ is a constant spinor. A straightforward calculationyields∫(1)〈ψ|F (y) φ〉 H =2π d⃗x χ (q/ + m) e iqx γ 0 P (x, y) φ(y)R∫3(5)= d 4 kδ 3 ( ⃗ k − ⃗q) χ (q/ + m)γ 0 (k/ + m) δ(k 2 − m 2 )Θ(−k 0 ) e iky φ(y)= 12|q 0 | χ (q/ + m)γ0 (q/ + m) e iqy φ(y)(∗)= −χ (q/ + m) e iqy φ(y) =−(ψφ)(y) ,where in (∗) we used the anti-commutation relations of the Dirac matrices.□This lemma gives an explicit solution to (3) <strong>and</strong> (2). The fact that F (y)φis merely a distribution shows that an ultraviolet regularization is needed inorder for F (y) to be a well-defined operator on H. We will come back to thistechnical point after (41) <strong>and</strong> refer to [14, Section 4] for details. For clarity,we now proceed simply by computing the eigenvalues of the operator productF (y)·F (x) formally (indeed, the following calculation is mathematically rigorousexcept if y lies on the boundary of the light cone centered at x, inwhichcase the expressions become singular). First of all, as the operators F (y) <strong>and</strong>F (x) have rank at most four, we know that their product F (y)·F (x) also
160 F. Finster, A. Grotz <strong>and</strong> D. Schiefenederhas at most four non-trivial eigenvalues, which counting with algebraic multiplicitieswe denote by λ 1 ...,λ 4 . Since this operator product is self-adjointonly if the factors F (x) <strong>and</strong>F (y) commute, the eigenvalues λ 1 ,...,λ 4 will ingeneral be complex. By iterating Lemma 1.1, we find that for any n ≥ 0,(F (x) ( F (y) F (x) ) nφ)(z) =P (z,x)(P (x, y) P (y, x)) nφ(x).Forming a characteristic polynomial, one sees that the non-trivial eigenvaluesof F (y)·F (x) coincide precisely with the eigenvalues of the (4×4)-matrix A xydefined byA xy = P (x, y) P (y, x) .ThequalitativepropertiesoftheeigenvaluesofA xy can easily be determinedeven without computing the Fourier integral (5): From Lorentz symmetry,we know that for all x <strong>and</strong> y for which the Fourier integral exists, P (x, y)can be written asP (x, y) = α (y − x) j γ j + β 1 (7)with two complex coefficients α <strong>and</strong> β. Taking the conjugate, we see thatAs a consequence,P (y, x) = α (y − x) j γ j + β 1 .A xy = P (x, y) P (y, x) = a (y − x) j γ j + b 1 (8)with two real parameters a <strong>and</strong> b given bya = αβ + βα, b = |α| 2 (y − x) 2 + |β| 2 . (9)Applying the formula (A xy − b1) 2 = a 2 (y − x) 2 1, we find that the roots ofthe characteristic polynomial of A xy are given byb ± √ a 2 (y − x) 2 .Thus if the vector (y − x) is timelike, the term (y − x) 2 is positive, so thatthe λ j are all real. If conversely the vector (y − x) is spacelike, the term(y − x) 2 is negative, <strong>and</strong> the λ j form a complex conjugate pair. We concludethat the the causal structure of Minkowski space has the following spectralcorrespondence:The non-trivial eigenvalues of F (x)·F (y){}{ }are realtimelikeif x <strong>and</strong> y areseparated.form a complex conjugate pairspacelike(10)1.2. Causal fermion systemsCausal fermion systems have two formulations, referred to as the particle<strong>and</strong> the space-time representation. We now introduce both formulations <strong>and</strong>explain their relation. After that, we introduce the setting of the fermionicprojector as a special case.
Causal Fermion Systems 1611.2.1. From the particle to the space-time representation.Definition 1.2. Given a complex Hilbert space (H, 〈.|.〉 H ) (the particle space)<strong>and</strong> a parameter n ∈ N (the spin dimension), we let F ⊂ L(H) be the set of allself-adjoint operators on H of finite rank which (counting with multiplicities)have at most n positive <strong>and</strong> at most n negative eigenvalues. On F we aregiven a positive measure ρ (defined on a σ-algebra of subsets of F), the socalleduniversal measure. We refer to (H, F,ρ)asacausal fermion system inthe particle representation.Vectors in the particle space have the interpretation as the occupiedfermionic states of our system. The name “universal measure” is motivatedby the fact that ρ describes the distribution of the fermions in a space-time“universe”, with causal relations defined as follows.Definition 1.3. (causal structure) For any x, y ∈ F, the product xy is anoperator of rank at most 2n. We denote its non-trivial eigenvalues (countingwith algebraic multiplicities) by λ xy1 ,...,λxy 2n .Thepointsx <strong>and</strong> y are calledtimelike separated if the λ xyjare all real. They are said to be spacelike separatedif all the λ xyj are complex <strong>and</strong> have the same absolute value. In allother cases, the points x <strong>and</strong> y are said to be lightlike separated.Since the operators xy <strong>and</strong> yx are isospectral (this follows from thematrix identity det(BC−λ1) = det(CB−λ1); see for example [7, Section 3]),this definition is symmetric in x <strong>and</strong> y.We now construct additional objects, leading us to the more familiarspace-time representation. First, on F we consider the topology induced bythe operator norm ‖A‖ := sup{‖Au‖ H with ‖u‖ H =1}. For every x ∈ F wedefine the spin space S x by S x = x(H); it is a subspace of H of dimension atmost 2n. OnS x we introduce the spin scalar product ≺.|.≻ x by≺u|v≻ x = −〈u|xu〉 H (for all u, v ∈ S x ) ; (11)it is an indefinite inner product of signature (p, q) withp, q ≤ n. Wedefinespace-time M as the support of the universal measure, M =suppρ. Itisaclosed subset of F, <strong>and</strong> by restricting the causal structure of F to M, wegetcausal relations in space-time. A wave function ψ is defined as a functionwhich to every x ∈ M associates a vector of the corresponding spin space,ψ : M → H with ψ(x) ∈ S x for all x ∈ M. (12)On the wave functions we introduce the indefinite inner product∫ = ≺ψ(x)|φ(x)≻ x dρ(x) . (13)MIn order to ensure that the last integral converges, we also introduce the norm||| . ||| by∫||| ψ ||| 2 〈 ∣= ψ(x) |x| ψ(x) 〉 dρ(x) (14)HM(where |x| is the absolute value of the operator x on H). The one-particlespace K is defined as the space of wave functions for which the norm ||| . |||
162 F. Finster, A. Grotz <strong>and</strong> D. Schiefenederis finite, with the topology induced by this norm, <strong>and</strong> endowed with theinner product . Then(K,) is a Krein space (see [2]). Next, for anyx, y ∈ M we define the kernel of the fermionic operator P (x, y) byP (x, y) =π x y : S y → S x , (15)where π x is the orthogonal projection onto the subspace S x ⊂ H. Theclosedchain is defined as the productA xy = P (x, y) P (y, x) : S x → S x .As it is an endomorphism of S x , we can compute its eigenvalues. The calculationA xy =(π x y)(π y x)=π x yx shows that these eigenvalues coincideprecisely with the non-trivial eigenvalues λ xy1 ,...,λxy 2n of the operator xy asconsidered in Definition 1.3. In this way, the kernel of the fermionic operatorencodes the causal structure of M. Choosing a suitable dense domain of definition1 D(P ), we can regard P (x, y) as the integral kernel of a correspondingoperator P ,∫P : D(P ) ⊂ K → K , (Pψ)(x) = P (x, y) ψ(y) dρ(y) , (16)referred to as the fermionic operator. We collect two properties of the fermionicoperator:(A) P is symmetric in the sense that = for all ψ, φ ∈D(P ):According to the definitions (15) <strong>and</strong> (11),≺P (x, y) ψ(y) | ψ(x)≻ x = −〈(π x yψ(y)) | xφ(x)〉 HM= −〈ψ(y) | yx φ(x)〉 H = ≺ψ(y) | P (y, x) ψ(x)≻ y .We now integrate over x <strong>and</strong> y <strong>and</strong> apply (16) <strong>and</strong> (13).(B) (−P )ispositive in the sense that ≥ 0 for all ψ ∈ D(P ):This follows immediately from the calculation∫∫ = − ≺ψ(x) | P (x, y) ψ(y)≻ x dρ(x) dρ(y)M×M∫∫= 〈ψ(x) | xπ x yψ(y)〉 H dρ(x) dρ(y) =〈φ|φ〉 H ≥ 0 ,M×Mwhere we again used (13) <strong>and</strong> (15) <strong>and</strong> set∫φ = xψ(x) dρ(x) .M1 For example, one may choose D(P ) as the set of all vectors ψ ∈ K satisfying the conditions∫φ := xψ(x) dρ(x) ∈ H <strong>and</strong> ||| φ ||| < ∞ .M
Causal Fermion Systems 163The space-time representation of the causal fermion system consists of theKrein space (K,), whose vectors are represented as functions on M (see(12), (13)), together with the fermionic operator P in the integral representation(16) with the above properties (A) <strong>and</strong> (B).Before going on, it is instructive to consider the symmetries of our framework.First of all, unitary transformationsψ → Uψ with U ∈ L(H) unitarygive rise to isomorphic systems. This symmetry corresponds to the fact thatthe fermions are indistinguishable particles (for details see [5, §3.2] <strong>and</strong> [11,Section 3]). Another symmetry becomes apparent if we choose basis representationsof the spin spaces <strong>and</strong> write the wave functions in components.Denoting the signature of (S x , ≺.|.≻ x )by(p(x),q(x)), we choose a pseudoorthonormalbasis (e α (x)) α=1,...,p+q of S x ,≺e α |e β ≻ = s α δ αβ with s 1 ,...,s p =1, s p+1 ,...,s p+q = −1 .Then a wave function ψ ∈ K can be represented asp+q∑ψ(x) = ψ α (x) e α (x)α=1with component functions ψ 1 ,...,ψ p+q . The freedom in choosing the basis(e α ) is described by the group U(p, q) of unitary transformations with respectto an inner product of signature (p, q),p+q∑e α → (U −1 ) β α e β with U ∈ U(p, q) . (17)β=1As the basis (e α ) can be chosen independently at each space-time point, thisgives rise to local unitary transformations of the wave functions,∑p+qψ α (x) → U(x) α β ψ β (x) . (18)β=1These transformations can be interpreted as local gauge transformations (seealso Section 5). Thus in our framework, the gauge group is the isometrygroup of the spin scalar product; it is a non-compact group whenever thespin scalar product is indefinite. Gauge invariance is incorporated in ourframework simply because the basic definitions are basis-independent.The fact that we have a distinguished representation of the wave functionsas functions on M can be expressed by the space-time projectors, definedas the operators of multiplication by a characteristic function. Thus for anymeasurable Ω ⊂ M, we define the space-time projector E Ω byE Ω : K → K , (E Ω ψ)(x) =χ Ω (x) ψ(x) .Obviously, the space-time projectors satisfy the relationsE U E V = E U∩V , E U + E V = E U∪V + E U∩V , E M =1 K , (19)
164 F. Finster, A. Grotz <strong>and</strong> D. Schiefenederwhich are familiar in functional analysis as the relations which characterizespectral projectors. We can now take the measure space (M,ρ) <strong>and</strong>theKreinspace (K,) together with the fermionic operator <strong>and</strong> the space-timeprojectors as the abstract starting point.1.2.2. From the space-time to the particle representation.Definition 1.4. Let (M,ρ) be a measure space (“space-time”) <strong>and</strong>(K,)a Krein space (the “one-particle space”). Furthermore, we let P : D(P ) ⊂K → K be an operator with dense domain of definition D(P ) (the “fermionicoperator”), such that P is symmetric <strong>and</strong> (−P ) is positive (see (A) <strong>and</strong> (B)on page 162). Moreover, to every ρ-measurable set Ω ⊂ M we associate aprojector E Ω onto a closed subspace E Ω (K) ⊂ K, such that the resultingfamily of operators (E Ω ) (the “space-time projectors”) satisfies the relations(19). We refer to (M,ρ) together with (K,, E Ω ,P)asacausal fermionsystem in the space-time representation.This definition is more general than the previous setting because it doesnot involve a notion of spin dimension. Before one can introduce this notion,one needs to “localize” the vectors in K with the help of the space-time projectorsto obtain wave functions on M. Ifρ were a discrete measure, this localizationcould be obtained by considering the vectors E x ψ with x ∈ supp ρ.If ψ could be expected to be a continuous function, we could consider thevectors E Ωn ψ for Ω n a decreasing sequence of neighborhoods of a single point.In the general setting of Definition 1.4, however, we must use a functionalanalytic construction, which in the easier Hilbert space setting was workedout in [4]. We now sketch how the essential parts of the construction can becarried over to Krein spaces. First, we need some technical assumptions.Definition 1.5. A causal fermion system in the space-time representation hasspin dimension at most n if there are vectors ψ 1 ,...,ψ 2n ∈ K with thefollowing properties:(i) For every measurable set Ω, the matrix S with components S ij = has at most n positive <strong>and</strong> at most n negative eigenvalues.(ii) The set{E Ω ψ k with Ω measurable <strong>and</strong> k =1,...,2n} (20)generates a dense subset of K.(iii) For all j, k ∈{1,...,2n}, the mappingμ jk :Ω→ defines a complex measure on M which is absolutely continuous withrespect to ρ.This definition allows us to use the following construction. In order tointroduce the spin scalar product between the vectors ψ 1 ,...ψ 2n ,weuse
Causal Fermion Systems 165property (iii) to form the Radon-Nikodym decomposition∫ = ≺ψ j |ψ k ≻ x dρ(x) with ≺ψ j |ψ k ≻∈L 1 (M,dρ) ,Ωvalid for any measurable set Ω ⊂ M. Setting≺E U ψ j |E V ψ k ≻ x = χ U (x) χ V (x) ≺ψ j |ψ k ≻ x ,we can extend the spin scalar product to the sets (20). Property (ii) allowsus to extend the spin scalar product by approximation to all of K. Property(i) ensures that the spin scalar product has the signature (p, q) withp, q ≤ n.Having introduced the spin scalar product, we can now get a simpleconnection to the particle representation: The range of the fermionic operatorI := P (D(P )) is a (not necessarily closed) subspace of K. By〈P (φ) | P (φ ′ )〉 := we introduce on I an inner product 〈.|.〉, which by the positivity property(B) is positive semi-definite. Thus its abstract completion H := I is a Hilbertspace (H, 〈.|.〉 H ). We again let F ⊂ L(H) be the set of all self-adjoint operatorsof finite rank which have at most n positive <strong>and</strong> at most n negativeeigenvalues. For any x ∈ M, the conditions〈ψ|Fφ〉 H = −≺ψ|φ≻ x for all ψ, φ ∈ I (21)uniquely define a self-adjoint operator F on I, which has finite rank <strong>and</strong>at most n positive <strong>and</strong> at most n negative eigenvalues. By continuity, thisoperator uniquely extends to an operator F ∈ F, referred to as the localcorrelation operator at x. We thus obtain a mapping F : M → F. Identifyingpoints of M which have the same image (see the discussion below), we canalso consider the subset F(M) ofF as our space-time. Replacing M by F (M)<strong>and</strong> ρ by the push-forward measure F ∗ ρ on F, we get back to the setting ofDefinition 1.2.We point out that, despite the fact that the particle <strong>and</strong> space-timerepresentations can be constructed from each other, the two representationsare not equivalent. The reason is that the construction of the particle representationinvolves an identification of points of M which have the samelocal correlation operators. Thus it is possible that two causal fermion systemsin the space-time representation which are not gauge-equivalent mayhave the same particle representation 2 . In this case, the two systems haveidentical causal structures <strong>and</strong> give rise to exactly the same densities <strong>and</strong>correlation functions. In other words, the two systems are indistinguishableby any measurements, <strong>and</strong> thus one can take the point of view that they areequivalent descriptions of the same physical system. Moreover, the particle2 As a simple example consider the case M = {0, 1} with ρ the counting measure, K = C 4with = 〈ψ, Sφ〉 C 4 <strong>and</strong> the signature matrix S = diag(1, −1, 1, −1). Moreover, wechoose the space-time projectors as E 1 = diag(1, 1, 0, 0), E 2 = diag(0, 0, 1, 1) <strong>and</strong> considera one-particle fermionic operator P = −|ψ>
166 F. Finster, A. Grotz <strong>and</strong> D. Schiefenederrepresentation gives a cleaner framework, without the need for technical assumptionsas in Definition 1.5. For these reasons, it seems preferable to takethe point of view that the particle representation is more fundamental, <strong>and</strong>to always deduce the space-time representation by the constructions givenafter Definition 1.2.1.2.3. The setting of the fermionic projector. A particularly appealing specialcase is the setting of the fermionic projector, which we now review. Beginningin the particle representation, we impose the additional constraint∫xdρ(x) =1 H , (22)Mwhere the integral is assumed to converge in the strong sense, meaning that∫‖xψ‖ dρ(x) < ∞ for all ψ ∈ HM(where ‖ψ‖ = √ 〈ψ|ψ〉 H is the norm on H). Under these assumptions, it isstraightforward to verify from (14) that the mappingι : H → K ,(ιψ)(x) =π x ψis well-defined. Moreover, the calculation∫∫ = ≺π x ψ | π x φ≻ x dρ(x) (11)= −MM〈ψ | xφ〉 H dρ(x) (22)= −〈ψ|φ〉 Hshows that ι is, up to a minus sign, an isometric embedding of H into K.Thus we may identify H with the subspace ι(H) ⊂ K, <strong>and</strong> on this closedsubspace the inner products ≺.|.≻ H <strong>and</strong> | H×H coincide up to a sign.Moreover, the calculation∫∫(Pιψ)(x) = π x yπ y ψdρ(y) = π x yψdρ(y) =π x ψ =(ιψ)(x)Myields that P restricted to H is the identity. Next, for every ψ ∈ D(P ), theestimate∫∫ ∫∫∥ yψ(y) dρ(y) ∥ 2 (22)= dρ(x) dρ(y) dρ(z) 〈yψ(y) | xz ψ(z)〉 HMMM×M= − < ∞shows (after a straightforward approximation argument) that∫φ := yψ(y) dρ(y) ∈ H .MOn the other h<strong>and</strong>, we know from (16) <strong>and</strong> (15) that Pψ = ιφ. This showsthat the image of P is contained in H. We conclude that P is a projectionoperator in K onto the negative definite, closed subspace H ⊂ K.M
Causal Fermion Systems 1671.3. An action principleWe now return to the general setting of Definitions 1.2 <strong>and</strong> 1.3. For twopoints x, y ∈ F we define the spectral weight |.| of the operator products xy<strong>and</strong> (xy) 2 by|xy| =2n∑i=1|λ xyi | <strong>and</strong>∣∣(xy) 2∣ ∣ =2n∑i=1|λ xyi | 2 .We also introduce theLagrangian L(x, y) = ∣ ∣(xy) 2∣ ∣ − 12n |xy|2 . (23)For a given universal measure ρ on F, we define the non-negative functionals∫∫action S[ρ] = L(x, y) dρ(x) dρ(y) (24)F×F∫∫constraint T [ρ] = |xy| 2 dρ(x) dρ(y) . (25)Our action principle is toF×Fminimize S for fixed T , (26)under variations of the universal measure. These variations should keep thetotal volume unchanged, which means that a variation (ρ(τ)) τ∈(−ε,ε) shouldfor all τ,τ ′ ∈ (−ε, ε) satisfy the conditions∣∣ρ(τ) − ρ(τ ′ ) ∣ (∣(F) < ∞ <strong>and</strong> ρ(τ) − ρ(τ ′ ) ) (F) =0(where |.| denotes the total variation of a measure; see [16, §28]). Dependingon the application, one may impose additional constraints. For example, inthe setting of the fermionic projector, the variations should obey the condition(22). Moreover, one may prescribe properties of the universal measureby choosing a measure space ( ˆM, ˆμ) <strong>and</strong> restricting attention to universalmeasures which can be represented as the push-forward of ˆμ,ρ = F ∗ ˆμ with F : ˆM → F measurable . (27)One then minimizes the action under variations of the mapping F .The Lagrangian (23) is compatible with our notion of causality in thefollowing sense. Suppose that two points x, y ∈ F are spacelike separated (seeDefinition 1.3). Then the eigenvalues λ xyi all have the same absolute value,so that the Lagrangian (23) vanishes. Thus pairs of points with spacelikeseparation do not enter the action. This can be seen in analogy to the usualnotion of causality where points with spacelike separation cannot influenceeach other.
168 F. Finster, A. Grotz <strong>and</strong> D. Schiefeneder1.4. Special casesWe now discuss modifications <strong>and</strong> special cases of the above setting as consideredearlier. First of all, in all previous papers except for [14] it was assumedthat the Hilbert space H is finite-dimensional <strong>and</strong> that the measure ρ isfinite. Then by rescaling, one can normalize ρ such that ρ(M) = 1. Moreover,the Hilbert space (H, 〈.|.〉 H ) can be replaced by C f with the canonicalscalar product (the parameter f ∈ N has the interpretation as the numberof particles of the system). These two simplifications lead to the setting ofcausal variational principles introduced in [10]. More precisely, the particle<strong>and</strong> space-time representations are considered in [10, Section 1 <strong>and</strong> 2] <strong>and</strong>[10, Section 3], respectively. The connection between the two representationsis established in [10, Section 3] by considering the relation (21) in a matrixrepresentation. In this context, F is referred to as the local correlation matrixat x. Moreover, in [10] the universal measure is mainly represented asin (27) as the push-forward by a mapping F . This procedure is of advantagewhen analyzing the variational principle, because by varying F while keeping( ˆM, ˆμ) fixed, one can prescribe properties of the measure ρ. For example, ifˆμ is a counting measure, then one varies ρ in the restricted class of measureswhose support consists of at most # ˆM points with integer weights. Moregenerally, if ˆμ is a discrete measure, then ρ is also discrete. However, if ˆμ is acontinuous measure (or in more technical terms a so-called non-atomic measure),then we do not get any constraints for ρ, so that varying F is equivalentto varying ρ in the class of positive normalized regular Borel measures.Another setting is to begin in the space-time representation (see Definition1.4), but assuming that ρ is a finite counting measure. Then the relations(19) becomeE x E y = δ xy E x<strong>and</strong>∑E x =1 K ,x∈Mwhereas the “localization” discussed after (19) reduces to multiplication withthe space-time projectors,ψ(x) =E x ψ, P(x, y) =E x PE y , ≺ψ(x) | φ(x)≻ x = .This is the setting of fermion systems in discrete space-time as consideredin [5] <strong>and</strong> [7, 6]. We point out that all the work before 2006 deals with thespace-time representation, simply because the particle representation had notyet been found.We finally remark that, in contrast to the settings considered previously,here the dimension <strong>and</strong> signature of the spin space S x may depend on x. Weonly know that it is finite-dimensional, <strong>and</strong> that its positive <strong>and</strong> negativesignatures are at most n. In order to get into the setting of constant spindimension, one can isometrically embed every S x into an indefinite innerproduct space of signature (n, n) (for details see [10, Section 3.3]).
2. Spontaneous structure formationCausal Fermion Systems 169For a given measure ρ, the structures of F induce corresponding structureson space-time M = supp ρ ⊂ F. Two of these structures are obvious: First,on M we have the relative topology inherited from F. Second, the causalstructure on F (see Definition 1.3) also induces a causal structure on M.Additional structures like a spin connection <strong>and</strong> curvature are less evident;their construction will be outlined in Section 3 below.The appearance of the above structures in space-time can also be understoodas an effect of spontaneous structure formation driven by our actionprinciple. We now explain this effect in the particle representation (for a discussionin the space-time representation see [12]). For clarity, we consider thesituation (27) where the universal measure is represented as the push-forwardof a given measure ˆμ on ˆM (this is no loss of generality because choosing anon-atomic measure space ( ˆM, ˆμ), any measure ρ on F can be represented inthis way; see [10, Lemma 1.4]). Thus our starting point is a measure space( ˆM, ˆμ), without any additional structures. The symmetries are described bythe group of mappings T of the formT : ˆM → ˆM is bijective <strong>and</strong> preserves the measure ˆμ. (28)We now consider measurable mappings F : ˆM → F <strong>and</strong> minimize S undervariations of F . The resulting minimizer gives rise to a measure ρ = F ∗ ˆμ on F.On M := supp ρ we then have the above structures inherited from F. Takingthe pull-back by F , we get corresponding structures on ˆM. The symmetrygroup reduces to the mappings T which in addition to (28) preserve thesestructures. In this way, minimizing our action principle triggers an effect ofspontaneous symmetry breaking, leading to additional structures in spacetime.3. A Lorentzian quantum geometryWe now outline constructions from [14] which give general notions of a connection<strong>and</strong> curvature (see Theorem 3.9, Definition 3.11 <strong>and</strong> Definition 3.12).We also explain how these notions correspond to the usual objects of differentialgeometry in Minkowski space (Theorem 3.15) <strong>and</strong> on a globally hyperbolicLorentzian manifold (Theorem 3.16).3.1. Construction of the spin connectionHaving Dirac spinors in a four-dimensional space-time in mind, we consideras in Section 1.1 a causal fermion system of spin dimension two. Moreover,we only consider space-time points x ∈ M which are regular in the sense thatthe corresponding spin spaces S x have the maximal dimension four.An important structure from spin geometry missing so far is Cliffordmultiplication. To this end, we need a Clifford algebra represented by symmetricoperators on S x . For convenience, we first consider Clifford algebras
170 F. Finster, A. Grotz <strong>and</strong> D. Schiefenederwith the maximal number of five generators; later we reduce to four spacetimedimensions (see Definition 3.14 below). We denote the set of symmetriclinear endomorphisms of S x by Symm(S x ); it is a 16-dimensional real vectorspace.Definition 3.1. A five-dimensional subspace K ⊂ Symm(S x ) is called a Cliffordsubspace if the following conditions hold:(i) For any u, v ∈ K, the anti-commutator {u, v} ≡uv + vu is a multipleof the identity on S x .(ii) The bilinear form 〈., .〉 on K defined by1{u, v} = 〈u, v〉 1 for all u, v ∈ K (29)2is non-degenerate <strong>and</strong> has signature (1, 4).In view of the situation in spin geometry, we would like to distinguisha specific Clifford subspace. In order to partially fix the freedom in choosingClifford subspaces, it is useful to impose that K should contain a given socalledsign operator.Definition 3.2. An operator v ∈ Symm(S x ) is called a sign operator if v 2 =1<strong>and</strong> if the inner product ≺.|v.≻ : S x × S x → C is positive definite.Definition 3.3. For a given sign operator v, the set of Clifford extensions T vis defined as the set of all Clifford subspaces containing v,T v = {K Clifford subspace with v ∈ K} .Considering x as an operator on S x , this operator has by definition ofthe spin dimension two positive <strong>and</strong> two negative eigenvalues. Moreover, thecalculation(11)≺u|(−x) u≻ x = 〈u|x 2 u〉 H > 0 for all u ∈ S x \{0}shows that the operator (−x) is positive definite on S x . Thus we can introducea unique sign operator s x by dem<strong>and</strong>ing that the eigenspaces of s xcorresponding to the eigenvalues ±1 are precisely the positive <strong>and</strong> negativespectral subspaces of the operator (−x). This sign operator is referred to asthe Euclidean sign operator.A straightforward calculation shows that for two Clifford extensionsK, ˜K ∈ T v , there is a unitary transformation U ∈ e iRv such that ˜K = UKU −1(for details see [14, Section 3]). By dividing out this group action, we obtaina five-dimensional vector space, endowed with the inner product 〈., 〉. Takingfor v the Euclidean signature operator, we regard this vector space as ageneralization of the usual tangent space.Definition 3.4. The tangent space T x is defined byT x = T s xx / exp(iRs x ) .It is endowed with an inner product 〈., .〉 of signature (1, 4).
Causal Fermion Systems 171We next consider two space-time points, for which we need to make thefollowing assumption.Definition 3.5. Two points x, y ∈ M are said to be properly time-like separatedif the closed chain A xy has a strictly positive spectrum <strong>and</strong> if thecorresponding eigenspaces are definite subspaces of S x .This definition clearly implies that x <strong>and</strong> y are time-like separated (seeDefinition 1.3). Moreover, the eigenspaces of A xy are definite if <strong>and</strong> only ifthose of A yx are, showing that Definition 3.5 is again symmetric in x <strong>and</strong>y. As a consequence, the spin space can be decomposed uniquely into anorthogonal direct sum S x = I + ⊕ I − of a positive definite subspace I + <strong>and</strong>a negative definite subspace I − of A xy . This allows us to introduce a uniquesign operator v xy by dem<strong>and</strong>ing that its eigenspaces corresponding to theeigenvalues ±1 are the subspaces I ± . This sign operator is referred to asthe directional sign operator of A xy . Having two sign operators s x <strong>and</strong> v xyat our disposal, we can distinguish unique corresponding Clifford extensions,provided that the two sign operators satisfy the following generic condition.Definition 3.6. Two sign operators v, ṽ are said to be generically separated iftheir commutator [v, ṽ] has rank four.Lemma 3.7. Assume that the sign operators s x <strong>and</strong> v xy are generically separated.Then there are unique Clifford extensions K x (y) ∈ T s x<strong>and</strong> K xy ∈ T v xy<strong>and</strong> a unique operator ρ ∈ K x(y) ∩ K xy with the following properties:(i) The relations {s x ,ρ} =0={v xy ,ρ} hold.(ii) The operator U xy := e iρ transforms one Clifford extension to the other,K xy = U xy K x(y) Uxy −1 . (30)(iii) If {s x ,v xy } is a multiple of the identity, then ρ =0.The operator ρ depends continuously on s x <strong>and</strong> v xy .We refer to U xy as the synchronization map. Exchanging the roles of x<strong>and</strong> y, we also have two sign operators s y <strong>and</strong> v yx at the point y. Assumingthat these sign operators are again generically separated, we also obtain aunique Clifford extension K yx ∈ T v yx.After these preparations, we can now explain the construction of thespin connection D (for details see [14, Section 3]). For two space-time pointsx, y ∈ M with the above properties, we want to introduce an operatorD x,y : S y → S x (31)(generally speaking, by the subscript xy we always denote an object at thepoint x, whereas the additional comma x,y denotes an operator which maps anobject at y to an object at x). It is natural to dem<strong>and</strong> that D x,y is unitary,that D y,x is its inverse, <strong>and</strong> that these operators map the directional sign
172 F. Finster, A. Grotz <strong>and</strong> D. Schiefenederoperators at x <strong>and</strong> y to each other,D x,y =(D y,x ) ∗ =(D y,x ) −1 (32)v xy = D x,y v yx D y,x . (33)The obvious idea for constructing an operator with these properties is to takea polar decomposition of P (x, y); this amounts to settingD x,y = A − 1 2xy P (x, y) . (34)This definition has the shortcoming that it is not compatible with the chosenClifford extensions. In particular, it does not give rise to a connection on thecorresponding tangent spaces. In order to resolve this problem, we modify(34) by the ansatzD x,y = e iϕ xy v xyA − 1 2xy P (x, y) (35)with a free real parameter ϕ xy . In order to comply with (32), we need todem<strong>and</strong> thatϕ xy = −ϕ yx mod 2π ; (36)then (33) is again satisfied. We can now use the freedom in choosing ϕ xy toarrange that the distinguished Clifford subspaces K xy <strong>and</strong> K yx are mappedonto each other,K xy = D x,y K yx D y,x . (37)It turns out that this condition determines ϕ xy up to multiples of π 2 .Inorderto fix ϕ xy uniquely in agreement with (36), we need to assume that ϕ xy isnot a multiple of π 4. This leads us to the following definition.Definition 3.8. Two points x, y ∈ M are called spin-connectable if the followingconditions hold:(a) The points x <strong>and</strong> y are properly timelike separated (note that this alreadyimplies that x <strong>and</strong> y are regular as defined at the beginning ofSection 3).(b) The Euclidean sign operators s x <strong>and</strong> s y are generically separated fromthe directional sign operators v xy <strong>and</strong> v yx , respectively.(c) Employing the ansatz (35), the phases ϕ xy which satisfy condition (37)are not multiples of π 4 .We denote the set of points which are spin-connectable to x by I(x). Itis straightforward to verify that I(x) isanopensubsetofM.Under these assumptions, we can fix ϕ xy uniquely by imposing thatϕ xy ∈(− π 2 , −π 4)∪( π4 , π 2), (38)giving the following result (for the proofs see [14, Section 3.3]).Theorem 3.9. Assume that two points x, y ∈ M are spin-connectable. Thenthere is a unique spin connection D x,y : S y → S x of the form (35) having theproperties (32), (33), (37) <strong>and</strong> (38).
Causal Fermion Systems 1733.2. A time direction, the metric connection <strong>and</strong> curvatureWe now outline a few further constructions from [14, Section 3]. First, forspin-connectable points we can distinguish a direction of time.Definition 3.10. Assume that the points x, y ∈ M are spin-connectable. Wesay that y lies in the future of x if the phase ϕ xy as defined by (35) <strong>and</strong> (38)is positive. Otherwise, y is said to lie in the past of x.According to (36), y lies in the future of x if <strong>and</strong> only if x lies in thepast of y. By distinguishing a direction of time, we get a structure similar toa causal set (see for example [3]). However, in contrast to a causal set, ournotion of “lies in the future of” is not necessarily transitive.The spin connection induces a connection on the corresponding tangentspaces, as we now explain. Suppose that u y ∈ T y . Then, according to Definition3.4 <strong>and</strong> Lemma 3.7, we can consider u y as a vector of the representativeK y (x) ∈ T s y. By applying the synchronization map, we obtain a vector in K yx ,u yx := U yx u y Uyx −1 ∈ K yx .According to (37), we can now “parallel-transport” the vector to the Cliffordsubspace K xy ,u xy := D x,y u yx D y,x ∈ K xy .Finally, we apply the inverse of the synchronization map to obtain the vectoru x := Uxy −1 u xy U xy ∈ K x (y) .As K x(y) is a representative of the tangent space T x <strong>and</strong> all transformationswere unitary, we obtain an isometry from T y to T x .Definition 3.11. The isometry between the tangent spaces defined by∇ x,y : T y → T x : u y ↦→ u xis referred to as the metric connection corresponding to the spin connectionD.We next introduce a notion of curvature.Definition 3.12. Suppose that three points x, y, z ∈ M are pairwise spinconnectable.Then the associated metric curvature R is defined byR(x, y, z) =∇ x,y ∇ y,z ∇ z,x : T x → T x . (39)ThemetriccurvatureR(x, y, z) can be thought of as a discrete analog ofthe holonomy of the Levi-Civita connection on a manifold, where a tangentvector is parallel-transported along a loop starting <strong>and</strong> ending at x. Onamanifold, the curvature at x is immediately obtained from the holonomyby considering the loops in a small neighborhood of x. With this in mind,Definition 3.12 indeed generalizes the usual notion of curvature to causalfermion systems.
174 F. Finster, A. Grotz <strong>and</strong> D. SchiefenederThe following construction relates directional sign operators to vectorsof the tangent space. Suppose that y is spin-connectable to x. Bysynchronizingthe directional sign operator v xy , we obtain the vectorŷ x := U −1xy v xy U xy ∈ K (y)x . (40)As K (y)x ∈ T s xis a representative of the tangent space, we can regard ŷ x asa tangent vector. We thus obtain a mappingI(x) → T x : y ↦→ ŷ x .We refer to ŷ x as the directional tangent vector of y in T x .Asv xy is a signoperator <strong>and</strong> the transformations in (40) are unitary, the directional tangentvector is a timelike unit vector with the additional property that the innerproduct ≺.|ŷ x .≻ x is positive definite.We finally explain how to reduce the dimension of the tangent space tofour, with the desired Lorentzian signature (1, 3).Definition 3.13. The fermion system is called chirally symmetric if to everyx ∈ M we can associate a spacelike vector u(x) ∈ T x which is orthogonal toall directional tangent vectors,〈u(x), ŷ x 〉 =0 forally ∈I(x) ,<strong>and</strong> is parallel with respect to the metric connection, i.e.u(x) =∇ x,y u(y) ∇ y,x for all y ∈I(x) .Definition 3.14. For a chirally symmetric fermion system, we introduce thereduced tangent space Txred byTx red = 〈u x 〉 ⊥ ⊂ T x .Clearly, the reduced tangent space has dimension four <strong>and</strong> signature(1, 3). Moreover, the operator ∇ x,y maps the reduced tangent spaces isometricallyto each other. The local operator γ 5 := −iu/ √ −u 2 takes the role ofthe pseudoscalar matrix.3.3. The correspondence to Lorentzian geometryWe now explain how the above spin connection is related to the usual spinconnection used in spin geometry (see for example [17, 1]). To this end,let (M,g) be a time-oriented Lorentzian spin manifold with spinor bundleSM (thus S x M is a 4-dimensional complex vector space endowed with aninner product ≺.|.≻ x of signature (2, 2)). Assume that γ(t) isasmooth,future-directed <strong>and</strong> timelike curve, for simplicity parametrized by the arclength, defined on the interval [0,T]withγ(0) = y <strong>and</strong> γ(T )=x. Then theparallel transport of tangent vectors along γ with respect to the Levi-Civitaconnection ∇ LC gives rise to the isometry∇ LCx,y : T y → T x .
Causal Fermion Systems 175In order to compare with the metric connection ∇ of Definition 3.11, wesubdivide γ (for simplicity with equal spacing, although a non-uniform spacingwould work just as well). Thus for any given N, we define the pointsx 0 ,...,x N byx n = γ(t n ) with t n = nT N .We define the parallel transport ∇ N x,y by successively composing the paralleltransport between neighboring points,∇ N x,y := ∇ xN ,x N−1∇ xN−1 ,x N−2 ···∇ x1 ,x 0: T y → T x .Our first theorem gives a connection to the Minkowski vacuum. For anyε>0 we regularize on the scale ε>0 by inserting a convergence-generatingfactor into the integr<strong>and</strong> in (5),∫P ε d 4 k(x, y) =(2π) (k/ + m) 4 δ(k2 − m 2 )Θ(−k 0 ) e εk0 e −ik(x−y) . (41)This function can indeed be realized as the kernel of the fermionic operator(15) corresponding to a causal fermion system (H, F,ρ ε ). Here the measureρ ε is the push-forward of the volume measure in Minkowski space by anoperator F ε , being an ultraviolet regularization of the operator F in (2)-(4)(for details see [14, Section 4]).Theorem 3.15. For given γ, we consider the family of regularized fermionicprojectors of the vacuum (P ε ) ε>0 as given by (41). Then for a generic curveγ <strong>and</strong> for every N ∈ N, thereisε 0 such that for all ε ∈ (0,ε 0 ] <strong>and</strong> alln =1,...,N, the points x n <strong>and</strong> x n−1 are spin-connectable, <strong>and</strong> x n+1 lies inthe future of x n (according to Definition 3.10). Moreover,∇ LCx,y = lim limN→∞ ε↘0 ∇N x,y .By a generic curve we mean that the admissible curves are dense in theC ∞ -topology (i.e., for any smooth γ <strong>and</strong> every K ∈ N, there is a sequence γ lof admissible curves such that D k γ l → D k γ uniformly for all k =0,...,K).The restriction to generic curves is needed in order to ensure that the Euclidean<strong>and</strong> directional sign operators are generically separated (see Definition3.8(b)). The proof of the above theorem is given in [14, Section 4].Clearly, in this theorem the connection ∇ LCx,y is trivial. In order to showthat our connection also coincides with the Levi-Civita connection in the casewith curvature, in [14, Section 5] a globally hyperbolic Lorentzian manifoldis considered. For technical simplicity, we assume that the manifold is flatMinkowski space in the past of a given Cauchy hypersurface.Theorem 3.16. Let (M,g) be a globally hyperbolic manifold which is isometricto Minkowski space in the past of a given Cauchy hypersurface N . For givenγ, we consider the family of regularized fermionic projectors (P ε ) ε>0 suchthat P ε (x, y) coincides with the distribution (41) if x <strong>and</strong> y lie in the past ofN . Then for a generic curve γ <strong>and</strong> for every sufficiently large N, thereisε 0such that for all ε ∈ (0,ε 0 ] <strong>and</strong> all n =1,...,N, the points x n <strong>and</strong> x n−1 are
176 F. Finster, A. Grotz <strong>and</strong> D. Schiefenederspin-connectable, <strong>and</strong> x n+1 lies in the future of x n (according to Definition3.10). Moreover,(lim limN→∞ ε↘0 ∇N x,y −∇ LCx,y = O L(γ) ∇R ) ( ( scal))m 2 1+Om 2 , (42)where R denotes the Riemann curvature tensor, scal is scalar curvature, <strong>and</strong>L(γ) is the length of the curve γ.Thus the metric connection of Definition 3.11 indeed coincides with theLevi-Civita connection, up to higher-order curvature corrections. For detailedexplanations <strong>and</strong> the proof we refer to [14, Section 5].At first sight, one might conjecture that Theorem 3.16 should also applyto the spin connection in the sense thatDx,y LC = lim limN→∞ ε↘0 DN x,y , (43)where D LC is the spin connection on SM induced by the Levi-Civita connection<strong>and</strong>Dx,y N := D xN ,x N−1D xN−1 ,x N−2 ···D x1 ,x 0: S y → S x (44)(<strong>and</strong> D is the spin connection of Theorem 3.9). It turns out that this conjectureis false. But the conjecture becomes true if we replace (44) by theoperator productD(x,y) N := D x N ,x N−1U (x N |x N−2 )x N−1D xN−1 ,x N−2U (x N−1|x N−3 )x N−2···U (x 2|x 0 )x 1D x1 ,x 0.Here the intermediate factors U . (.|.) are the so-called splice maps given byU (z|y)x= U xz VU −1xy ,where U xz <strong>and</strong> U xy are synchronization maps, <strong>and</strong> V ∈ exp(iRs x )isanoperator which identifies the representatives K xy ,K xz ∈ T x (for details see[14, Section 3.7 <strong>and</strong> Section 5]). The splice maps also enter the spin curvatureR, which is defined in analogy to the metric curvature (39) byR(x, y, z) =U (z|y)xD x,y U (x|z)yD y,z U (y|x)z D z,x : S x → S x .4. A “quantization effect” for the support of minimizersThe recent paper [15] contains a first numerical <strong>and</strong> analytical study of theminimizers of the action principle (26). We now explain a few results <strong>and</strong>discuss their potential physical significance. We return to the setting of causalvariational principles (see Section 1.4). In order to simplify the problem asmuch as possible, we only consider the case of spin dimension n =1<strong>and</strong>twoparticles f = 2 (although many results in [15] apply similarly to a generalfinite number of particles). Thus we identify the particle space (H, 〈.|.〉 H )with C 2 .EverypointF ∈ F is a Hermitian (2 × 2)-matrix with at most onepositive <strong>and</strong> at most one negative eigenvalue. We represent it in terms of thePauli matrices as F = α 1+⃗u⃗σ with |⃗u| ≥|α|. In order to further simplify theproblem, we prescribe the eigenvalues in the support of the universal measure
Causal Fermion Systems 177Figure 1. The function D.to be 1 + τ <strong>and</strong> 1 − τ, whereτ ≥ 1 is a given parameter. Then F can berepresented asF = τx·σ +1 with x ∈ S 2 ⊂ R 3 .Thus we may identify F with the unit sphere S 2 . The Lagrangian L(x, y)given in (23) simplifies to a function only of the angle ϑ between the vectorsx <strong>and</strong> y. More precisely,L(x, y) =max ( 0, D(〈x, y〉) )D(cos ϑ) =2τ 2 (1 + cos ϑ) ( 2 − τ 2 (1 − cos ϑ) ) .As shown in Figure 1 in typical examples, the function D is positive forsmall ϑ <strong>and</strong> becomes negative if ϑ exceeds a certain value ϑ max (τ). FollowingDefinition 1.3, two points x <strong>and</strong> y are timelike separated if ϑϑ max .Our action principle is to minimize the action (24) by varying the measureρ in the family of normalized Borel measures on the sphere. In order tosolve this problem numerically, we approximate the minimizing measure by aweighted counting measure. Thus for any given integer m, we choose pointsx 1 ,...,x m ∈ S 2 together with corresponding weights ρ 1 ,...,ρ m withm∑ρ i ≥ 0 <strong>and</strong> ρ i =1<strong>and</strong> introduce the measure ρ byρ =i=1m∑ρ i δ xi , (45)i=1where δ x denotes the Dirac measure. Fixing different values of m <strong>and</strong> seekingfor numerical minimizers by varying both the points x i <strong>and</strong> the weights ρ i ,we obtain the plots shown in Figure 2. It is surprising that for each fixed τ,the obtained minimal action no longer changes if m is increased beyond acertain value m 0 (τ). The numerics shows that if m>m 0 ,someofthepoints
178 F. Finster, A. Grotz <strong>and</strong> D. SchiefenederFigure 2. Numerical minima of the action on the twodimensionalsphere.x i coincide, so that the support of the minimizing measure never consists ofmore than m 0 points. Since this property remains true in the limit m →∞,our numerical analysis shows that the minimizing measure is discrete in thesense that its support consists of a finite number of m 0 points. Anotherinteresting effect is that the action seems to favor symmetric configurationson the sphere. Namely, the most distinct local minima in Figure 2 correspondto configurations where the points x i lie on the vertices of Platonic solids. Theanalysis in [15] gives an explanation for this “discreteness” of the minimizingmeasure, as is made precise in the following theorem (for more general resultsin the same spirit see [15, Theorems 4.15 <strong>and</strong> 4.17]).Theorem 4.1. If τ>τ c := √ 2, the support of every minimizing measure onthe two-dimensional sphere is singular in the sense that it has empty interior.Extrapolating to the general situation, this result indicates that ourvariational principle favors discrete over continuous configurations. Again interpretingM := supp ρ as our space-time, this might correspond to a mechanismdriven by our action principle which makes space-time discrete. Using amore graphic language, one could say that space-time “discretizes itself” onthe Planck scale, thus avoiding the ultraviolet divergences of quantum fieldtheory.Another possible interpretation gives a connection to field quantization:Our model on the two-sphere involves one continuous parameter τ. Ifwe
Causal Fermion Systems 179allow τ to be varied while minimizing the action (as is indeed possible if wedrop the constraint of prescribed eigenvalues), then the local minima of theaction attained at discrete values of τ (like the configurations of the Platonicsolids) are favored. Regarding τ as the amplitude of a “classical field”, ouraction principle gives rise to a “quantization” of this field, in the sense thatthe amplitude only takes discrete values.The observed “discreteness” might also account for effects related to thewave-particle duality <strong>and</strong> the collapse of the wave function in the measurementprocess (for details see [12]).5. The correspondence to quantum field theory <strong>and</strong> gaugetheoriesThe correspondence to Minkowski space mentioned in Section 3.3 can alsobe used to analyze our action principle for interacting systems in the socalledcontinuum limit. We now outline a few ideas <strong>and</strong> constructions (fordetails see [5, Chapter 4], [8] <strong>and</strong> the survey article [13]). We first observethat the vacuum fermionic projector (5) is a solution of the Dirac equation(iγ j ∂ j − m)P sea (x, y) = 0. To introduce the interaction, we replace the freeDirac operator by a more general Dirac operator, which may in particularinvolve gauge potentials or a gravitational field. For simplicity, we here onlyconsider an electromagnetic potential A,(iγ j (∂ j − ieA j ) − m ) P (x, y) =0. (46)Next, we introduce particles <strong>and</strong> anti-particles by occupying (suitably normalized)positive-energy states <strong>and</strong> removing states of the sea,P (x, y) =P sea (x, y) − 12πn f∑k=1|ψ k (x)≻≺ψ k (y)| + 12π∑n al=1|φ l (x)≻≺φ l (y)| .(47)Using the so-called causal perturbation expansion <strong>and</strong> light-cone expansion,the fermionic projector can be introduced uniquely from (46) <strong>and</strong> (47).It is important that our setting so far does not involve the field equations;in particular, the electromagnetic potential in the Dirac equation (46)does not need to satisfy the Maxwell equations. Instead, the field equationsshould be derived from our action principle (26). Indeed, analyzing the correspondingEuler-Lagrange equations, one finds that they are satisfied onlyif the potentials in the Dirac equation satisfy certain constraints. Some ofthese constraints are partial differential equations involving the potentials aswell as the wave functions of the particles <strong>and</strong> anti-particles in (47). In [8],such field equations are analyzed in detail for a system involving an axialfield. In order to keep the setting as simple as possible, we here consider theanalogous field equation for the electromagnetic field:n∑ f∑n a∂ jk A k − □A j = e ≺ψ k |γ j ψ k ≻−e ≺φ l |γ j φ l ≻ . (48)k=1l=1
180 F. Finster, A. Grotz <strong>and</strong> D. SchiefenederWith (46) <strong>and</strong> (48), the interaction as described by the action principle (26)reduces in the continuum limit to the coupled Dirac-Maxwell equations. Themany-fermion state is again described by the fermionic projector, which isbuilt up of one-particle wave functions. The electromagnetic field merely is aclassical bosonic field. Nevertheless, regarding (46) <strong>and</strong> (48) as a nonlinear hyperbolicsystem of partial differential equations <strong>and</strong> treating it perturbatively,one obtains all the Feynman diagrams which do not involve fermion loops.Taking into account that by exciting sea states we can describe pair creation<strong>and</strong> annihilation processes, we also get all diagrams involving fermion loops.In this way, we obtain agreement with perturbative quantum field theory (fordetails see [8, §8.4] <strong>and</strong> the references therein).We finally remark that in the continuum limit, the freedom in choosingthe spinor basis (17) can be described in the language of st<strong>and</strong>ard gaugetheories. Namely, introducing a gauge-covariant derivative D j = ∂ j − iC jwith gauge potentials C j (see for example [18]), the transformation (18) givesrise to the local gauge transformationsψ(x) → U(x) ψ(x) ,D j → UD j U −1C j (x) → U(x)C(x)U(x) −1 + iU(x)(∂ j U(x) −1 )(49)with U(x) ∈ U(p, q). The difference to st<strong>and</strong>ard gauge theories is that thegauge group cannot be chosen arbitrarily, but it is determined to be theisometry group of the spin space. In the case of spin dimension two, thecorresponding gauge group U(2, 2) allows for a unified description of electrodynamics<strong>and</strong> general relativity (see [5, Section 5.1]). By choosing a higherspin dimension (see [5, Section 5.1]), one gets a larger gauge group. Ourmathematical framework ensures that our action principle <strong>and</strong> thus also thecontinuum limit is gauge-symmetric in the sense that the transformations (49)with U(x) ∈ U(p, q) map solutions of the equations of the continuum limit toeach other. However, our action is not invariant under local transformationsof the form (49) if U(x) ∉ U(p, q) is not unitary. An important example ofsuch non-unitary transformations are chiral gauge transformations likeU(x) =χ L U L (x)+χ R U R (x) with U L/R ∈ U(1) , U L ≢ U R .Thus chiral gauge transformations do not describe a gauge symmetry in theabove sense. In the continuum limit, this leads to a mechanism which giveschiral gauge fields a rest mass (see [8, Section 8.5] <strong>and</strong> [13, Section 7]). Moreover,in systems of higher spin dimension, the presence of chiral gauge fieldsgives rise to a spontaneous breaking of the gauge symmetry, resulting in asmaller “effective” gauge group. As shown in [5, Chapters 6-8], these mechanismsmake it possible to realize the gauge groups <strong>and</strong> couplings of thest<strong>and</strong>ard model.AcknowledgmentWe thank the referee for helpful suggestions on the manuscript.
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182 F. Finster, A. Grotz <strong>and</strong> D. SchiefenederFelix Finster, Andreas Grotz <strong>and</strong> Daniela SchiefenederFakultät für MathematikUniversität RegensburgD-93040 Regensburg, Germanye-mail: Felix.Finster@mathematik.uni-regensburg.deAndreas.Grotz@mathematik.uni-regensburg.deDaniela.Schiefeneder@mathematik.uni-regensburg.de
CCR- versus CAR-Quantization on CurvedSpacetimesChristian Bär <strong>and</strong> Nicolas GinouxAbstract. We provide a systematic construction of bosonic <strong>and</strong> fermioniclocally covariant quantum field theories on curved backgrounds forlarge classes of free fields. It turns out that bosonic quantization is possibleunder much more general assumptions than fermionic quantization.Mathematics Subject Classification (2010). 58J45, 35Lxx, 81T20.Keywords. Wave operator, Dirac-type operator, globally hyperbolicspacetime, Green’s operator, CCR-algebra, CAR-algebra, locally covariantquantum field theory.1. IntroductionClassical fields on spacetime are mathematically modeled by sections of avector bundle over a Lorentzian manifold. The field equations are usually partialdifferential equations. We introduce a class of differential operators, calledGreen-hyperbolic operators, which have good analytical solubility properties.This class includes wave operators as well as Dirac-type operators but alsothe Proca <strong>and</strong> the Rarita-Schwinger operator.In order to quantize such a classical field theory on a curved background,we need local algebras of observables. They come in two flavors, bosonic algebrasencoding the canonical commutation relations <strong>and</strong> fermionic algebrasencoding the canonical anti-commutation relations. We show how such algebrascan be associated to manifolds equipped with suitable Green-hyperbolicoperators. We prove that we obtain locally covariant quantum field theoriesin the sense of [12]. There is a large literature where such constructions arecarried out for particular examples of fields, see e.g. [15, 16, 17, 22, 30]. In allthese papers the well-posedness of the Cauchy problem plays an importantrole. We avoid using the Cauchy problem altogether <strong>and</strong> only make use ofGreen’s operators. In this respect, our approach is similar to the one in [31].This allows us to deal with larger classes of fields, see Section 3.7, <strong>and</strong> toF. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_10,© Springer Basel AG 2012183
184 C. Bär <strong>and</strong> N. Ginouxtreat them systematically. Much of the work on particular examples can besubsumed under this general approach.It turns out that bosonic algebras can be obtained in much more generalsituations than fermionic algebras. For instance, for the classical Dirac fieldboth constructions are possible. Hence, on the level of observable algebras,there is no spin-statistics theorem.This is a condensed version of our paper [4] where full details are given.Here we confine ourselves to the results <strong>and</strong> the main arguments while weleave aside all technicalities. Moreover, [4] contains a discussion of states <strong>and</strong>the induced quantum fields.Acknowledgments. It is a pleasure to thank Alex<strong>and</strong>er Strohmaier forvery valuable discussion <strong>and</strong> the anonymous referee for his remarks. Theauthors would like to thank SPP 1154 “Globale Differentialgeometrie” <strong>and</strong>SFB 647 “Raum-Zeit-Materie”, both funded by Deutsche Forschungsgemeinschaft,for financial support. Last but not the least, the authors thank theorganizers of the international conference “<strong>Quantum</strong> field theory <strong>and</strong> gravity”.2. Algebras of canonical (anti-) commutation relationsWe start with algebraic preparations <strong>and</strong> collect the necessary algebraic factsabout CAR- <strong>and</strong> CCR-algebras.2.1. CAR-algebrasThe symbol “CAR” st<strong>and</strong>s for “canonical anti-commutation relations”. Thesealgebras are related to pre-Hilbert spaces. We always assume the Hermitianinner product (·,·) to be linear in the first argument <strong>and</strong> anti-linear in thesecond.Definition 2.1. ACAR-representation of a complex pre-Hilbert space (V,(·,·))is a pair (a,A), where A is a unital C ∗ -algebra <strong>and</strong> a : V → A is an anti-linearmap satisfying:(i) A = C ∗ (a(V )), that is, A is the C ∗ -algebra generated by A,(ii) {a(v 1 ), a(v 2 )} = 0 <strong>and</strong>(iii) {a(v 1 ) ∗ , a(v 2 )} =(v 1 ,v 2 ) · 1,for all v 1 ,v 2 ∈ V .As an example, for any complex pre-Hilbert vector space (V,(·,·)), theC ∗ -completion Cl(V C ,q C ) of the algebraic Clifford algebra of the complexification(V C ,q C ) of (V,(·,·)) is a CAR-representation of (V,(·,·)). See [4,App. A.1] for the details, in particular for the construction of the mapa : V → Cl(V C ,q C ).Theorem 2.2. Let (V,(·,·)) be an arbitrary complex pre-Hilbert space. Let Âbe any unital C ∗ -algebra <strong>and</strong> â : V → Â be any anti-linear map satisfying Axioms(ii) <strong>and</strong> (iii) of Definition 2.1. Then there exists a unique C ∗ -morphism
CCR- versus CAR-Quantization on Curved Spacetimes 185˜α :Cl(V C ,q C ) →  such thatVaCl(V C ,q C )commutes. Furthermore, ˜α is injective.In particular, (V,(·,·)) has, up to C ∗ -isomorphism, a unique CAR-representation.For an alternative description of the CAR-representation in terms ofcreation <strong>and</strong> annihilation operators on the fermionic Fock space we refer to[10, Prop. 5.2.2].From now on, given a complex pre-Hilbert space (V,(·,·)), we denote theC ∗ -algebra Cl(V C ,q C ) associated with the CAR-representation (a, Cl(V C ,q C ))of (V,(·,·)) by CAR(V,(·,·)). We list the properties of CAR-representationswhich are relevant for quantization, see also [10, Vol. II, Thm. 5.2.5, p. 15].Proposition 2.3. Let (a, CAR(V,(·,·))) be the CAR-representation of a complexpre-Hilbert space (V,(·,·)).(i) For every v ∈ V one has ‖a(v)‖ = |v| =(v, v) 1 2 ,where‖·‖ denotes theC ∗ -norm on CAR(V,(·,·)).(ii) The C ∗ -algebra CAR(V,(·,·)) is simple, i.e., it has no closed two-sided∗-ideals other than {0} <strong>and</strong> the algebra itself.(iii) The algebra CAR(V,(·,·)) is Z 2 -graded,˜αâÂCAR(V,(·,·)) = CAR even (V,(·,·)) ⊕ CAR odd (V,(·,·)),<strong>and</strong> a(V ) ⊂ CAR odd (V,(·,·)).(iv) Let f : V → V ′ be an isometric linear embedding, where (V ′ , (·,·) ′ ) isanother complex pre-Hilbert space. Then there exists a unique injectiveC ∗ -morphism CAR(f) :CAR(V,(·,·)) → CAR(V ′ , (·,·) ′ ) such thatVfcommutes.aCAR(V,(·,·))CAR(f)V ′ a ′ CAR(V ′ , (·,·) ′ )One easily sees that CAR(id) = id <strong>and</strong> that CAR(f ′ ◦ f) =CAR(f ′ ) ◦CAR(f) for all isometric linear embeddings V −→ f V ′ f−→ ′V ′′ . Therefore wehave constructed a covariant functorCAR : HILB −→ C ∗ Alg,where HILB denotes the category whose objects are the complex pre-Hilbertspaces <strong>and</strong> whose morphisms are the isometric linear embeddings <strong>and</strong> C ∗ Alg is
186 C. Bär <strong>and</strong> N. Ginouxthe category whose objects are the unital C ∗ -algebras <strong>and</strong> whose morphismsare the injective unit-preserving C ∗ -morphisms.For real pre-Hilbert spaces there is the concept of self-dual CAR-representations.Definition 2.4. A self-dual CAR-representation of a real pre-Hilbert space(V,(·,·)) is a pair (b,A), where A is a unital C ∗ -algebra <strong>and</strong> b : V → A is anR-linear map satisfying:(i) A = C ∗ (b(V )),(ii) b(v) =b(v) ∗ <strong>and</strong>(iii) {b(v 1 ), b(v 2 )} =(v 1 ,v 2 ) · 1,for all v, v 1 ,v 2 ∈ V .Note that a self-dual CAR-representation is not a CAR-representationin the sense of Definition 2.1. Given a self-dual CAR-representation, one canextend b to a C-linear map from the complexification V C to A. This extensionb : V C → A then satisfies b(¯v) =b(v) ∗ <strong>and</strong> {b(v 1 ), b(v 2 )} =(v 1 , ¯v 2 ) · 1forall v, v 1 ,v 2 ∈ V C . These are the axioms of a self-dual CAR-representation asin [1, p. 386].Theorem 2.5. For every real pre-Hilbert space (V,(·,·)), theC ∗ -Clifford algebraCl(V C ,q C ) provides a self-dual CAR-representation of (V,(·,·)) viab(v) = √ i2v.Moreover, self-dual CAR-representations have the following universalproperty: Let  be any unital C∗ -algebra <strong>and</strong> ̂b : V →  be any R-linear mapsatisfying Axioms (ii) <strong>and</strong> (iii) of Definition 2.4. Then there exists a uniqueC ∗ -morphism ˜β :Cl(V C ,q C ) →  such thatVbCl(V C ,q C )˜β̂bÂcommutes. Furthermore, ˜β is injective.In particular, (V,(·,·)) has, up to C ∗ -isomorphism, a unique self-dualCAR-representation.From now on, given a real pre-Hilbert space (V,(·,·)), we denotethe C ∗ -algebra Cl(V C ,q C ) associated with the self-dual CAR-representation(b, Cl(V C ,q C )) of (V,(·,·)) by CAR sd (V,(·,·)).Proposition 2.6. Let (b, CAR sd (V,(·,·))) be the self-dual CAR-representationof a real pre-Hilbert space (V,(·,·)).(i) For every v ∈ V one has ‖b(v)‖ = √ 1 2|v|, where‖·‖ denotes the C ∗ -norm on CAR sd (V,(·,·)).(ii) The C ∗ -algebra CAR sd (V,(·,·)) is simple.
CCR- versus CAR-Quantization on Curved Spacetimes 187(iii) The algebra CAR sd (V,(·,·)) is Z 2 -graded,CAR sd (V,(·,·)) = CAR even (V,(·,·)) ⊕ CARodd (V,(·,·)),sd<strong>and</strong> b(V ) ⊂ CAR oddsd (V,(·,·)).(iv) Let f : V → V ′ be an isometric linear embedding, where (V ′ , (·,·) ′ )is another real pre-Hilbert space. Then there exists a unique injectiveC ∗ -morphism CAR sd (f) :CAR sd (V,(·,·)) → CAR sd (V ′ , (·,·) ′ ) such thatsdVfcommutes.bCAR sd (V,(·,·))CAR sd (f)V ′ b ′ CARsd (V ′ , (·,·) ′ )The proofs are similar to the ones for CAR-representations of complexpre-Hilbert spaces as given in [4, App. A]. We have constructed a functorCAR sd : HILB R −→ C ∗ Alg,where HILB R denotes the category whose objects are the real pre-Hilbertspaces <strong>and</strong> whose morphisms are the isometric linear embeddings.Remark 2.7. Let (V,(·,·)) be a complex pre-Hilbert space. If we consider Vas a real vector space, then we have the real pre-Hilbert space (V,Re(·,·)).For the corresponding CAR-representations we have<strong>and</strong>CAR(V,(·,·)) = CAR sd (V,Re(·,·)) = Cl(V C ,q C )b(v) =i √2(a(v) − a(v) ∗ ).2.2. CCR-algebrasIn this section, we recall the construction of the representation of any (real)symplectic vector space by the so-called canonical commutation relations(CCR). Proofs can be found in [5, Sec. 4.2].Definition 2.8. A CCR-representation of a symplectic vector space (V,ω) isapair (w, A), where A is a unital C ∗ -algebra <strong>and</strong> w is a map V → A satisfying:(i) A = C ∗ (w(V )),(ii) w(0) = 1,(iii) w(−ϕ) =w(ϕ) ∗ ,(iv) w(ϕ + ψ) =e iω(ϕ,ψ)/2 w(ϕ) · w(ψ),for all ϕ, ψ ∈ V .The map w is in general neither linear, nor any kind of group homomorphism,nor continuous as soon as V carries a topology which is differentfrom the discrete one [5, Prop. 4.2.3].
188 C. Bär <strong>and</strong> N. GinouxExample 2.9. Given any symplectic vector space (V,ω), consider the Hilbertspace H := L 2 (V,C), where V is endowed with the counting measure. Definethe map w from V into the space L(H) of bounded endomorphisms of H by(w(ϕ)F )(ψ) :=e iω(ϕ,ψ)/2 F (ϕ + ψ),for all ϕ, ψ ∈ V <strong>and</strong> F ∈ H. It is well-known that L(H) isaC ∗ -algebra withthe operator norm as C ∗ -norm, <strong>and</strong> that the map w satisfies the Axioms (ii)-(iv) from Definition 2.8, see e.g. [5, Ex. 4.2.2]. Hence setting A := C ∗ (w(V )),the pair (w, A) provides a CCR-representation of (V,ω).This is essentially the only CCR-representation:Theorem 2.10. Let (V,ω) be a symplectic vector space <strong>and</strong> (ŵ, Â) be a pairsatisfying the Axioms (ii)-(iv) of Definition 2.8. Then there exists a uniqueC ∗ -morphism Φ:A → Â such that Φ ◦ w = ŵ, where(w, A) is the CCRrepresentationfrom Example 2.9. Moreover, Φ is injective.In particular, (V,ω) has a CCR-representation, unique up to C ∗ -isomorphism.We denote the C ∗ -algebra associated to the CCR-representation of(V,ω) from Example 2.9 by CCR(V,ω). As a consequence of Theorem 2.10,we obtain the following important corollary.Corollary 2.11. Let (V,ω) be a symplectic vector space <strong>and</strong> (w, CCR(V,ω))its CCR-representation.(i) The C ∗ -algebra CCR(V,ω) is simple, i.e., it has no closed two-sided∗-ideals other than {0} <strong>and</strong> the algebra itself.(ii) Let (V ′ ,ω ′ ) be another symplectic vector space <strong>and</strong> f : V → V ′ asymplecticlinear map. Then there exists a unique injective C ∗ -morphismCCR(f) :CCR(V,ω) → CCR(V ′ ,ω ′ ) such thatVfwCCR(f)CCR(V,ω) CCR(V ′ ,ω ′ )commutes.Obviously CCR(id) = id <strong>and</strong> CCR(f ′ ◦ f) =CCR(f ′ ) ◦ CCR(f) forall symplectic linear maps V → f V ′ f→ ′V ′′ , so that we have constructed acovariant functorCCR : Sympl −→ C ∗ Alg.V ′ w ′3. <strong>Field</strong> equations on Lorentzian manifolds3.1. Globally hyperbolic manifoldsWe begin by fixing notation <strong>and</strong> recalling general facts about Lorentzianmanifolds, see e.g. [26] or [5] for more details. Unless mentioned otherwise, the
CCR- versus CAR-Quantization on Curved Spacetimes 189pair (M,g) will st<strong>and</strong> for a smooth m-dimensional manifold M equipped witha smooth Lorentzian metric g, where our convention for Lorentzian signatureis (−+···+). The associated volume element will be denoted by dV. We shallalso assume our Lorentzian manifold (M,g) to be time-orientable, i.e., thatthere exists a smooth timelike vector field on M. Time-oriented Lorentzianmanifolds will be also referred to as spacetimes. Note that in contrast toconventions found elsewhere, we do not assume that a spacetime be connectednor that its dimension be m =4.For every subset A of a spacetime M we denote the causal future <strong>and</strong>past of A in M by J + (A) <strong>and</strong>J − (A), respectively. If we want to emphasizethe ambient space M in which the causal future or past of A is considered,we write J M ± (A) instead of J ± (A). Causal curves will always be implicitlyassumed (future or past) oriented.Definition 3.1. A Cauchy hypersurface in a spacetime (M,g) is a subset ofM which is met exactly once by every inextensible timelike curve.Cauchy hypersurfaces are always topological hypersurfaces but need notbe smooth. All Cauchy hypersurfaces of a spacetime are homeomorphic.Definition 3.2. A spacetime (M,g) is called globally hyperbolic if <strong>and</strong> only ifit contains a Cauchy hypersurface.A classical result of R. Geroch [18] says that a globally hyperbolic spacetimecan be foliated by Cauchy hypersurfaces. It is a rather recent <strong>and</strong> veryimportant result that this also holds in the smooth category: any globally hyperbolicspacetime is of the form (R × Σ, −βdt 2 ⊕ g t ), where each {t}×Σisasmooth spacelike Cauchy hypersurface, β a smooth positive function <strong>and</strong> (g t ) ta smooth one-parameter family of Riemannian metrics on Σ [7, Thm. 1.1].The hypersurface Σ can be even chosen such that {0} ×Σcoincideswitha given smooth spacelike Cauchy hypersurface [8, Thm. 1.2]. Moreover, anycompact acausal smooth spacelike submanifold with boundary in a globallyhyperbolic spacetime is contained in a smooth spacelike Cauchy hypersurface[8, Thm. 1.1].Definition 3.3. A closed subset A ⊂ M is called• spacelike compact if there exists a compact subset K ⊂ M such thatA ⊂ J M (K) :=J− M (K) ∪ J+ M (K),• future-compact if A ∩ J + (x) is compact for any x ∈ M,• past-compact if A ∩ J − (x) is compact for any x ∈ M.A spacelike compact subset is in general not compact, but its intersectionwith any Cauchy hypersurface is compact, see e.g. [5, Cor. A.5.4].Definition 3.4. A subset Ω of a spacetime M is called causally compatible if<strong>and</strong> only if J Ω ±(x) =J M ± (x) ∩ Ω for every x ∈ Ω.This means that every causal curve joining two points in Ω must becontained entirely in Ω.
190 C. Bär <strong>and</strong> N. Ginoux3.2. Differential operators <strong>and</strong> Green’s functionsA differential operator of order (at most) k on a vector bundle S → M overK = R or K = C is a linear map P : C ∞ (M,S) → C ∞ (M,S) which in localcoordinates x =(x 1 ,...,x m )ofM <strong>and</strong> with respect to a local trivializationlooks likeP = ∑A α (x) ∂α∂x . α|α|≤kHere C ∞ (M,S) denotes the space of smooth sections of S → M, α =(α 1 ,...,α m ) ∈ N 0 ×···×N 0 runs over multi-indices, |α| = ∑ mj=1 α j <strong>and</strong>∂ α∂= |α|∂x α ∂(x 1 ) α 1 ···∂(x m ) .Theprincipal symbol σ αm P of P associates to eachcovector ξ ∈ Tx ∗ M a linear map σ P (ξ) :S x → S x . Locally, it is given byσ P (ξ) = ∑A α (x)ξ α ,|α|=kwhere ξ α = ξ α 11 ···ξα m m <strong>and</strong> ξ = ∑ j ξ jdx j .IfP <strong>and</strong> Q are two differentialoperators of order k <strong>and</strong> l respectively, then Q ◦ P is a differential operatorof order k + l <strong>and</strong>σ Q◦P (ξ) =σ Q (ξ) ◦ σ P (ξ).For any linear differential operator P : C ∞ (M,S) → C ∞ (M,S) thereisaunique formally dual operator P ∗ : C ∞ (M,S ∗ ) → C ∞ (M,S ∗ )ofthesameorder characterized by ∫ ∫〈ϕ, P ψ〉 dV = 〈P ∗ ϕ, ψ〉 dVMMfor all ψ ∈ C ∞ (M,S) <strong>and</strong>ϕ ∈ C ∞ (M,S ∗ ) with supp(ϕ) ∩ supp(ψ) compact.Here 〈·, ·〉 : S ∗ ⊗ S → K denotes the canonical pairing, i.e., the evaluation ofa linear form in Sx ∗ on an element of S x ,wherex ∈ M. Wehaveσ P ∗(ξ) =(−1) k σ P (ξ) ∗ where k is the order of P .Definition 3.5. Let a vector bundle S → M be endowed with a non-degenerateinner product 〈· , ·〉. A linear differential operator P on S is called formallyself-adjoint if <strong>and</strong> only if∫∫〈Pϕ,ψ〉 dV = 〈ϕ, P ψ〉 dVMholds for all ϕ, ψ ∈ C ∞ (M,S) with supp(ϕ) ∩ supp(ψ) compact.Similarly, we call P formally skew-adjoint if instead∫∫〈Pϕ,ψ〉 dV = − 〈ϕ, P ψ〉 dV .MWe recall the definition of advanced <strong>and</strong> retarded Green’s operators fora linear differential operator.Definition 3.6. Let P be a linear differential operator acting on the sectionsof a vector bundle S over a Lorentzian manifold M. Anadvanced Green’soperator for P on M is a linear mapG + : Cc∞ (M,S) → C ∞ (M,S)MM
CCR- versus CAR-Quantization on Curved Spacetimes 191satisfying:(G 1 ) P ◦ G + =id ;C ∞ c (M,S)(G 2 ) G + ◦ P | =id C ∞ c (M,S) Cc ∞(M,S);(G + 3 ) supp(G +ϕ) ⊂ J+ M (supp(ϕ)) for any ϕ ∈ Cc ∞ (M,S).A retarded Green’s operator for P on M is a linear map G − : Cc∞C ∞ (M,S) satisfying (G 1 ), (G 2 ), <strong>and</strong>(G − 3 ) supp(G −ϕ) ⊂ J− M (supp(ϕ)) for any ϕ ∈ Cc ∞ (M,S).(M,S) →Here we denote by Cc∞sections of S.(M,S) the space of compactly supported smoothDefinition 3.7. Let P : C ∞ (M,S) → C ∞ (M,S) be a linear differential operator.We call P Green-hyperbolic if the restriction of P to any globallyhyperbolic subregion of M has advanced <strong>and</strong> retarded Green’s operators.The Green’s operators for a given Green-hyperbolic operator P providesolutions ϕ of Pϕ = 0. More precisely, denoting by C ∞ sc (M,S) the set ofsmooth sections in S with spacelike compact support, we have the followingTheorem 3.8. Let M be a Lorentzian manifold, let S → M be a vector bundle,<strong>and</strong> let P be a Green-hyperbolic operator acting on sections of S. LetG ± beadvanced <strong>and</strong> retarded Green’s operators for P , respectively. PutG := G + − G − : Cc∞ (M,S) → Csc ∞ (M,S).Then the following linear maps form a complex:{0} →C ∞ c (M,S) P −→ C ∞ c (M,S) G −→ C ∞ sc (M,S) P −→ C ∞ sc (M,S). (1)This complex is always exact at the first Cc∞ (M,S). IfM is globally hyperbolic,then the complex is exact everywhere.We refer to [4, Theorem 3.5] for the proof. Note that exactness at thefirst Cc∞ (M,S) in sequence (1) says that there are no non-trivial smooth solutionsof Pϕ = 0 with compact support. Indeed, if M is globally hyperbolic,more is true. Namely, if ϕ ∈ C ∞ (M,S) solves Pϕ = 0 <strong>and</strong> supp(ϕ) is futureor past-compact, then ϕ = 0 (see e.g. [4, Remark 3.6] for a proof). As astraightforward consequence, the Green’s operators for a Green-hyperbolicoperator on a globally hyperbolic spacetime are unique [4, Remark 3.7].3.3. Wave operatorsThe most prominent class of Green-hyperbolic operators are wave operators,sometimes also called normally hyperbolic operators.Definition 3.9. A linear differential operator of second order P : C ∞ (M,S) →C ∞ (M,S) is called a wave operator if its principal symbol is given by theLorentzian metric, i.e., for all ξ ∈ T ∗ M we haveσ P (ξ) =−〈ξ,ξ〉·id.
192 C. Bär <strong>and</strong> N. GinouxIn other words, if we choose local coordinates x 1 ,...,x m on M <strong>and</strong> alocal trivialization of S, thenm∑P = − g ij ∂ 2(x)∂x i ∂x j + ∑m A j (x) ∂∂x j + B(x),i,j=1j=1where A j <strong>and</strong> B are matrix-valued coefficients depending smoothly on x <strong>and</strong>(g ij )istheinversematrixof(g ij )withg ij = 〈 ∂ ∂∂x, i ∂x〉.IfP is a wavejoperator, then so is its dual operator P ∗ . In [5, Cor. 3.4.3] it has been shownthat wave operators are Green-hyperbolic.Example 3.10 (d’Alembert operator). Let S be the trivial line bundle so thatsections of S are just functions. The d’Alembert operator P = = −div◦gradis a formally self-adjoint wave operator, see e.g. [5, p. 26].Example 3.11 (connection-d’Alembert operator). More generally, let S be avector bundle <strong>and</strong> let ∇ be a connection on S. This connection <strong>and</strong> the Levi-Civita connection on T ∗ M induce a connection on T ∗ M ⊗ S, again denoted∇. We define the connection-d’Alembert operator ∇ to be the compositionof the following three mapsC ∞ (M,S) −→ ∇ C ∞ (M,T ∗ M⊗S) −→ ∇ C ∞ (M,T ∗ M⊗T ∗ M⊗S) −tr⊗id S−−−−−→ C ∞ (M,S),where tr : T ∗ M ⊗ T ∗ M → R denotes the metric trace, tr(ξ ⊗ η) =〈ξ,η〉. Wecompute the principal symbol,σ ∇(ξ)ϕ = −(tr⊗id S )◦σ ∇ (ξ)◦σ ∇ (ξ)(ϕ) =−(tr⊗id S )(ξ ⊗ξ ⊗ϕ) =−〈ξ,ξ〉 ϕ.Hence ∇ is a wave operator.Example 3.12 (Hodge-d’Alembert operator). For S =Λ k T ∗ M, exterior differentiationd : C ∞ (M,Λ k T ∗ M) → C ∞ (M,Λ k+1 T ∗ M) increases the degreeby one, the codifferential δ = d ∗ : C ∞ (M,Λ k T ∗ M) → C ∞ (M,Λ k−1 T ∗ M)decreases the degree by one. While d is independent of the metric, the codifferentialδ does depend on the Lorentzian metric. The operator P = −dδ −δdis a formally self-adjoint wave operator.3.4. The Proca equationThe Proca operator is an example of a Green-hyperbolic operator of secondorder which is not a wave operator.Example 3.13 (Proca operator). The discussion of this example follows [31,p. 116f]. The Proca equation describes massive vector bosons. We take S =T ∗ M <strong>and</strong> let m 0 > 0. The Proca equation isPϕ := δdϕ + m 2 0ϕ =0, (2)where ϕ ∈ C ∞ (M,S). Applying δ to (2) we obtain, using δ 2 = 0 <strong>and</strong> m 0 ≠0,δϕ =0 (3)<strong>and</strong> hence(dδ + δd)ϕ + m 2 0ϕ =0. (4)Conversely, (3) <strong>and</strong> (4) clearly imply (2).
CCR- versus CAR-Quantization on Curved Spacetimes 193Since ˜P := dδ + δd + m 2 0 is minus a wave operator, it has Green’soperators ˜G ± . We define G ± : Cc∞ (M,S) → Csc ∞ (M,S) byG ± := (m −20 dδ +id)◦ ˜G ± = ˜G ± ◦ (m −20 dδ +id).The last equality holds because d <strong>and</strong> δ commute with ˜P , see [4, Lemma 2.16].For ϕ ∈ Cc∞ (M,S) we computeG ± Pϕ = ˜G ± (m −20 dδ +id)(δd + m2 0)ϕ = ˜G ± ˜Pϕ = ϕ<strong>and</strong> similarly PG ± ϕ = ϕ. Since the differential operator m −20 dδ +id does notincrease supports, the third axiom in the definition of advanced <strong>and</strong> retardedGreen’s operators holds as well.This shows that G + <strong>and</strong> G − are advanced <strong>and</strong> retarded Green’s operatorsfor P , respectively. Thus P is not a wave operator but Green-hyperbolic.3.5. Dirac-type operatorsThe most important Green-hyperbolic operators of first order are the socalledDirac-type operators.Definition 3.14. A linear differential operator D : C ∞ (M,S) → C ∞ (M,S)of first order is called of Dirac type if −D 2 is a wave operator.Remark 3.15. If D is of Dirac type, then i times its principal symbol satisfiesthe Clifford relationshence by polarization(iσ D (ξ)) 2 = −σ D 2(ξ) =−〈ξ,ξ〉·id,(iσ D (ξ))(iσ D (η)) + (iσ D (η))(iσ D (ξ)) = −2〈ξ,η〉·id.The bundle S thus becomes a module over the bundle of Clifford algebrasCl(TM) associated with (TM,〈· , ·〉). See [6, Sec. 1.1] or [23, Ch. I] for the definition<strong>and</strong> properties of the Clifford algebra Cl(V ) associated with a vectorspace V with inner product.Remark 3.16. If D is of Dirac type, then so is its dual operator D ∗ .Onaglobally hyperbolic region let G + be the advanced Green’s operator for D 2 ,which exists since −D 2 is a wave operator. Then it is not hard to check thatD ◦ G + is an advanced Green’s operator for D, see [25, Thm. 3.2]. The samediscussion applies to the retarded Green’s operator. Hence any Dirac-typeoperator is Green-hyperbolic.Example 3.17 (classical Dirac operator). If the spacetime M carries a spinstructure, then one can define the spinor bundle S =ΣM <strong>and</strong> the classicalDirac operatorm∑D : C ∞ (M,ΣM) → C ∞ (M,ΣM), Dϕ := i ε j e j ·∇ ej ϕ.j=1
194 C. Bär <strong>and</strong> N. GinouxHere (e j ) 1≤j≤m is a local orthonormal basis of the tangent bundle, ε j =〈e j ,e j 〉 = ±1 <strong>and</strong>“·” denotes the Clifford multiplication, see e.g. [6] or [3,Sec. 2]. The principal symbol of D is given byσ D (ξ)ψ = iξ ♯ · ψ.Here ξ ♯ denotes the tangent vector dual to the 1-form ξ via the Lorentzianmetric, i.e., 〈ξ ♯ ,Y〉 = ξ(Y ) for all tangent vectors Y over the same point ofthe manifold. Henceσ D 2(ξ)ψ = σ D (ξ)σ D (ξ)ψ = −ξ ♯ · ξ ♯ · ψ = 〈ξ,ξ〉 ψ.Thus P = −D 2 is a wave operator. Moreover, D is formally self-adjoint, seee.g. [3, p. 552].Example 3.18 (twisted Dirac operators). More generally, let E → M bea complex vector bundle equipped with a non-degenerate Hermitian innerproduct <strong>and</strong> a metric connection ∇ E over a spin spacetime M. In the notationof Example 3.17, one may define the Dirac operator of M twisted with E byD E := im∑j=1ε j e j ·∇ ΣM⊗Ee j: C ∞ (M,ΣM ⊗ E) → C ∞ (M,ΣM ⊗ E),where ∇ ΣM⊗E is the tensor product connection on ΣM ⊗ E. Again, D E is aformally self-adjoint Dirac-type operator.Example 3.19 (Euler operator). In Example 3.12, replacing Λ k T ∗ M byS := ΛT ∗ M ⊗ C = ⊕ m k=0 Λk T ∗ M ⊗ C, the Euler operator D = i(d − δ)defines a formally self-adjoint Dirac-type operator. In case M is spin, theEuler operator coincides with the Dirac operator of M twisted with ΣM ifm is even <strong>and</strong> twisted with ΣM ⊕ ΣM if m is odd.Example 3.20 (Buchdahl operators). On a 4-dimensional spin spacetime M,consider the st<strong>and</strong>ard orthogonal <strong>and</strong> parallel splitting ΣM =Σ + M ⊕ Σ − Mof the complex spinor bundle of M into spinors of positive <strong>and</strong> negativechirality. The finite dimensional irreducible representations of the simplyconnectedLie group Spin 0 (3, 1) are given by Σ (k/2)+ ⊗ Σ (l/2)− where k, l ∈ N.Here Σ (k/2)+ =Σ ⊙k+ is the k-th symmetric tensor product of the positive halfspinorrepresentation Σ + <strong>and</strong> similarly for Σ (l/2)− . Let the associated vectorbundles Σ (k/2)± M carry the induced inner product <strong>and</strong> connection.For s ∈ N, s ≥ 1, consider the twisted Dirac operator D (s) acting onsections of ΣM ⊗ Σ ((s−1)/2)+ M. In the induced splittingΣM ⊗ Σ ((s−1)/2)+ M =Σ + M ⊗ Σ ((s−1)/2)+ M ⊕ Σ − M ⊗ Σ ((s−1)/2)+ M(0 D (s)−D (s)+ 0)
CCR- versus CAR-Quantization on Curved Spacetimes 195because Clifford multiplication by vectors exchanges the chiralities. TheClebsch-Gordan formulas [11, Prop. II.5.5] tell us that the representationΣ + ⊗ Σ ( s−12 )+ splits asΣ + ⊗ Σ ( s−12 )+ =Σ ( s 2 )+ ⊕ s Σ( 2 −1)+ .Hence we have the corresponding parallel orthogonal projectionsπ s :Σ + M ⊗ Σ ( s−12 )+ M → Σ ( s 2 )+ M <strong>and</strong> π′ s :Σ + M ⊗ Σ ( s−12 )+ M → Σ ( s 2 −1)+ M.On the other h<strong>and</strong>, the representation Σ − ⊗ Σ ( s−12 )+ is irreducible. Now Buchdahloperators are the operators of the formB (s)μ 1 ,μ 2 ,μ 3:=(μ 1 · π s + μ 2 · π ′ sD (s)−D (s)+ μ 3 · id),where μ 1 ,μ 2 ,μ 3 ∈ C are constants. By definition, B μ (s)1 ,μ 2 ,μ 3is of the formD (s) + b, whereb is of order zero. In particular, B μ (s)1 ,μ 2 ,μ 3is a Dirac-typeoperator, hence it is Green-hyperbolic. For a definition of Buchdahl operatorsusing indices we refer to [13, 14, 35] <strong>and</strong> to [24, Def. 8.1.4, p. 104].3.6. The Rarita-Schwinger operatorFor the Rarita-Schwinger operator on Riemannian manifolds, we refer to [34,Sec. 2], see also [9, Sec. 2]. In this section let the spacetime M be spin <strong>and</strong> considerthe Clifford multiplication γ : T ∗ M ⊗ΣM → ΣM, θ ⊗ψ ↦→ θ ♯ ·ψ, whereΣM is the complex spinor bundle of M. Then there is the representationtheoreticsplitting of T ∗ M ⊗ ΣM into the orthogonal <strong>and</strong> parallel sumT ∗ M ⊗ ΣM = ι(ΣM) ⊕ Σ 3/2 M,where Σ 3/2 M := ker(γ) <strong>and</strong> ι(ψ) := − 1 m∑ mj=1 e∗ j ⊗ e j · ψ. Here again(e j ) 1≤j≤m is a local orthonormal basis of the tangent bundle. Let D be thetwisted Dirac operator on T ∗ M ⊗ ΣM, thatis,D := i · (id ⊗ γ) ◦∇,where∇denotes the induced covariant derivative on T ∗ M ⊗ ΣM.Definition 3.21. The Rarita-Schwinger operator on the spin spacetime M isdefined by Q := (id − ι ◦ γ) ◦D: C ∞ (M,Σ 3/2 M) → C ∞ (M,Σ 3/2 M).By definition, the Rarita-Schwinger operator is pointwise obtained asthe orthogonal projection onto Σ 3/2 M of the twisted Dirac operator D restrictedto a section of Σ 3/2 M. As for the Dirac operator, its characteristicvariety coincides with the set of lightlike covectors, at least when m ≥ 3,see [4, Lemma 2.26]. In particular, [21, Thms. 23.2.4 & 23.2.7] imply thatthe Cauchy problem for Q is well-posed in case M is globally hyperbolic.Since the well-posedness of the Cauchy problem implies the existence of advanced<strong>and</strong> retarded Green’s operators (compare e.g. [4, Theorem 3.3.1 &Prop. 3.4.2] for wave operators), the operator Q has advanced <strong>and</strong> retardedGreen’s operators. Hence Q is not of Dirac type but is Green-hyperbolic.
196 C. Bär <strong>and</strong> N. GinouxRemark 3.22. The equations originally considered by Rarita <strong>and</strong> Schwingerin [28] correspond to the twisted Dirac operator D restricted to Σ 3/2 M butnot projected back to Σ 3/2 M. In other words, they considered the operatorD| C ∞ (M,Σ 3/2 M) : C ∞ (M,Σ 3/2 M) → C ∞ (M,T ∗ M ⊗ ΣM).These equations are over-determined. Therefore it is not a surprise that nontrivialsolutions restrict the geometry of the underlying manifold as observedby Gibbons [19] <strong>and</strong> that this operator has no Green’s operators.3.7. Combining given operators into a new oneGiven two Green-hyperbolic operators we can form the direct sum <strong>and</strong> obtaina new operator in a trivial fashion. Namely, let S 1 ,S 2 → M be two vectorbundles over a globally hyperbolic manifold M <strong>and</strong> let P 1 <strong>and</strong> P 2 be twoGreen-hyperbolic operators acting on sections of S 1 <strong>and</strong> S 2 respectively. Then( )P1 0P 1 ⊕ P 2 :=: C ∞ (M,S0 P 1 ⊕ S 2 ) → C ∞ (M,S 1 ⊕ S 2 )2is Green-hyperbolic [5, Lemma 2.27]. Note that the two operators need nothave the same order. Hence Green-hyperbolic operators need not be hyperbolicin the usual sense.4. Algebras of observablesOur next aim is to quantize the classical fields governed by Green-hyperbolicdifferential operators. We construct local algebras of observables <strong>and</strong> we provethat we obtain locally covariant quantum field theories in the sense of [12].4.1. Bosonic quantizationIn this section we show how a quantization process based on canonical commutationrelations (CCR) can be carried out for formally self-adjoint Greenhyperbolicoperators. This is a functorial procedure. We define the first categoryinvolved in the quantization process.Definition 4.1. The category GlobHypGreen consists of the following objects<strong>and</strong> morphisms:• An object in GlobHypGreen is a triple (M,S,P), where M is a globally hyperbolic spacetime, S is a real vector bundle over M endowed with a non-degenerateinner product 〈· , ·〉 <strong>and</strong> P is a formally self-adjoint Green-hyperbolic operator acting onsections of S.• A morphism between objects (M 1 ,S 1 ,P 1 ), (M 2 ,S 2 ,P 2 )ofGlobHypGreenis a pair (f,F), where f is a time-orientation preserving isometric embedding M 1 → M 2with f(M 1 ) causally compatible <strong>and</strong> open in M 2 ,
CCR- versus CAR-Quantization on Curved Spacetimes 197 F is a fiberwise isometric vector bundle isomorphism over f suchthat the following diagram commutes:C ∞ (M 2 ,S 2 )P 2C ∞ (M 2 ,S 2 )(5)resresC ∞ (M 1 ,S 1 )P 1 C ∞ (M 1 ,S 1 ),where res(ϕ) :=F −1 ◦ ϕ ◦ f for every ϕ ∈ C ∞ (M 2 ,S 2 ).Note that morphisms exist only if the manifolds have equal dimension<strong>and</strong> the vector bundles have the same rank. Note, furthermore, that the innerproduct 〈· , ·〉 on S is not required to be positive or negative definite.The causal compatibility condition, which is not automatically satisfied(see e.g. [5, Fig. 33]), ensures the commutation of the extension <strong>and</strong> restrictionmaps with the Green’s operators. Namely, if (f,F) be a morphism betweentwo objects (M 1 ,S 1 ,P 1 )<strong>and</strong>(M 2 ,S 2 ,P 2 ) in the category GlobHypGreen, <strong>and</strong>if (G 1 ) ± <strong>and</strong> (G 2 ) ± denote the respective Green’s operators for P 1 <strong>and</strong> P 2 ,then we haveres ◦ (G 2 ) ± ◦ ext = (G 1 ) ± .Here ext(ϕ) ∈ Cc ∞ (M 2 ,S 2 ) is the extension by 0 of F ◦ ϕ ◦ f −1 : f(M 1 ) → S 2to M 2 , for every ϕ ∈ Cc ∞ (M 1 ,S 1 ), see [4, Lemma 3.2].What is most important for our purpose is that the Green’s operatorsfor a formally self-adjoint Green-hyperbolic operator provide a symplecticvector space in a canonical way. First recall how the Green’s operators of anoperator <strong>and</strong> of its formally dual operator are related: if M is a globally hyperbolicspacetime, G + ,G − are the advanced <strong>and</strong> retarded Green’s operatorsfor a Green-hyperbolic operator P acting on sections of S → M <strong>and</strong> G ∗ +,G ∗ −denote the advanced <strong>and</strong> retarded Green’s operators for P ∗ ,then∫∫〈G ∗ ±ϕ, ψ〉 dV = 〈ϕ, G ∓ ψ〉 dV (6)Mfor all ϕ ∈ C ∞ c (M,S ∗ )<strong>and</strong>ψ ∈ C ∞ c (M,S), see e.g. [4, Lemma 3.3]. Thisimplies:Proposition 4.2. Let (M,S,P) be an object in the category GlobHypGreen.Set G := G + − G − ,whereG + ,G − are the advanced <strong>and</strong> retarded Green’soperator for P , respectively.Then the pair (SYMPL(M,S,P),ω) is a symplectic vector space, where∫SYMPL(M,S,P) :=Cc ∞ (M,S)/ ker(G) <strong>and</strong> ω([ϕ], [ψ]) := 〈Gϕ, ψ〉 dV.Here the square brackets [·] denote residue classes modulo ker(G).MM
198 C. Bär <strong>and</strong> N. GinouxProof. The bilinear form (ϕ, ψ) ↦→ ∫ M 〈Gϕ, ψ〉 dV on C∞ c (M,S) is skewsymmetricas a consequence of (6) because P is formally self-adjoint. Itsnull space is exactly ker(G). Therefore the induced bilinear form ω on thequotient space SYMPL(M,S,P) is non-degenerate <strong>and</strong> hence a symplecticform.□Theorem 3.8 shows that G(Cc∞ (M,S)) coincides with the space ofsmooth solutions of the equation Pϕ = 0 which have spacelike compactsupport. In particular, given an object (M,S,P) inGlobHypGreen, themapG induces an isomorphismSYMPL(M,S,P) =Cc∞ (M,S)/ ker(G) −→ ∼= ker(P ) ∩ Csc ∞ (M,S).Hence we may think of SYMPL(M,S,P) as the space of classical solutionsof the equation Pϕ = 0 with spacelike compact support.Now, let (f,F) be a morphism between objects (M 1 ,S 1 ,P 1 ) <strong>and</strong>(M 2 ,S 2 ,P 2 ) in the category GlobHypGreen. Then the extension by zeroinduces a symplectic linear map SYMPL(f,F) : SYMPL(M 1 ,S 1 ,P 1 ) →SYMPL(M 2 ,S 2 ,P 2 )withSYMPL(id M , id S )=id SYMPL(M,S,P) (7)<strong>and</strong>, for any further morphism (f ′ ,F ′ ):(M 2 ,S 2 ,P 2 ) → (M 3 ,S 3 ,P 3 ),SYMPL((f ′ ,F ′ ) ◦ (f,F)) = SYMPL(f ′ ,F ′ ) ◦ SYMPL(f,F). (8)Remark 4.3. Under the isomorphism SYMPL(M,S,P) → ker(P )∩C ∞ sc (M,S)induced by G, the extension by zero corresponds to an extension as a smoothsolution of Pϕ = 0 with spacelike compact support. In other words, for anymorphism (f,F) from(M 1 ,S 1 ,P 1 )to(M 2 ,S 2 ,P 2 )inGlobHypGreen we havethe following commutative diagram:SYMPL(M 1 ,S 1 ,P 1 )∼ =ker(P 1 ) ∩ C ∞ sc (M 1 ,S 1 )SYMPL(f,F)extension asa solutionSYMPL(M 2 ,S 2 ,P 2 )∼ = ker(P2 ) ∩ Csc ∞ (M 2 ,S 2 ).Summarizing, we have constructed a covariant functorSYMPL : GlobHypGreen −→ Sympl,where Sympl denotes the category of real symplectic vector spaces with symplecticlinear maps as morphisms. In order to obtain an algebra-valued functor,we compose SYMPL with the functor CCR which associates to anysymplectic vector space its Weyl algebra. Here “CCR” st<strong>and</strong>s for “canonicalcommutation relations”. This is a general algebraic construction which is independentof the context of Green-hyperbolic operators <strong>and</strong> which is carriedout in Section 2.2. As a result, we obtain the functorA bos := CCR ◦ SYMPL : GlobHypGreen −→ C ∗ Alg,
CCR- versus CAR-Quantization on Curved Spacetimes 199where C ∗ Alg is the category whose objects are the unital C ∗ -algebras <strong>and</strong>whose morphisms are the injective unit-preserving C ∗ -morphisms.In the remainder of this section we show that the functor A bos is abosonic locally covariant quantum field theory. We call two subregions M 1<strong>and</strong> M 2 of a spacetime M causally disjoint if <strong>and</strong> only if J M (M 1 ) ∩ M 2 = ∅.In other words, there are no causal curves joining M 1 <strong>and</strong> M 2 .Theorem 4.4. The functor A bos : GlobHypGreen −→ C ∗ Alg is a bosonic locallycovariant quantum field theory, i.e., the following axioms hold:(i) (<strong>Quantum</strong> causality) Let (M j ,S j ,P j ) be objects in GlobHypGreen,j =1, 2, 3, <strong>and</strong>(f j ,F j ) morphisms from (M j ,S j ,P j ) to (M 3 ,S 3 ,P 3 ),j = 1, 2, such that f 1 (M 1 ) <strong>and</strong> f 2 (M 2 ) are causally disjoint regionsin M 3 . Then the subalgebras A bos (f 1 ,F 1 )(A bos (M 1 ,S 1 ,P 1 )) <strong>and</strong>A bos (f 2 ,F 2 )(A bos (M 2 ,S 2 ,P 2 )) of A bos (M 3 ,S 3 ,P 3 ) commute.(ii) (Time-slice axiom) Let (M j ,S j ,P j ) be objects in GlobHypGreen, j =1, 2, <strong>and</strong>(f,F) a morphism from (M 1 ,S 1 ,P 1 ) to (M 2 ,S 2 ,P 2 ) such thatthere is a Cauchy hypersurface Σ ⊂ M 1 for which f(Σ) is a Cauchyhypersurface of M 2 .ThenA bos (f,F) :A bos (M 1 ,S 1 ,P 1 ) → A bos (M 2 ,S 2 ,P 2 )is an isomorphism.Proof. We first show (i). For notational simplicity we assume without lossof generality that f j <strong>and</strong> F j are inclusions, j =1, 2. Let ϕ j ∈ Cc ∞ (M j ,S j ).Since M 1 <strong>and</strong> M 2 are causally disjoint, the sections Gϕ 1 <strong>and</strong> ϕ 2 have disjointsupport, thus∫ω([ϕ 1 ], [ϕ 2 ]) = 〈Gϕ 1 ,ϕ 2 〉 dV = 0.MNow relation (iv) in Definition 2.8 tells usw([ϕ 1 ]) · w([ϕ 2 ]) = w([ϕ 1 ]+[ϕ 2 ]) = w([ϕ 2 ]) · w([ϕ 1 ]).Since A bos (f 1 ,F 1 )(A bos (M 1 ,S 1 ,P 1 )) is generated by elements of the formw([ϕ 1 ]) <strong>and</strong> A bos (f 2 ,F 2 )(A bos (M 2 ,S 2 ,P 2 )) by elements of the form w([ϕ 2 ]),the assertion follows.In order to prove (ii) we show that SYMPL(f,F) is an isomorphismof symplectic vector spaces provided f maps a Cauchy hypersurface of M 1onto a Cauchy hypersurface of M 2 . Since symplectic linear maps are alwaysinjective, we only need to show surjectivity of SYMPL(f,F). This is mosteasily seen by replacing SYMPL(M j ,S j ,P j )byker(P j ) ∩ Csc ∞ (M j ,S j )asinRemark 4.3. Again we assume without loss of generality that f <strong>and</strong> F areinclusions.Let ψ ∈ Csc ∞ (M 2 ,S 2 ) be a solution of P 2 ψ =0.Letϕ be the restrictionof ψ to M 1 .Thenϕ solves P 1 ϕ = 0 <strong>and</strong> has spacelike compact support inM 1 , see [4, Lemma 3.11]. We will show that there is only one solution in M 2with spacelike compact support extending ϕ. It will then follow that ψ isthe image of ϕ under the extension map corresponding to SYMPL(f,F) <strong>and</strong>surjectivity will be shown.
200 C. Bär <strong>and</strong> N. GinouxTo prove uniqueness of the extension, we may, by linearity, assume thatϕ =0.Thenψ + defined by{ψ(x), if x ∈ J M 2+ψ + (x) :=(Σ),0, otherwise,is smooth since ψ vanishes in an open neighborhood of Σ. Now ψ + solvesP 2 ψ + = 0 <strong>and</strong> has past-compact support. As noticed just below Theorem 3.8,this implies ψ + ≡ 0, i.e., ψ vanishes on J M 2+ (Σ). One shows similarly that ψvanishes on J M 2− (Σ), hence ψ =0. □The quantization process described in this subsection applies in particularto formally self-adjoint wave <strong>and</strong> Dirac-type operators.4.2. Fermionic quantizationNext we construct a fermionic quantization. For this we need a functorial constructionof Hilbert spaces rather than symplectic vector spaces. As we shallsee this seems to be possible only under much more restrictive assumptions.The underlying Lorentzian manifold M is assumed to be a globally hyperbolicspacetime as before. The vector bundle S is assumed to be complexwith Hermitian inner product 〈· , ·〉 which may be indefinite. The formallyself-adjoint Green-hyperbolic operator P is assumed to be of first order.Definition 4.5. A formally self-adjoint Green-hyperbolic operator P of firstorder acting on sections of a complex vector bundle S over a spacetime M isof definite type if <strong>and</strong> only if for any x ∈ M <strong>and</strong> any future-directed timeliketangent vector n ∈ T x M, the bilinear mapS x × S x → C, (ϕ, ψ) ↦→ 〈iσ P (n ♭ ) · ϕ, ψ〉,yields a positive definite Hermitian scalar product on S x .Example 4.6. The classical Dirac operator P from Example 3.17 is, whendefined with the correct sign, of definite type, see e.g. [6, Sec. 1.1.5] or [3,Sec. 2].Example 4.7. If E → M is a semi-Riemannian or semi-Hermitian vectorbundle endowed with a metric connection over a spin spacetime M, then thetwisted Dirac operator from Example 3.18 is of definite type if <strong>and</strong> only if themetric on E is positive definite. This can be seen by evaluating the tensorizedinner product on elements of the form σ ⊗ v, wherev ∈ E x is null.Example 4.8. The operator P = i(d−δ)onS =ΛT ∗ M⊗C is of Dirac type butnot of definite type. This follows from Example 4.7 applied to Example 3.19,since the natural inner product on ΣM is not positive definite. An alternativeelementary proof is the following: for any timelike tangent vector n on M <strong>and</strong>the corresponding covector n ♭ , one has〈iσ P (n ♭ )n ♭ , n ♭ 〉 = −〈n ♭ ∧ n ♭ − nn ♭ , n ♭ 〉 = 〈n, n〉〈1, n ♭ 〉 =0.Example 4.9. An elementary computation shows that the Rarita-Schwingeroperator defined in Section 3.6 is not of definite type if m ≥ 3, see [4, Ex. 3.16].
CCR- versus CAR-Quantization on Curved Spacetimes 201We define the category GlobHypDef, whose objects are triples (M,S,P),where M is a globally hyperbolic spacetime, S is a complex vector bundleequipped with a complex inner product 〈· , ·〉, <strong>and</strong>P is a formally self-adjointGreen-hyperbolic operator of definite type acting on sections of S. ThemorphismsarethesameasinthecategoryGlobHypGreen.We construct a covariant functor from GlobHypDef to HILB, whereHILBdenotes the category whose objects are complex pre-Hilbert spaces <strong>and</strong> whosemorphisms are isometric linear embeddings. As in Section 4.1, the underlyingvector space is the space of classical solutions to the equation Pϕ =0withspacelike compact support. We putSOL(M,S,P) :=ker(P ) ∩ Csc ∞ (M,S).Here “SOL” st<strong>and</strong>s for classical solutions of the equation Pϕ = 0 with spacelikecompact support. We endow SOL(M,S,P) with a positive definite Hermitianscalar product as follows: consider a smooth spacelike Cauchy hypersurfaceΣ ⊂ M with its future-oriented unit normal vector field n <strong>and</strong> itsinduced volume element dA <strong>and</strong> set∫(ϕ, ψ) := 〈iσ P (n ♭ ) · ϕ| Σ,ψ| Σ〉 dA, (9)Σfor all ϕ, ψ ∈ C ∞ sc (M,S). The Green’s formula for formally self-adjoint firstorderdifferential operators [32, p. 160, Prop. 9.1] (see also [4, Lemma 3.17])implies that (·,·) does not depend on the choice of Σ. Of course, it is positivedefinite because of the assumption that P is of definite type. In case P is notof definite type, the sesquilinear form (·,·) is still independent of the choiceof Σ but may be degenerate, see [4, Remark 3.18].For any object (M,S,P) inGlobHypDef we equip SOL(M,S,P) withthe Hermitian scalar product in (9) <strong>and</strong> thus turn SOL(M,S,P) intoapre-Hilbert space.Given a morphism (f,F): (M 1 ,S 1 ,P 1 ) → (M 2 ,S 2 ,P 2 )inGlobHypDef,then this is also a morphism in GlobHypGreen <strong>and</strong> hence induces a homomorphismSYMPL(f,F) :SYMPL(M 1 ,S 1 ,P 1 ) → SYMPL(M 2 ,S 2 ,P 2 ). Asexplained in Remark 4.3, there is a corresponding extension homomorphismSOL(f,F) :SOL(M 1 ,S 1 ,P 1 ) → SOL(M 2 ,S 2 ,P 2 ). In other words, SOL(f,F)is defined such that the diagramSYMPL(M 1 ,S 1 ,P 1 ) SYMPL(f,F) SYMPL(M 2 ,S 2 ,P 2 )(10)∼ =SOL(M 1 ,S 1 ,P 1 )SOL(f,F)∼ = SOL(M2 ,S 2 ,P 2 )commutes. The vertical arrows are the vector space isomorphisms induced bethe Green’s propagators G 1 <strong>and</strong> G 2 , respectively.Lemma 4.10. The vector space homomorphism SOL(f,F ): SOL(M 1 ,S 1 ,P 1 )→SOL(M 2 ,S 2 ,P 2 ) preserves the scalar products, i.e., it is an isometric linearembedding of pre-Hilbert spaces.
202 C. Bär <strong>and</strong> N. GinouxWe refer to [4, Lemma 3.19] for a proof. The functoriality of SYMPL<strong>and</strong> diagram (10) show that SOL is a functor from GlobHypDef to HILB, thecategory of pre-Hilbert spaces with isometric linear embeddings. Composingwith the functor CAR (see Section 2.1), we obtain the covariant functorA ferm := CAR ◦ SOL : GlobHypDef −→ C ∗ Alg.The fermionic algebras A ferm (M,S,P) are actually Z 2 -graded algebras, seeProposition 2.3 (iii).Theorem 4.11. The functor A ferm : GlobHypDef −→ C ∗ Alg is a fermioniclocally covariant quantum field theory, i.e., the following axioms hold:(i) (<strong>Quantum</strong> causality) Let (M j ,S j ,P j ) be objects in GlobHypDef, j =1, 2, 3, <strong>and</strong> (f j ,F j ) morphisms from (M j ,S j ,P j ) to (M 3 ,S 3 ,P 3 ),j = 1, 2, such that f 1 (M 1 ) <strong>and</strong> f 2 (M 2 ) are causally disjoint regionsin M 3 . Then the subalgebras A ferm (f 1 ,F 1 )(A ferm (M 1 ,S 1 ,P 1 )) <strong>and</strong>A ferm (f 2 ,F 2 )(A ferm (M 2 ,S 2 ,P 2 )) of A ferm (M 3 ,S 3 ,P 3 ) super-commute 1 .(ii) (Time-slice axiom) Let (M j ,S j ,P j ) be objects in GlobHypDef, j =1, 2,<strong>and</strong> (f,F) a morphism from (M 1 ,S 1 ,P 1 ) to (M 2 ,S 2 ,P 2 ) such that thereis a Cauchy hypersurface Σ ⊂ M 1 for which f(Σ) is a Cauchy hypersurfaceof M 2 .ThenA ferm (f,F) :A ferm (M 1 ,S 1 ,P 1 ) → A ferm (M 2 ,S 2 ,P 2 )is an isomorphism.Proof. To show (i), we assume without loss of generality that f j <strong>and</strong> F j areinclusions. Let ϕ 1 ∈ SOL(M 1 ,S 1 ,P 1 )<strong>and</strong>ψ 1 ∈ SOL(M 2 ,S 2 ,P 2 ). Denotethe extensions to M 3 by ϕ 2 := SOL(f 1 ,F 1 )(ϕ 1 )<strong>and</strong>ψ 2 := SOL(f 2 ,F 2 )(ψ 1 ).Choose a compact submanifold K 1 (with boundary) in a spacelike Cauchyhypersurface Σ 1 of M 1 such that supp(ϕ 1 ) ∩ Σ 1 ⊂ K 1 <strong>and</strong> similarly K 2for ψ 1 .SinceM 1 <strong>and</strong> M 2 are causally disjoint, K 1 ∪ K 2 is acausal. Hence,by [8, Thm. 1.1], there exists a Cauchy hypersurface Σ 3 of M 3 containingK 1 <strong>and</strong> K 2 . As in the proof of Lemma 4.10 one sees that supp(ϕ 2 ) ∩ Σ 3 =supp(ϕ 1 ) ∩ Σ 1 <strong>and</strong> similarly for ψ 2 . Thus, when restricted to Σ 3 , ϕ 2 <strong>and</strong>ψ 2have disjoint support. Hence (ϕ 2 ,ψ 2 ) = 0. This shows that the subspacesSOL(f 1 ,F 1 )(SOL(M 1 ,S 1 ,P 1 )) <strong>and</strong> SOL(f 2 ,F 2 )(SOL(M 2 ,S 2 ,P 2 )) ofSOL(M 3 ,S 3 ,P 3 ) are perpendicular. Since the even (resp. odd) part of theClifford algebra of a vector space V with quadratic form is linearly spannedby the even (resp. odd) products of vectors in V , Definition 2.1 shows thatthe corresponding CAR-algebras must super-commute.To see (ii) we recall that (f,F) is also a morphism in GlobHypGreen <strong>and</strong>that we know from Theorem 4.4 that SYMPL(f,F) is an isomorphism. Fromdiagram (10) we see that SOL(f,F) is an isomorphism. Hence A ferm (f,F) isalso an isomorphism.□1 This means that the odd parts of the algebras anti-commute while the even parts commutewith everything.
CCR- versus CAR-Quantization on Curved Spacetimes 203Remark 4.12. Since causally disjoint regions should lead to commuting observablesalso in the fermionic case, one usually considers only the even partA evenferm (M,S,P) as the observable algebra while the full algebra A ferm(M,S,P)is called the field algebra.There is a slightly different description of the functor A ferm .LetHILB Rdenote the category whose objects are the real pre-Hilbert spaces <strong>and</strong> whosemorphisms are the isometric linear embeddings. We have the functor REAL :HILB → HILB R which associates to each complex pre-Hilbert space (V,(·,·))its underlying real pre-Hilbert space (V,Re(·,·)). By Remark 2.7,A ferm =CAR sd ◦ REAL ◦ SOL.Since the self-dual CAR-algebra of a real pre-Hilbert space is the Cliffordalgebra of its complexification <strong>and</strong> since for any complex pre-Hilbert spaceV we haveREAL(V ) ⊗ R C = V ⊕ V ∗ ,A ferm (M,S,P ) is also the Clifford algebra of SOL(M,S,P ) ⊕ SOL(M,S,P ) ∗ =SOL(M,S ⊕ S ∗ ,P ⊕ P ∗ ). This is the way this functor is often described inthe physics literature, see e.g. [31, p. 115f].Self-dual CAR-representations are more natural for real fields. Let Mbe globally hyperbolic <strong>and</strong> let S → M be a real vector bundle equipped witha real inner product 〈· , ·〉. A formally skew-adjoint 2 differential operator Pacting on sections of S is called of definite type if <strong>and</strong> only if for any x ∈ M<strong>and</strong> any future-directed timelike tangent vector n ∈ T x M, the bilinear mapS x × S x → R, (ϕ, ψ) ↦→ 〈σ P (n ♭ ) · ϕ, ψ〉,yields a positive definite Euclidean scalar product on S x . An example is givenby the real Dirac operatorm∑D := ε j e j ·∇ ejj=1acting on sections of the real spinor bundle Σ R M.Given a smooth spacelike Cauchy hypersurface Σ ⊂ M with futuredirectedtimelike unit normal field n, we define a scalar product onSOL(M,S,P) =ker(P ) ∩ Csc ∞ (M,S,P) by∫(ϕ, ψ) := 〈σ P (n ♭ ) · ϕ| Σ,ψ| Σ〉 dA.ΣWith essentially the same proofs as before, one sees that this scalar productdoes not depend on the choice of Cauchy hypersurface Σ <strong>and</strong> that a morphism(f,F) :(M 1 ,S 1 ,P 1 ) → (M 2 ,S 2 ,P 2 ) gives rise to an extension operatorSOL(f,F) :SOL(M 1 ,S 1 ,P 1 ) → SOL(M 2 ,S 2 ,P 2 ) preserving the scalar product.We have constructed a functorSOL : GlobHypSkewDef −→ HILB R ,2 instead of self-adjoint!
204 C. Bär <strong>and</strong> N. Ginouxwhere GlobHypSkewDef denotes the category whose objects are triples(M,S,P) withM globally hyperbolic, S → M a real vector bundle withreal inner product <strong>and</strong> P a formally skew-adjoint, Green-hyperbolic differentialoperator of definite type acting on sections of S. The morphisms are thesame as before.Now the functorA sdferm := CAR sd ◦ SOL : GlobHypSkewDef −→ C ∗ Algis a locally covariant quantum field theory in the sense that Theorem 4.11holds with A ferm replaced by A sdferm .5. ConclusionWe have constructed three functors,A bos : GlobHypGreen −→ C ∗ Alg,A ferm : GlobHypDef −→ C ∗ Alg,A sdferm : GlobHypSkewDef −→ C ∗ Alg.The first functor turns out to be a bosonic locally covariant quantum fieldtheory while the second <strong>and</strong> third are fermionic locally covariant quantumfield theories.The category GlobHypGreen seems to contain basically all physically relevantfree fields such as fields governed by wave equations, Dirac equations,the Proca equation <strong>and</strong> the Rarita-Schwinger equation. It contains operatorsof all orders. Bosonic quantization of Dirac fields might be considered unphysicalbut the discussion shows that there is no spin-statistics theorem onthe level of observable algebras. In order to obtain results like Theorem 5.1in [33] one needs more structure, namely representations of the observablealgebras with good properties.The categories GlobHypDef <strong>and</strong> GlobHypSkewDef are much smaller.They contain only operators of first order with Dirac operators as main examples.But even certain twisted Dirac operators such as the Euler operatordo not belong to this class. The category GlobHypSkewDef is essentially thereal analogue of GlobHypDef.References[1] H. Araki: On quasifree states of CAR <strong>and</strong> Bogoliubov automorphisms. Publ.Res. Inst. Math. Sci. 6 (1970/71), 385–442.[2] C. Bär <strong>and</strong> C. Becker: C ∗ -algebras. In: C. Bär <strong>and</strong> K. Fredenhagen (Eds.):<strong>Quantum</strong> field theory on curved spacetimes. 1–37, Lecture Notes in Phys. 786,Springer-Verlag, Berlin, 2009.[3] C. Bär, P. Gauduchon, <strong>and</strong> A. Moroianu: Generalized cylinders in semi-Riemannian <strong>and</strong> spin geometry. Math. Zeitschr. 249 (2005), 545–580.
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206 C. Bär <strong>and</strong> N. Ginoux[24] R. Mühlhoff: Higher spin fields on curved spacetimes. Diplomarbeit, UniversitätLeipzig, 2007.[25] R. Mühlhoff: Cauchy problem <strong>and</strong> Green’s functions for first order differentialoperators <strong>and</strong> algebraic quantization. J. Math. Phys. 52 (2011), 022303.[26] B. O’Neill: Semi-Riemannian geometry. Academic Press, San Diego, 1983.[27] R. J. Plymen <strong>and</strong> P.L. Robinson: Spinors in Hilbert space. Cambridge Tractsin Mathematics 114, Cambridge University Press, Cambridge, 1994.[28] W. Rarita <strong>and</strong> J. Schwinger: On a theory of particles with half-integral spin.Phys. Rev. 60 (1941), 61.[29] M. Reed <strong>and</strong> B. Simon: Methods of modern mathematical physics I: Functionalanalysis. Academic Press, Orl<strong>and</strong>o, 1980.[30] K. S<strong>and</strong>ers: The locally covariant Dirac field. Rev. Math. Phys. 22 (2010),381–430.[31] A. Strohmaier: The Reeh-Schlieder property for quantum fields on stationaryspacetimes. Commun. Math. Phys. 215 (2000), 105–118.[32] M. E. Taylor: Partial differential equations I - Basic theory. Springer-Verlag,New York - Berlin - Heidelberg, 1996.[33] R. Verch: A spin-statistics theorem for quantum fields on curved spacetimemanifolds in a generally covariant framework. Commun. Math. Phys. 223(2001), 261–288.[34] McK. Y. Wang: Preserving parallel spinors under metric deformations. IndianaUniv. Math. J. 40 (1991), 815–844.[35] V. Wünsch: Cauchy’s problem <strong>and</strong> Huygens’ principle for relativistic higherspin wave equations in an arbitrary curved space-time. Gen.RelativityGravitation17 (1985), 15–38.Christian BärUniversität PotsdamInstitut für MathematikAm Neuen Palais 10Haus 814469 PotsdamGermanye-mail: baer@math.uni-potsdam.deNicolas GinouxFakultät für MathematikUniversität Regensburg93040 RegensburgGermanye-mail: nicolas.ginoux@mathematik.uni-regensburg.de
On the Notion of ‘the Same Physicsin All Spacetimes’Christopher J. FewsterAbstract. Brunetti, Fredenhagen <strong>and</strong> Verch (BFV) have shown how thenotion of local covariance for quantum field theories can be formulatedin terms of category theory: a theory being described as a functor froma category of spacetimes to a category of (C) ∗ -algebras. We discusswhether this condition is sufficient to guarantee that a theory represents‘the same physics’ in all spacetimes, giving examples to show thatit does not. A new criterion, dynamical locality, is formulated, whichrequires that descriptions of local physics based on kinematical <strong>and</strong> dynamicalconsiderations should coincide. Various applications are given,including a proof that dynamical locality for quantum fields is incompatiblewith the possibility of covariantly choosing a preferred state ineach spacetime.As part of this discussion we state a precise condition that shouldhold on any class of theories each representing the same physics in allspacetimes. This condition holds for the dynamically local theories butis violated by the full class of locally covariant theories in the BFV sense.The majority of results described form part of forthcoming paperswith Rainer Verch [16].Mathematics Subject Classification (2010). 81T05, 81T20, 81P99.Keywords. Locality, covariance, quantum field theory in curved spacetime.1. IntroductionThis contribution is devoted to the issue of how a physical theory shouldbe formulated in arbitrary spacetime backgrounds in such a way that thephysical content is preserved. Our motivation arises from various directions.First, it is essential for the extension of axiomatic quantum field theory tocurved spacetimes. Second, one would expect that any quantization of gravitycoupled to matter should, in certain regimes, resemble a common theory ofmatter on different fixed backgrounds. Third, as our universe appears to beF. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_11,© Springer Basel AG 2012207
208 C. J. Fewsterwell-described by a curved spacetime on many scales, one may well wonderwhat physical relevance should be ascribed to a theory that could only beformulated in Minkowski space.In Lagrangian theories, there is an apparently satisfactory answer to ourquestion: namely that the Lagrangian should transform covariantly undercoordinate transformations. Actually, even here there are subtleties, as asimple example reveals. Consider the nonminimally coupled scalar field withLagrangian densityL = 1 2√ −g(∇ a φ∇ a φ − ξRφ 2) .In Minkowski space the equation of motion does not depend on the couplingconstant ξ but theories with different coupling constant can still bedistinguished by their stress-tensors, which contain terms proportional to ξ.Suppose, however, that ξR is replaced by ζ(R), where ζ is a smooth functionvanishing in a neighbourhood of the origin <strong>and</strong> taking a constant value ξ 0 ≠0for all sufficiently large |R|. This gives a new theory that coincides with theξ = 0 theory in Minkowski space in terms of the field equation, stress-energytensor <strong>and</strong> any other quantity formed by functional differentiation of the action<strong>and</strong> then evaluated in Minkowski space. But in de Sitter space (withsufficiently large cosmological constant) the theory coincides, in the samesense, with the ξ = ξ 0 theory. Of course, it is unlikely that a theory of thistype would be physically relevant, but the example serves to illustrate thatcovariance of the Lagrangian does not guarantee that the physical content ofa theory will be the same in all spacetimes. Additional conditions would berequired; perhaps that the (undensitized) Lagrangian should depend analyticallyon the metric <strong>and</strong> curvature quantities.The situation is more acute when one attempts to generalize axiomaticquantum field theory to the curved spacetime context. After all, these approachesdo not take a classical action as their starting point, but focusinstead on properties that ought to hold for any reasonable quantum fieldtheory. In Minkowski space, Poincarécovariance<strong>and</strong>theexistenceofauniqueinvariant vacuum state obeying the spectrum condition provide strong constraintswhich ultimately account to a large part for the successes of axiomaticQFT in both its Wightman–Gårding [30] <strong>and</strong> Araki–Haag–Kastler [20] formulations.As a generic spacetime has no symmetries, <strong>and</strong> moreover attemptsto define distinguished vacuum states in general curved spacetimes lead tofailure even for free fields, 1 there are severe difficulties associated with generalizingthe axiomatic setting to curved spacetime.Significant progress was made by Brunetti, Fredenhagen <strong>and</strong> Verch [5](henceforth BFV) who formalized the idea of a locally covariant quantumfield theory in the language of category theory (see [26, 1] as general references).As we will see, this paper opens up new ways to analyze physicaltheories; in this sense it justifies its subtitle, ‘A new paradigm for local quantumphysics’. At the structural level, the ideas involved have led to a number1 Theorem 5.2 below strengthens this to a general no-go result.
The Same Physics in All Spacetimes 209of new model-independent results for QFT in curved spacetime such as thespin-statistics connection [31], Reeh–Schlieder type results [27] <strong>and</strong> the analysisof superselection sectors [6, 7]; they were also crucial in completing theperturbative construction of interacting QFT in curved spacetime [4, 22, 23].It should also be mentioned that antecedents of the ideas underlying theBFV formalism can be found in [17, 24, 11]. However, we stress that theBFV approach is not simply a matter of formalism: it suggests <strong>and</strong> facilitatesnew calculations that can lead to concrete physical predictions such asa priori bounds on Casimir energy densities [15, 14] <strong>and</strong> new viewpoints incosmology [8, 9] (see also Verch’s contribution to these proceedings [32]).The focus in this paper is on the physical content of local covariancein the BFV formulation: in particular, is it sufficiently strong to enforce thesame physics in all spacetimes? And, if not, what does? We confine ourselveshere to the main ideas, referring to forthcoming papers [16] for the details.2. Locally covariant physical theoriesTo begin, let us consider what is required for a local, causal description ofphysics. A minimal list might be the following:• Experiments can be conducted over a finite timespan <strong>and</strong> spatial extentin reasonable isolation from the rest of the world.• We may distinguish a ‘before’ <strong>and</strong> an ‘after’ for such experiments.• The ‘same’ experiment could, in principle, be conducted at other locations<strong>and</strong> times with the ‘same’ outcomes (to within experimentalaccuracy, <strong>and</strong> possibly in a statistical sense).• The theoretical account of such experiments should be independent (asfar as possible) of the rest of the world.Stated simply, we should not need to know where in the Universe our laboratoryis, nor whether the Universe has any extent much beyond its walls(depending on the duration of the experiment <strong>and</strong> the degree of shielding thewalls provide). Of course, these statements contain a number of undefinedterms; <strong>and</strong>, when referring to the ‘same’ experiment, we should take into accountthe motion of the apparatus relative to local inertial frames (compare aFoucault pendulum in Regensburg (latitude 49 ◦ N) with one in York (54 ◦ N)).Anticipating Machian objections to the last requirement, we assume thatsufficient structures are present to permit identification of (approximately)inertial frames.If spacetime is modelled as a Lorentzian manifold, the minimal requirementscan be met by restricting to the globally hyperbolic spacetimes. Recallthat an orientable <strong>and</strong> time-orientable Lorentzian manifold M (the symbolM will encompass the manifold, choice of metric, 2 orientation <strong>and</strong> timeorientation)is said to be globally hyperbolic if it has no closed causal curves2 We adopt signature + −···−.
210 C. J. Fewster<strong>and</strong> J + M (p) ∩ J − M(q) is compact for all p, q ∈ M (see [3, Thm 3.2] for equivalencewith the older definition [21]). 3 Compactness of these regions ensuresthat experiments can be conducted within bounded spacetime regions, whichcould in principle be shielded, <strong>and</strong> that communication within the region canbe achieved by finitely many radio links of finite range. Technically, of course,the main use of global hyperbolicity is that it guarantees well-posedness ofthe Klein–Gordon equation <strong>and</strong> other field equations with metric principalpart (as described, for example, in Prof. Bär’s contribution to this workshop).Henceforth we will assume that all spacetimes are globally hyperbolic<strong>and</strong> have at most finitely many connected components.The globally hyperbolic spacetimes constitute the objects of a categoryLoc, in which the morphisms are taken to be hyperbolic embeddings.Definition 2.1. A hyperbolic embedding of M into N is an isometry ψ : M →N preserving time <strong>and</strong> space orientations <strong>and</strong> such that ψ(M) is a causallyconvex 4 (<strong>and</strong> hence globally hyperbolic) subset of N.A hyperbolic embedding ψ : M → N allows us to regard N as anenlarged version of M: any experiment taking place in M should have ananalogue in ψ(M) which should yield indistinguishable results – at least if‘physics is the same’ in both spacetimes.BFV implemented this idea by regarding a physical theory as a functorfrom the category Loc to a category Phys, which encodes the type ofphysical system under consideration: the objects representing systems <strong>and</strong>the morphisms representing embeddings of one system in a larger one. BFVwere interested in quantum field theories, described in terms of their algebrasof observables. Here, natural c<strong>and</strong>idates for Phys are provided by Alg (resp.,C ∗ -Alg), whose objects are unital (C) ∗ -algebras with unit-preserving injective∗-homomorphisms as the morphisms. However, the idea applies more widely.For example, in the context of linear Hamiltonian systems, a natural choiceof Phys would be the category Sympl of real symplectic spaces with symplecticmaps as morphisms. As a more elaborate example, take as objects of acategory Sys all pairs (A,S)whereA∈Alg (or C ∗ -Alg) <strong>and</strong>S is a nonemptyconvex subset of the states on A, closed under operations induced by A. 5Whenever α : A→Bis an Alg-morphism such that α ∗ (T ) ⊂ S, we will saythat α induces a morphism (A,S) → (B,T)inSys. It may be shown thatSys is a category under the composition inherited from Alg. This categoryprovides an arena for discussing algebras of observables equipped with a distinguishedclass of states. The beauty of the categorical description is that itallows us to treat different types of physical theory in very similar ways.3 Here, J ± M (p) denotes the set of points that can be reached by future (+) or past (−)directed causal curves originating from a point p in M (including p itself). If S is a subsetof M, we write J ± (S) for the union J ± (S) = ⋃ p∈S J ± M (p), <strong>and</strong> J M (S) =J + M (S)∪J− M (S).4 A subset S of N is causally convex if every causal curve in N with endpoints in S iscontained wholly in S.5 That is, if σ ∈ S <strong>and</strong> A ∈Ahas σ(A ∗ A) = 1, then σ A (X) :=σ(A ∗ XA) defines σ A ∈ S.
The Same Physics in All Spacetimes 211According to BFV, then, a theory should assign to each spacetime M ∈Loc a mathematical object A (M) modelling ‘the physics on M’ <strong>and</strong>toeachhyperbolic embedding ψ : M → N a means of embedding the physics on Minside the physics on N, expressed as a morphism A (ψ) :A (M) → A (N).The natural conditions that A (id M )=id A (M) <strong>and</strong> A (ϕ◦ψ) =A (ϕ)◦A (ψ)are precisely those that make A a (covariant) functor.In particular, this gives an immediate definition of the ‘local physics’ inany nonempty open globally hyperbolic subset O of M asA kin (M; O) :=A (M| O ),where M| O denotes the region O with geometry induced from M <strong>and</strong> consideredas a spacetime in its own right; this embeds in A (M) byA kin (M; O) A (ι M;O)−−−−−−→ A (M),where ι M;O : M| O → M is the obvious embedding. Anticipating futuredevelopments, we refer to this as a kinematic description of the local physics.The kinematic description has many nice properties. Restricted to Minkowskispace, <strong>and</strong> with Phys = C ∗ -Alg, BFV were able to show that the netO ↦→ A kin (M; O) (for nonempty open, relatively compact, globally hyperbolicO) satisfies the Haag–Kastler axioms of algebraic QFT. 6 In generalspacetimes, it therefore provides a generalisation of the Haag–Kastler framework.It is worth focussing on one particular issue. Dimock [11] also articulateda version of Haag–Kastler axioms for curved spacetime QFT <strong>and</strong> alsoexpressed this partly in functorial language. However, Dimock’s covarianceaxiom was global in nature: it required that when spacetimes M <strong>and</strong> N areisometric then there should be an isomorphism between the nets of algebrason M <strong>and</strong> N. In the BFV framework this idea is localised <strong>and</strong> extended tosituations in which M is hyperbolically embedded in N but not necessarilyglobally isometric to it. To be specific, suppose that there is a morphismψ : M → N <strong>and</strong> that O is a nonempty, open globally hyperbolic subset ofM with finitely many connected components. Then ψ(O) also obeys theseconditions in N <strong>and</strong>, moreover, restricting the domain of ψ to O <strong>and</strong> thecodomain to ψ(O), we obtain an isomorphism ˆψ : M| O → N| ψ(O) makingthe diagramˆψ∼ =M| O N| ψ(O)ι M;OMψNι N;ψ(O)6 More precisely, BFV study the images A (ι M;O )(A kin (M; O)) as sub-C ∗ -algebras ofA (M).
212 C. J. Fewstercommute. As functors always map commuting diagrams to commuting diagrams,<strong>and</strong> isomorphisms to isomorphisms, we obtain the commuting diagramA kin (M; O)A ( ˆψ)A kin (N; ψ(O))∼ =(2.1)α kinM;OA (M)A (ψ)A (N)α kinN;ψ(O)in which, again anticipating future developments, we have written αM;O kin forthe morphism A (ι M;O ) embedding A kin (M; O) inA (M). The significanceof this diagram is that it shows that the kinematic description of local physicsis truly local: it assigns equivalent descriptions of the physics to isometricsubregions of the ambient spacetimes M <strong>and</strong> N. This holds even when thereis no hyperbolic embedding of one of these ambient spacetimes in the other,as can be seen if we consider a further morphism ϕ : M → L, thus obtainingan isomorphism A ( ˆψ) ◦ A (ˆϕ −1 ):A kin (L; ϕ(O)) → A kin (N; ψ(O)) eventhough there need be no morphism between L <strong>and</strong> N.Example: the real scalar field. Take Phys = Alg. The quantization of theKlein–Gordon equation (□ M +m 2 )φ = 0 is well understood in arbitrary globallyhyperbolic spacetimes. To each spacetime M ∈ Loc we assign A (M) ∈Alg with generators Φ M (f), labelled by test functions f ∈ C0 ∞ (M) <strong>and</strong>interpretedas smeared fields, subject to the following relations (which hold forarbitrary f,f ′ ∈ C0 ∞ (M)):• f ↦→ Φ M (f) is complex linear• Φ M (f) ∗ =Φ M (f)• Φ M ((□ M + m 2 )f) =0• [Φ M (f), Φ M (f ′ )] = iE M (f,f ′ )1 A (M) .Here, E M is the advanced-minus-retarded Green function for □ M +m 2 ,whoseexistence is guaranteed by global hyperbolicity of M. Nowifψ : M → N isa hyperbolic embedding we haveψ ∗ □ M f = □ N ψ ∗ f <strong>and</strong> E N (ψ ∗ f,ψ ∗ f ′ )=E M (f,f ′ ) (2.2)for all f,f ′ ∈ C0∞ (M), where{f(ψ −1 (p)) p ∈ ψ(M)(ψ ∗ f)(p) =0 otherwise.The second assertion in (2.2) follows from the first, together with the uniquenessof advanced/retarded solutions to the inhomogeneous Klein–Gordonequation. In consequence, the mapA (ψ)(Φ M (f)) = Φ N (ψ ∗ f), A (ψ)1 A (M) = 1 A (N)extends to a ∗-algebra homomorphism A (ψ) :A (M) → A (N). Furthermore,this is a monomorphism because A (M) is simple, so A (ψ) isindeeda morphism in Alg. It is clear that A (ψ ◦ ϕ) = A (ψ) ◦ A (ϕ) <strong>and</strong>A (id M )=id A (M) because these equations hold on the generators, so A
The Same Physics in All Spacetimes 213M I NFigure 1. Schematic representation of the deformation constructionin [18]: globally hyperbolic spacetimes M <strong>and</strong> N,whose Cauchy surfaces are related by an orientation preservingdiffeomorphism, can be linked by a chain of Cauchymorphisms <strong>and</strong> an interpolating spacetime I.indeed defines a functor from Loc to Alg. Finally, if O is a nonempty openglobally hyperbolic subset of M, A kin (M; O) is defined as A (M| O ); its imageunder A (ι M;O ) may be characterized as the unital subalgebra of A (M)generated by those Φ M (f) withf ∈ C0∞ (O).3. The time-slice axiom <strong>and</strong> relative Cauchy evolutionThe structures described so far may be regarded as kinematic. To introducedynamics we need a replacement for the idea that evolution is determinedby data on a Cauchy surface. With this in mind, we say that a hyperbolicembedding ψ : M → N is Cauchy if ψ(M) contains a Cauchy surface ofN <strong>and</strong> say that a locally covariant theory A satisfies the time-slice axiom ifA (ψ) is an isomorphism whenever ψ is Cauchy. The power of the time-sliceaxiom arises as follows. Any Cauchy surface naturally inherits an orientationfrom the ambient spacetime; if two spacetimes M <strong>and</strong> N have Cauchy surfacesthat are equivalent modulo an orientation-preserving diffeomorphism(not necessarily an isometry) then the two spacetimes may be linked by achain of Cauchy morphisms [18] (see Fig. 1). If the time-slice axiom is satisfied,then the theory A assigns isomorphisms to each Cauchy morphism;this can often be used to infer that the theory on N obeys some property,given that it holds in M.One of the innovative features of the BFV framework is its ability toquantify the response of the theory to a perturbation in the metric. We writeH(M) for the set of compactly supported smooth metric perturbations h onM that are sufficiently mild so as to modify M to another globally hyperbolicspacetime M[h]. In the schematic diagram of Fig. 2, the metric perturbationlies between two Cauchy surfaces. By taking a globally hyperbolic neighbourhoodof each Cauchy surface we are able to find Cauchy morphisms ι ± <strong>and</strong>ι ± [h]asshown.NowifA obeys the time-slice axiom, any metric perturbationh ∈ H(M) defines an automorphism of A (M),rce M [h] =A (ι − ) ◦ A (ι − [h]) −1 ◦ A (ι + [h]) ◦ A (ι + ) −1 .BFV showed that the functional derivative of the relative Cauchy evolutiondefines a derivation on the algebra of observables which can be interpreted
214 C. J. Fewsterι +ι −M +ι + [h]ι − [h]hMM −M[h]Figure 2. Morphisms involved in relative Cauchy evolutionas a commutator with a stress-energy tensor so that[T M (f),A]=2i d ds rce M [h(s)]A∣ ,s=0where h(s) is a smooth one-parameter family of metric perturbations inH(M) <strong>and</strong>f = ḣ(0). In support of the interpretation of T as a stress-energytensor we may cite the fact that T is symmetric <strong>and</strong> conserved. Moreover,the stress-energy tensor in concrete models can be shown to satisfy the aboverelation (see BFV <strong>and</strong> [28] for the scalar <strong>and</strong> Dirac fields respectively) <strong>and</strong>we will see further evidence for this link below in Sect. 7.4. The same physics in all spacetimes4.1. The meaning of SPASsWe can now begin to analyse the question posed at the start of this paper.However, is not yet clear what should be understood by saying that atheory represents the same physics in all spacetimes (SPASs). Rather thangive a direct answer we instead posit a property that should be valid for anyreasonable notion of SPASs: If two theories individually represent the samephysics in all spacetimes, <strong>and</strong> there is some spacetime in which the theoriescoincide then they should coincide in all spacetimes. In particular, we obtainthe following necessary condition:Definition 4.1. A class of theories T has the SPASs property if no propersubtheory T ′ of T in T can fully account for the physics of T in any singlespacetime.This can be made precise by introducing a new category: the categoryof locally covariant theories, LCT, whose objects are functors from Loc toPhys <strong>and</strong> in which the morphisms are natural transformations between suchfunctors. Recall that a natural transformation ζ : A → . B between theoriesA <strong>and</strong> B assigns to each M a morphism A (M) ζ M−→ B(M) sothatforeach
The Same Physics in All Spacetimes 215hyperbolic embedding ψ, the following diagramζ MM A (M) B(M)ψA (ψ)B(ψ)ζ NN A (N) B(N)commutes; that is, ζ N ◦ A (ψ) =B(ψ) ◦ ζ M . The interpretation is that ζembeds A as a sub-theory of B. Ifeveryζ M is an isomorphism, ζ is anequivalence of A <strong>and</strong> B; if some ζ M is an isomorphism we will say that ζ isa partial equivalence.Example 4.2. With Phys = Alg: given any A ∈ LCT, define A ⊗k (k ∈ N) byA ⊗k (M) =A (M) ⊗k ,A ⊗k (ψ) =A (ψ) ⊗ki.e., k independent copies of A , where the algebraic tensor product is used.Thenη k,lM : A ⊗k (M) → A ⊗l (M)A ↦→ A ⊗ 1 ⊗(l−k)A (M)defines a natural η k,l : A ⊗k .→ A ⊗l for k ≤ l <strong>and</strong> we have the obviousrelations that η k,m = η l,m ◦ η k,l if k ≤ l ≤ m.We can now formulate the issue of SPASs precisely:Definition 4.3. AclassT of theories in LCT has the SPASs property if allpartial equivalences between theories in T are equivalences.That is, if A <strong>and</strong> B are theories in T, such that A is a subtheory of Bwith ζ : A → . B in LCT, <strong>and</strong>ζ M : A (M) → B(M) is an isomorphism forsome M, thenζ is an equivalence.4.2. Failure of SPASs in LCTPerhaps surprisingly, theories in LCT need not represent the same physicsin all spacetimes. Indeed, a large class of pathological theories may be constructedas follows. Let us take any functor ϕ : Loc → LCT, i.e., a locallycovariant choice of locally covariant theory [we will give an example below].Thus for each M, ϕ(M) ∈ LCT is a choice of theory defined in all spacetimes,<strong>and</strong> each hyperbolic embedding ψ : M → N corresponds to an embeddingϕ(ψ) ofϕ(M) as a sub-theory of ϕ(N).Given ϕ as input, we may define a diagonal theory ϕ Δ ∈ LCT by settingϕ Δ (M) =ϕ(M)(M) ϕ Δ (ψ) =ϕ(ψ) N ◦ ϕ(M)(ψ)for each spacetime M <strong>and</strong> hyperbolic embedding ψ : M → N. The definitionof ϕ Δ (ψ) is made more comprehensible by realising that it is the diagonal ofa natural square
216 C. J. Fewsterϕ(ψ) MM ϕ(M)(M) ϕ(N)(M)ψϕ(M)(ψ)ϕ Δ(ψ)ϕ(N)(ψ)N ϕ(M)(N) ϕ(N)(N)ϕ(ψ) Narising from the subtheory embedding ϕ(ψ) :ϕ(M) . → ϕ(N). The key pointis that, while ϕ Δ is easily shown to be a functor <strong>and</strong> therefore defines alocally covariant theory, there seems no reason to believe that ϕ Δ shouldrepresent the same physics in all spacetimes. Nonetheless, ϕ Δ may have manyreasonable properties. For example, it satisfies the time-slice axiom if eachϕ(M) does <strong>and</strong> in addition ϕ does (i.e., ϕ(ψ) is an equivalence for everyCauchy morphism ψ).To demonstrate the existence of nontrivial functors ϕ : Loc → LCT [inthe case Phys = Alg] letuswriteΣ(M) to denote the equivalence class ofthe Cauchy surface of M modulo orientation-preserving diffeomorphisms <strong>and</strong>suppose that λ : Loc → N = {1, 2,...} is constant on such equivalence classes<strong>and</strong> obeys λ(M) =1ifΣ(M) is noncompact. Defining μ(M) =max{λ(C) :C a component of M}, for example, one can show:Proposition 4.4. Fix any theory A ∈ LCT (Phys = Alg). Then the assignmentsϕ(M) =A ⊗μ(M) ϕ(M → ψ N) =η μ(M),μ(N)for all M ∈ Loc <strong>and</strong> hyperbolic embeddings ψ : M → N define a functor ϕ ∈Fun(Loc, LCT), whereη k,l are the natural transformations in Example 4.2. IfA obeys the time-slice axiom, then so does ϕ Δ .Proof. Thekeypointistoshowthatμ is monotone with respect to hyperbolicembeddings: if ψ : M → N then μ(M) ≤ μ(N). This in turn follows froma result in Lorentzian geometry which [for connected spacetimes] asserts: ifψ : M → N is a hyperbolic embedding <strong>and</strong> M has compact Cauchy surfacethen ψ is Cauchy <strong>and</strong> the Cauchy surfaces of N are equivalent to those of Munder an orientation preserving diffeomorphism. The functorial properties ofϕ follow immediately from properties of η k,l .IfA obeys the time-slice axiom,then so do its powers A ⊗k .Moreover,ifψ is Cauchy, then M <strong>and</strong> N haveoriented-diffeomorphic Cauchy surfaces <strong>and</strong> therefore ϕ(ψ) is an identity, <strong>and</strong>ϕ Δ (ψ) =A ⊗μ(M) (ψ) is an isomorphism.□putAs a concrete example, which we call the one-field-two-field model, ifweλ(M) ={1 Σ(M) noncompact2 otherwisethen the resulting diagonal theory ϕ Δ represents one copy of the underlyingtheory A in spacetimes whose Cauchy surfaces are purely noncompact
The Same Physics in All Spacetimes 217ϕ Δ (D) =A (D)ϕ Δ (C) =A ⊗2 (C)ϕ Δ (C ⊔ D) =A ⊗2 (C ⊔ D)Figure 3. The one-field-two-field model assigns one copyof the theory A to a diamond spacetime D with noncompactCauchy surface, but two copies to a spacetime C withcompact Cauchy surface, or spacetimes such as the disjointunion C ⊔ D which have at least one component with compactCauchy surface.(i.e., have no compact connected components), but two copies in all otherspacetimes (see Fig. 3). Moreover, there are obvious subtheory embeddingsA −→ . .ϕ Δ −→ A ⊗2 (4.1)which are isomorphisms in some spacetimes but not in others; the left-h<strong>and</strong>in spacetimes with purely noncompact Cauchy surfaces but not otherwise,<strong>and</strong> vice versa for the right-h<strong>and</strong> embedding. As we have exhibited partialequivalences that are not equivalences, we conclude that SPASs fails in LCT.Meditating on this example, we can begin to see the root of the problem.If O is any nonempty open globally hyperbolic subset of any M such that Ohas noncompact Cauchy surfaces then the kinematic local algebras obeyϕ kinΔ (M; O) =ϕ kinΔ (M| O )=A kin (M; O), (4.2)so the kinematic local algebras are insensitive to the ambient algebra. Inone-field-two-field model, for example, the kinematical algebras of relativelycompact regions with nontrivial causal complement are always embeddingsof the corresponding algebra for a single field, even when the ambient algebracorresponds to two independent fields. This leads to a surprising diagnosis:the framework described so far is insufficiently local, <strong>and</strong> it is necessary tofind another way of describing the local physics.The problem, therefore, is to detect the existence of the ambient localdegrees of freedom that are ignored in the kinematical local algebras. Thesolution is simple: where there are physical degrees of freedom, there oughtto be energy. In other words, we should turn our attention to dynamics.Some immediate support for the idea that dynamics has a role to playcan be obtained by computing the relative Cauchy evolution of a diagonal
218 C. J. Fewstertheory. The result isrce (ϕ Δ)M[h] =(rce(ϕ)M [h]) M ◦ rce (ϕ(M))M[h], (4.3)where rce (ϕ)M[h] ∈ Aut(ϕ(M)) is the relative Cauchy evolution of ϕ in LCT.In known examples this is trivial, but it would be intriguing to find examplesin which the choice of theory as a function of the spacetime contributesnontrivially to the total stress-energy tensor of the theory. Provided thatrce (ϕ) is indeed trivial <strong>and</strong> stress-energy tensors exist, it follows that[T (ϕ Δ)M(f),A]=[T(ϕ(M)) (f),A]for all A ∈ ϕ Δ (M), which means that the stress-energy tensor correctlydetects the ambient degrees of freedom that are missed in the kinematicdescription.M5. Local physics from dynamicsLet A be a locally covariant theory obeying the time-slice axiom <strong>and</strong> O be aregion in spacetime M. Thekinematical description of the physical contentof O was given in terms of the physics assigned to O when considered as aspacetime in its own right, i.e., A kin (M; O) =A (M| O ). For the dynamicaldescription, we focus on that portion of the physical content on M thatis unaffected by the geometry in the causal complement of O. Thiscanbeisolated using the relative Cauchy evolution, which precisely encapsulates theresponse of the system under a change in the geometry.Accordingly, for any compact set K ⊂ M, we define A • (M; K) tobe the maximal subobject of A (M) invariant under rce M [h] for all h ∈H(M; K ⊥ ), where H(M; K ⊥ ) is the set of metric perturbations in H(M)supported in the causal complement K ⊥ = M \ J M (K) ofK. Subobjects ofthis type will exist provided the category Phys has equalizers (at least for pairsof automorphisms) <strong>and</strong> arbitrary set-indexed intersections. If Phys = Alg, ofcourse, A • (M; K) is simply the invariant subalgebra.From the categorical perspective, it is usually not objects that are of interestbut rather the morphisms between them. Strictly speaking, a subobjectof an object A in a general category is an equivalence class of monomorphismswith codomain A, whereα <strong>and</strong> α ′ are regarded as equivalent if there is a (necessarilyunique) isomorphism β such that α ′ = α◦β. In our case, we are reallyinterested in a morphism α M;K • with codomain A (M), characterized (up toisomorphism) by the requirements that (a)rce M [h] ◦ α M;K • = α M;K • ∀h ∈ H(M; K ⊥ )<strong>and</strong> (b) if β is any other morphism with this property then there existsa unique morphism ˆβ such that β = α M;K • ◦ ˆβ. The notation A • (M; K)denotes the domain of α M;K • . For simplicity <strong>and</strong> familiarity, however, we willignore this issue <strong>and</strong> write everything in terms of the objects, rather than themorphisms. This is completely legitimate in categories such as Alg or C ∗ -Alg,in which the objects are sets with additional structure <strong>and</strong> A • (M; K) canbe
The Same Physics in All Spacetimes 219concretely constructed as the subset of A (M) left fixed by the appropriaterelative Cauchy evolution automorphisms. (At the level of detail, however,it is not simply a matter of precision to work with the morphisms: universaldefinitions, such as that given above for α M;K • , permit efficient proofs thatare ‘portable’ between different choices of the category Phys.)The subobjects A • (M; K) have many features reminiscent of a netof local algebras in local quantum physics: for example, isotony (K 1 ⊂ K 2implies A • (M; K 1 ) ⊂ A • (M; K 2 )) together withif K ⊥⊥ is also compact, <strong>and</strong>A • (M; K) =A • (M; K ⊥⊥ )A • (M; K 1 ) ∨ A • (M; K 2 ) ⊂ A • (M; K 1 ∪ K 2 )A • (M; K 1 ∩ K 2 ) ⊂ A • (M; K 1 ) ∩ A • (M; K 2 )A • (M; ∅) ⊂ A • (M; K)for any compact K, K 1 ,K 2 . A rather striking absence is that there is no localcovariance property: it is not true in general that a morphism ψ : M → Ninduces isomorphisms between A • (M; K) <strong>and</strong>A • (N; ψ(K)) so as to makethe diagramA • (M; K)∼ =?A • (N; ψ(K))α • M;Kα • N;ψ(K)(5.1)A (M)A (ψ)A (N)commute (unless ψ is itself an isomorphism). Again, this can be seen fromthe example of diagonal theories, for which we haveϕ • Δ(M; K) =ϕ(M) • (M; K)as a consequence of (4.3). In the one-field-two-field model, for example, ahyperbolic embedding of ψ : D → C (where D <strong>and</strong> C areasinFig.3)we have ϕ • Δ (D; K) =A • (D; K) but ϕ • Δ (C; ψ(K)) = A (⊗2)• (C; ψ(K)), forany compact subset K of D. In general, these subobjects will not be isomorphicin the required sense. This situation should be compared with the localcovariance property enjoyed by the kinematic description, expressed by thecommuting diagram (2.1).The discussion so far has shown how dynamics allows us to associatea subobject A • (M; K) ofA (M) with every compact subset K of M. Asmentioned above, for simplicity we are taking a concrete viewpoint in whichA • (M; K) is a subset of A (M). For purposes of comparison with the kinematicviewpoint, we must also associate subobjects of A (M) with open subsetsof M. Given an open subset O of M, the basic idea will be to take thesubobject generated by all A • (M; K) indexed over a suitable class of compactsubsets K of M. TodefinetheclassofK involved, we must introducesome terminology.
220 C. J. FewsterFollowing [7], a diamond is defined to be an open relatively compactsubset of M, taking the form D M (B) whereB, thebase of the diamond,is a subset of a Cauchy surface Σ so that in a suitable chart (U, φ) ofΣ,φ(B) is a nonempty open ball in R n−1 whose closure is contained in φ(U). (Agiven diamond has many possible bases.) By a multi-diamond, we underst<strong>and</strong>any finite union of mutually causally disjoint diamonds; the base of a multidiamondis formed by any union of bases for each component diamond.Given these definitions, for any open set O in M, letK (M; O) betheset of compact subsets of O that have a multi-diamond neighbourhood witha base contained in O. We then define∨A dyn (M; O) =A • (M; K) ;K∈K (M;O)that is, the smallest Phys-subobject of A (M)thatcontainsalltheA • (M; K)indexed by K ∈ K (M; O). Of course, the nature of the category Phys entershere: if it is Alg (resp., C ∗ -Alg) thenA dyn (M; O) isthe(C) ∗ -subalgebra ofA (M) generated by the A • (M; K); if Phys is Sympl, then the linear spanis formed. More abstractly, we are employing the categorical join of subobjects,which can be given a universal definition in terms of the morphismsαM;K • , <strong>and</strong> gives a morphism (characterized up to isomorphism) αdynM;O withcodomain A (M); A dyn (M; O) is the domain of some such morphism. In Alg(<strong>and</strong> more generally, if Phys has pullbacks, <strong>and</strong> class-indexed intersections)the join has the stronger property of being a categorical union [10], <strong>and</strong> thisis used in our general setup.It is instructive to consider these algebras for diagonal theories ϕ Δ ,under the assumption that rce (ϕ) is trivial. Using (4.3), it follows almostimmediately thatϕ • Δ(M; K) =ϕ(M) • (M; K), ϕ dynΔ (M; O) =ϕ(M)dyn (M; O)for all compact K ⊂ M <strong>and</strong> open O ⊂ M. This should be contrasted withthe corresponding calculations for kinematic algebras in (4.2); we see that thedynamical algebras sense the ambient spacetime in a way that the kinematicalgebras do not. In the light of this example, the following definition is verynatural.Definition 5.1. A obeys dynamical locality ifA dyn (M; O) ∼ = A kin (M; O)for all nonempty open globally hyperbolic subsets O of M with finitely manycomponents, where the isomorphism should be understood as asserting theexistence of isomorphisms β M;O such that α dynM;O = αkin M;O ◦ β M;O; i.e., equivalenceas subobjects of A (M).In the next section, we will see that the class of dynamically local theoriesin LCT has the SPASs property, our main focus in this contribution.However, we note that dynamical locality has a number of other appealing
The Same Physics in All Spacetimes 221consequences, which suggest that it may prove to be a fruitful assumption inother contexts.First, any dynamically local theory A is necessarily additive, in thesense thatA (M) = ∨ A dyn (M; O),Owhere the categorical union is taken over a sufficiently large class of openglobally hyperbolic sets O; for example, the truncated multi-diamonds (intersectionsof a multi-diamond with a globally hyperbolic neighbourhood ofa Cauchy surface containing its base).Second, the ‘net’ K ↦→ A • (M; K) is locally covariant: if ψ : M → N<strong>and</strong> K ⊂ M is outer regular then there is a unique isomorphism making thediagram (5.1) commute. Here, a compact set K is said to be outer regularif there exist relatively compact nonempty open globally hyperbolic subsetsO n (n ∈ N) with finitely many components such that (a) cl(O n+1 ) ⊂ O n <strong>and</strong>K ∈ K (M; O n )foreachn <strong>and</strong> (b) K = ⋂ n∈N O n.Third, in the case Phys = Alg or C ∗ -Alg, 7 if A • (M; ∅) = C1 A (M)<strong>and</strong> A is dynamically local then the theory obeys extended locality [25], i.e.,if O 1 ,O 2 are causally disjoint nonempty globally hyperbolic subsets thenA kin (M; O 1 ) ∩ A kin (M; O 2 )=C1 A (M) . Conversely, we can infer trivialityof A • (M; ∅) from extended locality.Fourth, there is an interesting application to the old question of whetherthere can be any preferred state in general spacetimes. Suppose that A isa theory in LCT with Phys taken to be Alg or C ∗ -Alg. Anatural state ω ofthetheoryisanassignmentM ↦→ ω M of states ω M on A (M), subject tothe contravariance requirement that ω M = ω N ◦ A (ψ) for all ψ : M → N.Holl<strong>and</strong>s <strong>and</strong> Wald [22] give an argument to show that this is not possiblefor the free scalar field; likewise, BFV also sketch an argument to this effect(also phrased concretely in terms of the free field). The following brings thesearguments to a sharper <strong>and</strong> model-independent form:Theorem 5.2. Suppose A is a dynamically local theory in LCT equippedwithanatural state ω. IfthereisaspacetimeM with noncompact Cauchy surfacessuch that ω M induces a faithful representation of A (M) with the Reeh–Schlieder property, then the relative Cauchy evolution in M is trivial. Moreover,if extended locality holds in M then A is equivalent to the trivial theory.By the Reeh–Schlieder property, we mean that the GNS vector Ω M correspondingto ω M is cyclic for the induced representation of each A kin (M; O)for any open, globally hyperbolic, relatively compact O ⊂ M. This requirementwill be satisfied if, for example, the theory in Minkowski spaceobeys st<strong>and</strong>ard assumptions of AQFT with the natural state reducing to theMinkowski vacuum state. The trivial theory I is the theory assigning thetrivial unital algebra to each M <strong>and</strong> its identity morphism to any morphismin LCT.7 One may also formulate an analogous statement for more general categories Phys.
222 C. J. FewsterSketch of proof. We first argue that ω M ◦ rce M [h] =ω M for all h ∈ H(M)<strong>and</strong> all M ∈ Loc by virtue of the natural state assumption <strong>and</strong> the definitionof the relative Cauchy evolution. In the GNS representation π M of ω M ,therelative Cauchy evolution can be unitarily represented by unitaries leavingthe vacuum vector Ω M invariant. Choose any h ∈ H(M) <strong>and</strong> an open,relatively compact, globally hyperbolic O spacelike separated from supp h.Bygeneral properties of the relative Cauchy evolution, we have rce M [h]◦α kinα kinM;O =M;O <strong>and</strong> hence that the unitary U M [h] implementing rce M [h] isequaltothe identity on the subspace π M (A (ι M;O )A)Ω M (A ∈ A (M| O )) which isdense by the Reeh–Schlieder property. As π M is faithful, the relative Cauchyevolution is trivial as claimed.It follows that A • (M; K) =A (M) for all compact K;consequently,bydynamical locality, A kin (M; O) =A dyn (M; O) =A (M) for all nonemptyopen, globally hyperbolic O. Taking two such O at spacelike separation <strong>and</strong>applying extended locality, we conclude that A (M) =C1. The remainderof the proof will be given in the next section.□6. SPASs at lastOur aim is to show that the category of dynamically local theories has theSPASs property.Theorem 6.1. Suppose A <strong>and</strong> B are theories in LCT obeying dynamical locality.Then any partial equivalence ζ : A. → B is an equivalence.Sketch of proof. Themainideasareasfollows.First,weshowthatifζ N isan isomorphism then ζ L is an isomorphism for every spacetime L for whichthereisamorphismψ : L → N. Second, if ψ : L → N is Cauchy, then ζ L isan isomorphism if <strong>and</strong> only if ζ N is.As ζ is a partial equivalence, there exists a spacetime M for which ζ M isan isomorphism. It follows that ζ D is an isomorphism for every multi-diamondspacetime D thatcanbeembeddedinM. Using deformation arguments[18] there is a chain of Cauchy morphisms linking each such multi-diamondspacetime to any other multi-diamond spacetime with the same number ofconnected components. Hence ζ D is an isomorphism for all multi-diamondspacetimes D. Now consider any other spacetime N. Owing to dynamicallocality, both A (N) <strong>and</strong>B(N) are generated over subobjects that correspondto diamond spacetimes, which (it transpires) are isomorphic under therestriction of ζ N . By the properties of the categorical union, it follows thatζ N is an isomorphism.□We remark that the chain of partial equivalences in (4.1) shows that wecannot relax the condition that both A <strong>and</strong> B be dynamically local.The foregoing result allows us to conclude the discussion of naturalstates:
The Same Physics in All Spacetimes 223End of the proof of Theorem 5.2. Note that there is a natural transformationζ : I → . A , such that each ζ N simply embeds the trivial unital algebra inA (N). For the spacetime M given in the hypotheses, we already know thatA (M) =C1, soζ M is an isomorphism. Hence by dynamical locality <strong>and</strong>Theorem 6.1, ζ is an equivalence, so A ∼ = I .□Dynamical locality also significantly reduces our freedom to constructpathological theories.Theorem 6.2. If ϕ Δ is a diagonal theory (with rce ϕ trivial) such that ϕ Δ <strong>and</strong>every ϕ(M) are dynamically local, then all ϕ(M) are equivalent. If the ϕ(M)have trivial automorphism group then ϕ Δ is equivalent to each of them.Sketch of proof. Consider any morphism ψ : M → N in Loc. Using dynamicallocality of ϕ Δ <strong>and</strong> ϕ(N), we may deduce that ϕ(ψ) M is an isomorphism<strong>and</strong> hence ϕ(ψ) is an equivalence. As any two spacetimes in Loc can be connectedby a chain of (not necessarily composable) morphisms, it follows thatall the ϕ(M) areequivalent.In particular, writing M 0 for Minkowski space, the previous argumentallows us to choose an equivalence ζ (M) : ϕ(M 0 ) → . ϕ(M) for each spacetimeM (no uniqueness is assumed). Every morphism ψ : M → N then inducesan automorphism η(ψ) =ζ −1(M) ◦ ϕ(ψ) ◦ ζ (N) of ϕ(M 0 ). As Aut(ϕ(M 0 )) isassumed trivial, it follows that ϕ(ψ) =ζ (M) ◦ ζ −1(N)<strong>and</strong> it is easy to deducethat there is an equivalence ζ : ϕ(M 0 ) → . ϕ Δ with components (ζ (M) ) M . □We remark that the automorphism group of a theory in LCT can beinterpreted as its group of global gauge invariances [13] <strong>and</strong> that a theory inwhich the A (M) are algebras of observables will have trivial gauge group.The argument just given has a cohomological flavour, which can also be madeprecise <strong>and</strong> indicates that a cohomological study of Loc is worthy of furtherstudy.7. Example: Klein–Gordon theoryThe abstract considerations of previous sections would be of questionable relevanceif they were not satisfied in concrete models. We consider the st<strong>and</strong>ardexample of the free scalar field, discussed above.First let us consider the classical theory. Let Sympl be the categoryof real symplectic spaces with symplectomorphisms as the morphisms. Toeach M, we assign the space L (M) of smooth, real-valued solutions to(□ + m 2 )φ = 0, with compact support on Cauchy surfaces; we endow L (M)with the st<strong>and</strong>ard (weakly nondegenerate) symplectic product∫σ M (φ, φ ′ )= (φn a ∇ a φ ′ − φ ′ n a ∇ a φ)dΣΣfor any Cauchy surface Σ. To each hyperbolic embedding ψ : M → N thereis a symplectic map L (ψ) :L (M) → L (N) sothatE N ψ ∗ f = L (ψ)E M ffor all f ∈ C ∞ 0 (M) (see BFV or [2]).
224 C. J. FewsterThe relative Cauchy evolution for L was computed in BFV (theirEq. (15)) as was its functional derivative with respect to metric perturbations(see pp. 61-62 of BFV). An important observation is that this can beput into a nicer form: writingF M [f]φ = d ∣ds rce(L ) ∣∣∣s=0M[f]φit turns out thatwhere∫σ M (F M [f]φ, φ) =Mf ab T ab [φ] dvol M , (7.1)T ab [φ] =∇ a φ∇ b φ − 1 2 g ab∇ c φ∇ c φ + 1 2 m2 g ab φ 2is the classical stress-energy tensor of the solution φ. (In passing, we note that(7.1) provides an explanation, for the free scalar field <strong>and</strong> similar theories,as to why the relative Cauchy evolution can be regarded as equivalent to thespecification of the classical action.)We may use these results to compute L • (M; K) <strong>and</strong>L dyn (M; O):they consist of solutions whose stress-energy tensors vanish in K ⊥ (resp.,O ′ = M \cl J M (O)). For nonzero mass, the solution must vanish wherever thestress-energy does <strong>and</strong> we may conclude that L dyn (M; O) =L kin (M; O),i.e., we have dynamical locality of the classical theory L .At zero mass, however, we can deduce only that the solution is constantin regions where its stress-energy tensor vanishes, so L • (M; K) consists ofsolutions in L (M) that are constant on each connected component of K ⊥ .In particular, if K ⊥ is connected <strong>and</strong> M has noncompact Cauchy surfacesthis forces the solutions to vanish in K ⊥ as in the massive case. However, ifM has compact Cauchy surfaces this argument does not apply <strong>and</strong> indeedthe constant solution φ ≡ 1 belongs to every L • (M; K) <strong>and</strong>L dyn (M; O),but it does not belong to L kin (M; O) unless O contains a Cauchy surfaceof M. Hence the massless theory fails to be dynamically local. The sourceof this problem is easily identified: it arises from the global gauge symmetryφ ↦→ φ + const in the classical action.At the level of the quantized theory, one may show that similar resultshold: the m > 0 theory is dynamically local, while the m = 0 theory isnot. We argue that this should be taken seriously as indicating a (fairlymild) pathology of the massless minimally coupled scalar field, rather than alimitation of dynamical locality. In support of this position we note:• Taking the gauge symmetry seriously, we can alternatively quantize thetheory of currents j = dφ; this turns out to be a well-defined locallycovariant <strong>and</strong> dynamically local theory in dimensions n > 2. Whiledynamical locality fails for this model in n = 2 dimensions in the presentsetting, it may be restored by restricting the scope of the theory toconnected spacetimes.• The constant solution is also the source of another well-known problem:there is no ground state for the theory in ultrastatic spacetimes
The Same Physics in All Spacetimes 225with compact spatial section (see, for example [19]). The same problemafflicts the massless scalar field in two-dimensional Minkowski space,whereitiscommonplacetorejectthealgebraoffieldsinfavourofthealgebra of currents.• The nonminimally coupled scalar field (which does not have the gaugesymmetry) is dynamically local even at zero mass (a result due to Ferguson[12]).Actually, this symmetry has other interesting aspects: it is spontaneouslybroken in Minkowski space [29], for example, <strong>and</strong> the automorphism groupof the functor A is noncompact: Aut(A )=Z 2 ⋉R[13].8. Summary <strong>and</strong> outlookWe have shown that the notion of the ‘same physics in all spacetimes’ can begiven a formal meaning (at least in part) <strong>and</strong> can be analysed in the contextof the BFV framework of locally covariant theories. While local covariance initself does not guarantee the SPASs property, the dynamically local theoriesdo form a class of theories with SPASs; moreover, dynamical locality seems tobe a natural <strong>and</strong> useful property in other contexts. Relative Cauchy evolutionenters the discussion in an essential way, <strong>and</strong> seems to be the replacementof the classical action in the axiomatic setting. A key question, given ourstarting point, is the extent to which the SPASs condition is sufficient aswell as necessary for a class of theories to represent the same physics in allspacetimes. As a class of theories that had no subtheory embeddings otherthan equivalences would satisfy SPASs, there is clearly scope for further workon this issue. In particular, is it possible to formulate a notion of ‘the samephysics on all spacetimes’ in terms of individual theories rather than classesof theories?In closing, we remark that the categorical framework opens a completelynew way of analysing quantum field theories, namely at the functorial level. Itis conceivable that all structural properties of QFT should have a formulationat this level, with the instantiations of the theory in particular spacetimestaking a secondary place. Lest this be seen as a flight to abstraction, we emphasizeagain that this framework is currently leading to new viewpoints <strong>and</strong>concrete calculations in cosmology <strong>and</strong> elsewhere. This provides all the morereason to underst<strong>and</strong> why theories based on our experience with terrestrialparticle physics can be used in very different spacetime environments whilepreserving the same physical content.References[1] Adámek, J., Herrlich, H., Strecker, G.E.: Abstract <strong>and</strong> concrete categories: thejoy of cats. Repr. <strong>Theory</strong> Appl. Categ. (17), 1–507 (2006), reprint of the 1990original [Wiley, New York]
226 C. J. Fewster[2] Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on Lorentzian manifolds <strong>and</strong>quantization. European <strong>Mathematical</strong> Society (EMS), Zürich (2007)[3] Bernal, A.N., Sánchez, M.: Globally hyperbolic spacetimes can be defined ascausal instead of strongly causal. Class. Quant. Grav. 24, 745–750 (2007), grqc/0611138[4] Brunetti, R., Fredenhagen, K.: Microlocal analysis <strong>and</strong> interacting quantumfield theories: Renormalization on physical backgrounds. Comm. Math. Phys.208(3), 623–661 (2000)[5] Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle:A new paradigm for local quantum physics. Commun. Math. Phys. 237,31–68 (2003)[6] Brunetti, R., Ruzzi, G.: Superselection sectors <strong>and</strong> general covariance. I. Commun.Math. Phys. 270, 69–108 (2007)[7] Brunetti, R., Ruzzi, G.: <strong>Quantum</strong> charges <strong>and</strong> spacetime topology: The emergenceof new superselection sectors. Comm. Math. Phys. 287(2), 523–563 (2009)[8] Dappiaggi, C., Fredenhagen, K., Pinamonti, N.: Stable cosmological modelsdriven by a free quantum scalar field. Phys. Rev. D77, 104015 (2008)[9] Degner, A., Verch, R.: Cosmological particle creation in states of low energy.J. Math. Phys. 51(2), 022302 (2010)[10] Dikranjan, D., Tholen, W.: Categorical structure of closure operators, Mathematics<strong>and</strong> its Applications, vol. 346. Kluwer Academic Publishers Group,Dordrecht (1995)[11] Dimock, J.: Algebras of local observables on a manifold. Commun. Math. Phys.77, 219–228 (1980)[12] Ferguson, M.: In preparation[13] Fewster, C.J.: In preparation[14] Fewster, C.J.: <strong>Quantum</strong> energy inequalities <strong>and</strong> local covariance. II. Categoricalformulation. Gen. Relativity Gravitation 39(11), 1855–1890 (2007)[15] Fewster, C.J., Pfenning, M.J.: <strong>Quantum</strong> energy inequalities <strong>and</strong> local covariance.I: Globally hyperbolic spacetimes. J. Math. Phys. 47, 082303 (2006)[16] Fewster, C.J., Verch, R.: Dynamical locality; Dynamical locality for the freescalar field. In preparation[17] Fulling, S.A.: Nonuniqueness of canonical field quantization in Riemannianspace-time. Phys. Rev. D7, 2850–2862 (1973)[18] Fulling, S.A., Narcowich, F.J., Wald, R.M.: Singularity structure of the twopointfunction in quantum field theory in curved spacetime. II. Ann. Physics136(2), 243–272 (1981)[19] Fulling, S.A., Ruijsenaars, S.N.M.: Temperature, periodicity <strong>and</strong> horizons.Phys. Rept. 152, 135–176 (1987)[20] Haag, R.: Local <strong>Quantum</strong> Physics: <strong>Field</strong>s, Particles, Algebras. Springer-Verlag,Berlin (1992)[21] Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. CambridgeUniversity Press, London (1973)[22] Holl<strong>and</strong>s, S., Wald, R.M.: Local Wick polynomials <strong>and</strong> time ordered productsof quantum fields in curved spacetime. Commun. Math. Phys. 223, 289–326(2001)
The Same Physics in All Spacetimes 227[23] Holl<strong>and</strong>s, S., Wald, R.M.: Existence of local covariant time ordered productsof quantum fields in curved spacetime. Commun. Math. Phys. 231, 309–345(2002)[24] Kay, B.S.: Casimir effect in quantum field theory. Phys. Rev. D20, 3052–3062(1979)[25] L<strong>and</strong>au, L.J.: A note on extended locality. Comm. Math. Phys. 13, 246–253(1969)[26] Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer-Verlag, New York (1998)[27] S<strong>and</strong>ers, K.: On the Reeh-Schlieder property in curved spacetime. Comm.Math. Phys. 288(1), 271–285 (2009)[28] S<strong>and</strong>ers, K.: The locally covariant Dirac field. Rev. Math. Phys. 22(4), 381–430(2010)[29] Streater, R.F.: Spontaneous breakdown of symmetry in axiomatic theory. Proc.Roy. Soc. Ser. A 287, 510–518 (1965)[30] Streater, R.F., Wightman, A.S.: PCT, spin <strong>and</strong> statistics, <strong>and</strong> all that. PrincetonL<strong>and</strong>marks in Physics. Princeton University Press, Princeton, NJ (2000).Corrected third printing of the 1978 edition[31] Verch, R.: A spin-statistics theorem for quantum fields on curved spacetimemanifolds in a generally covariant framework. Commun. Math. Phys. 223, 261–288 (2001)[32] Verch, R.: Local covariance, renormalization ambiguity, <strong>and</strong> local thermal equilibriumin cosmology (2011). These proceeedings <strong>and</strong> arXiv:1105.6249.Christopher J. FewsterDepartment of MathematicsUniversity of YorkHeslington, York YO10 5DDUnited Kingdome-mail: chris.fewster@york.ac.uk
Local Covariance, RenormalizationAmbiguity, <strong>and</strong> Local Thermal Equilibriumin CosmologyRainer VerchAbstract. This article reviews some aspects of local covariance <strong>and</strong> ofthe ambiguities <strong>and</strong> anomalies involved in the definition of the stressenergytensor of quantum field theory in curved spacetime. Then, a summaryis given of the approach proposed by Buchholz et al. to define localthermal equilibrium states in quantum field theory, i.e., non-equilibriumstates to which, locally, one can assign thermal parameters, such astemperature or thermal stress-energy. The extension of that concept tocurved spacetime is discussed <strong>and</strong> some related results are presented. Finally,the recent approach to cosmology by Dappiaggi, Fredenhagen <strong>and</strong>Pinamonti, based on a distinguished fixing of the stress-energy renormalizationambiguity in the setting of the semiclassical Einstein equations,is briefly described. The concept of local thermal equilibrium states isthen applied, to yield the result that the temperature behaviour of aquantized, massless, conformally coupled linear scalar field at early cosmologicaltimes is more singular than that of classical radiation.Mathematics Subject Classification (2010). 83F05, 81T05, 81T16.Keywords. Cosmology, stress-energy-tensor, renormalization ambiguity,local thermal equilibrium.1. Local covariant quantum field theoryThe main theme of this contribution is the concept of local thermal equilibriumstates <strong>and</strong> their properties in quantum field theory on cosmologicalspacetimes. The starting point for our considerations is the notion of localcovariant quantum field theory which has been developed in [36, 5, 22] <strong>and</strong>which is also reviewed in [1] <strong>and</strong> is, furthermore, discussed in Chris Fewster’scontribution to these conference proceedings [12]. I will therefore attempt tobe as far as possible consistent with Fewster’s notation <strong>and</strong> shall at severalpoints refer to his contribution for precise definitions <strong>and</strong> further discussionof matters related to local covariant quantum field theory.F. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_12,© Springer Basel AG 2012229
230 R. VerchThe basic idea of local covariant quantum field theory is to considernot just a quantum field on some fixed spacetime, but simultaneously the“same” type of quantum field theory on all — sufficiently nice — spacetimesat once. In making this more precise, one collects all spacetimes which onewants to consider in a category Loc. Bydefinition,anobjectinLoc is afour-dimensional, globally hyperbolic spacetime with chosen orientation <strong>and</strong>time-orientation. An arrow, or morphism, M −→ ψN of Loc is an isometric,hyperbolic embedding which preserves orientation <strong>and</strong> time-orientation. (SeeFewster’s contribution for more details.)Additionally, we also assume that we have, for each object M of Loc,a quantum field Φ M , thought of as describing some physics happening inM. More precisely, we assume that there is a topological ∗-algebra A (M)(with unit) <strong>and</strong> that f ↦→ Φ M (f), f ∈ C0∞ (M), is an operator-valued distributiontaking values in A (M). (This would correspond to a “quantizedscalar field” which we treat here for simplicity, but everything can be generalizedto more general tensor- <strong>and</strong> spinor-type fields, as e.g. in [36].) Onewould usually require that the Φ M (f) generate A (M) in a suitable sense,if necessary allowing suitable completions. The unital topological ∗-algebrasalso form a category which shall be denoted by Alg, where the morphismsA −→ αB are injective, unital, topological ∗-algebraic morphisms betweenthe objects. Then one says that the family (Φ M ), as M ranges over theobjects of Loc, isalocal covariant quantum field theory if the assignmentsM → A (M) <strong>and</strong>A (ψ)(Φ M (f)) = Φ N (ψ ∗ f), for any morphism M −→ ψNof Loc, induce a functor A : Loc → Alg. (See again Fewster’s contribution [12]for a fuller dish.) Actually, a generally covariant quantum field theory shouldmore appropriately be viewed as a natural transformation <strong>and</strong> we refer to[5] for a discussion of that point. The degree to which the present notion oflocal covariant quantum field theory describes the “same” quantum field onall spacetimes is analyzed in Fewster’s contribution to these proceedings.We write Φ =(Φ M ) to denote a local covariant quantum field theory,<strong>and</strong> we remark that examples known so far include the scalar field, the Diracfield, the Proca field, <strong>and</strong> — with some restrictions — the electromagneticfield, as well as perturbatively constructed P (φ) <strong>and</strong> Yang-Mills models [22,20, 36]. A state of a local covariant quantum field theory Φ is defined as afamily ω =(ω M ), M ranging over the objects of Loc, whereeachω M isa state on A (M) — i.e. an expectation value functional. It might appearnatural to assume that there is an invariant state ω, defined byω N ◦ A (ψ) =ω M (1.1)for all morphisms M −→ ψN of Loc, but it has been shown that this propertyis in conflict with the dynamics <strong>and</strong> stability properties one would dem<strong>and</strong> ofthe ω M for each M [5, 22]; the outline of an argument against an invariantstate with the property (1.1) is given in Fewster’s contribution [12].
Local Thermal Equilibrium <strong>and</strong> Cosmology 2312. The quantized stress-energy tensorA general property of quantum field theories is the existence of a causaldynamical law, or time-slice axiom. In a local covariant quantum field theoryΦ, this leads to a covariant dynamics [5] termed “relative Cauchy evolution”in Fewster’s contribution [12], to which we refer again for further details. Therelative Cauchy evolution consists, for each M in Loc, of an isomorphismrce M (h) :A (M) → A (M)which describes the effect of an additive perturbation of the metric of Mby a symmetric 2-tensor field h on the propagation of the quantum field,akin to a scattering transformation brought about by perturbation of thebackground metric. One may consider the derivation – assuming it exists –which is obtained as d/ds| s=0rce M (h(s)) where h(s) is any smooth familyof metric perturbations with h(s=0) = 0. Following the spirit of Bogoliubov’sformula [3], this gives rise (under fairly general assumptions) to a localcovariant quantum field (T M ) which generates the derivation upon takingcommutators, <strong>and</strong> can be identified (up to some multiplicative constant) withthe quantized stress-energy tensor (<strong>and</strong> has been shown in examples to agreeindeed with the derivation induced by forming the commutator with thequantized stress-energy tensor [5]). Assuming for the moment that T M (f)takes values in A (M), we see that in a local covariant quantum field theorywhich fulfills the time-slice axiom the stress-energy tensor is characterized bythe following properties:LCSE-1 Generating property for the relative Cauchy evolution:2i[T M (f),A]=d ds∣ rce M (h(s))(A) , A ∈ A (M) , f = d s=0ds∣ h(s)s=0(2.1)Note that f = f ab is a smooth, symmetric C0∞ two-tensor field on M. Usingthe more ornamental abstract index notation, one would therefore writeT Mab (f ab )=T M (f) .LCSE-2 Local covariance:A (ψ)(T M (f)) = T N (ψ ∗ f) (2.2)whenever M −→ ψN is an arrow in Loc.LCSE-3 Symmetry:T Mab = T Mba . (2.3)LCSE-4 Vanishing divergence:∇ a T Mab =0, (2.4)where ∇ is the covariant derivative of the metric of M.Conditions LCSE-3,4 are (not strictly, but morally) consequences of theprevious two conditions, see [5] for discussion.
232 R. VerchThe conditions LCSE-1...4 ought to be taken as characterizing for anystress-energy tensor of a given local covariant quantum field theory Φ. However,they don’t fix T M completely. Suppose that we have made some choice,), of a local covariant stress-energy tensor for our local covariant quantumfield theory Φ. Now choose a local covariant family (C M )of(number-smooth tensor fields C M = C Mab on M, <strong>and</strong> write C M (f) =∫valued)M C Mabf ab dvol M where dvol M denotes the metric-induced volume formon M. If(C M ) is not only local covariant, but also symmetric <strong>and</strong> has vanishingdivergence, then we may set(T [1]MT [2]M(f) =T[1]M (f)+C M (f)1(where 1 is the unit of A (M)) to obtain in this way another choice, (T [2]M ),of a stress-energy tensor for Φ whichisasgoodasthepreviousonesinceit satisfies the conditions LCSE-1...4 equally well. Again under quite generalconditions, we may expect that this is actually the complete freedom forthe stress-energy tensor that is left by the above conditions, in particular thefreedom should in fact be given by a multiple of unity (be state-independent).Since the C M must be local covariant <strong>and</strong> divergence-free, they should belocal functionals of the spacetime metric of M, <strong>and</strong> one can prove this uponadding some technical conditions.The freedom which is left by the conditions LCSE-1...4 for the quantizedstress-energy tensor of a local covariant quantum field theory can be tracedback to the circumstance that even in such a simple theory as the linearscalar field, the stress-energy tensor is a renormalized quantity. Let us explainthis briefly, using the example of the minimally coupled linear scalar field,following the path largely developed by Wald [37, 40]. On any spacetime, theclassical minimally coupled linear scalar field φ obeys the field equation∇ a ∇ a φ + m 2 φ =0. (2.5)The stress-energy tensor for a classical solution φ of the field equation canbe presented in the formT ab (x) = limy→xP ab (x, y; ∇ (x) , ∇ (y) )φ(x)φ(y)(where x, y are points in spacetime) with some partial differential operatorP ab (x, y; ∇ (x) , ∇ (y) ). This serves as starting point for defining T Mab throughreplacing φ by the quantized linear scalar field Φ M . However, one finds alreadyin Minkowski spacetime that upon performing this replacement, the behaviourof the resulting expression is singular in the coincidence limit y → x.As usual with non-linear expressions in a quantum field at coinciding points,one must prescribe a renormalization procedure in order to obtain a welldefinedquantity. In Minkowski spacetime, this is usually achieved by normalordering with respect to the vacuum. This makes explicit reference to the vacuumstate in Minkowski spacetime, the counterpart of which is not availablein case of generic spacetimes. Therefore, one proceeds in a different manner
Local Thermal Equilibrium <strong>and</strong> Cosmology 233on generic curved spacetimes (but in Minkowski spacetime, the result coincideswith what one obtains from normal ordering, up to a renormalizationfreedom).Since one is mainly interested in expectation values of the quantizedstress-energy tensor — as these are the quantities entering on the right-h<strong>and</strong>side of the semiclassical Einstein equations, discussed in the next section —one may concentrate on a class of states S(M) onA (M) for which the expectationvalue of the stress-energy tensor can be defined as unambiguouslyas possible, <strong>and</strong> in a manner consistent with the conditions LCSE-1...4 above.Defining only the expectation value of the stress-energy tensor has the additionalbenefit that one need not consider the issue if, or in which sense,T M (f) is contained in A (M) (or its suitable extensions). Of course, it hasthe drawback that non-linear expressions in T M (f), as they would appearin the variance of the stress-energy, need extra definition, but we may regardthis as an additional issue. Following the idea above for defining thequantized stress-energy tensor, one can see that the starting point for theexpectation value of the stress-energy tensor in a state ω M on A (M) isthecorresponding two-point functionw M (x, y) =ω M (Φ M (x)Φ M (y)) ,written here symbolically as a function, although properly it is a distributionon C0 ∞ (M × M). One now considers a particular set S μsc (M), the states onA (M) whose two-point functions fulfill the microlocal spectrum condition[28, 4, 35]. The microlocal spectrum condition specifies the wavefront set ofthe distribution w M in a particular, asymmetric way, which is reminiscent ofthe spectrum condition for the vacuum state in Minkowski spacetime. Usingthe transformation properties of the wavefront set under diffeomorphisms,one can show (1) the sets S μsc (M) transform contravariantly under arrowsM −→ ψN in Loc, meaning that every state in S μsc (N) restricts to a state ofS μsc (ψ(M)) <strong>and</strong> S μsc (ψ(M)) ◦ A (ψ) =S μsc (M), <strong>and</strong> moreover (2) that themicrolocal spectrum condition is equivalent to the Hadamard condition onw M [28, 29]. This condition says [23] that the two-point function w M splitsin the formw M (x, y) =h M (x, y)+u(x, y) ,where h M is a Hadamard parametrix for the wave equation (2.5), <strong>and</strong> u is aC ∞ integral kernel. Consequently, the singular behaviour of h M <strong>and</strong> hence ofw M is determined by the spacetime geometry of M <strong>and</strong> is state-independentwithin the set S μsc (M) whereas different states are distinguished by differentsmooth terms u(x, y). With respect to such a splitting of a two-point function,one can then define the expectation value of the stress-energy tensor in a stateω M in S(M) as〈˜T Mab (x)〉 ωM = lim P ab (x, y; ∇ (x) , ∇ (y) )(w M − h M ) . (2.6)y→xNote that 〈˜T Mab (x)〉 ωM is, in fact, smooth in x <strong>and</strong> therefore is a C ∞ tensorfield on M. The reason for the appearance of the twiddle on top of the symbol
234 R. Verchfor the just defined stress-energy tensor is due to the circumstance that withthis definition, the stress-energy tensor is (apart from exceptional cases) notdivergence-free. However, Wald [37, 40] has shown that this can be repairedby subtracting the divergence-causing term. More precisely, there is a smoothfunction Q M on M, constructed locally from the metric of M (so that thefamily (Q M ) is local covariant), such that〈T Mab (x)〉 ωM = 〈˜T Mab (x)〉 ωM − Q M (x)g ab (x) , (2.7)where g ab denotes the spacetime metric of M, defines an expectation value ofthe stress-energy tensor which is divergence-free. Moreover, with this definition,the conditions LCSE-1...4 hold when interpreted as valid for expectationvalues of states fulfilling the Hadamard condition [40, 5] (possibly after suitablesymmetrization in order to obtain LCSE-4). This may actually be seen asone of the main motivations for introducing the Hadamard condition in [37],<strong>and</strong> in the same paper, Wald proposed conditions on the expectation valuesof the stress-energy tensor which are variants of our LCSE-1...4 above. Wald[37, 40] also proved that, if there are two differing definitions for the expectationvalues of the stress-energy tensor complying with the conditions, thenthe difference is given by a state-independent, local covariant family (C Mab )of smooth, symmetric, divergence-free tensor fields, in complete analogy toour discussion above. Actually, the formulation of the local covariance conditionfor the expectation values of the stress-energy tensor in [40] was animportant starting point for the later development of local covariant quantumfield theory, so the agreement between Wald’s result on the freedom indefining the stress-energy tensor <strong>and</strong> ours, discussed above, are in no waycoincidental.How does this freedom come about? To see this, note that only the singularitiesof the Hadamard parametrix h M are completely fixed. One has thefreedom of altering the smooth contributions to that Hadamard parametrix,<strong>and</strong> as long as this leads to an expected stress-energy tensor which is stillconsistent with conditions LCSE-1...4, this yields an equally good definitionof 〈T Mab (x)〉 ωM . Even requiring as in [37, 40] that the expectation value ofthe stress-energy tensor agrees in Minkowski spacetime with the expressionobtained by normal ordering — implying that in Minkowski spacetime, thestress-energy expectation value of the vacuum vanishes — doesn’t fully solvethe problem. For if another definition is chosen so that the resulting differenceterm C Mab is made of curvature terms, that difference vanishes on Minkowskispacetime. The problem could possibly be solved if there was a distinguishedstate ω =(ω M ) for which the expectation value of the stress-energy tensorcould be specified in some way, but as was mentioned before, the most likelyc<strong>and</strong>idate for such a state, the invariant state, doesn’t exist (at least not asa state fulfilling the microlocal spectrum condition). This means that apparentlythe setting of local covariant quantum field theory does not, at leastwithout using further ingredients, specify intrinsically the absolute value ofthe local stress-energy content of a quantum field state, since the ambiguityofbeingabletoaddadifferencetermC Mab always remains. In fact, this
Local Thermal Equilibrium <strong>and</strong> Cosmology 235difference term ambiguity has the character of a renormalization ambiguitysince it occurs in the process of renormalization (by means of discarding thesingularities of a Hadamard parametrix), as is typical in quantum field theory.The fact that local covariant quantum field theory does not fix the localstress-energy content may come as a surprise — <strong>and</strong> disappointment — sinceit differs from what one would be inclined to expect from quantum field theoryon Minkowski spacetime with a distinguished vacuum state. This justgoes to show that one cannot too naively transfer concepts from quantumfield theory on Minkowski spacetime to local covariant quantum field theory(a fact which, incidentally, has already been demonstrated by the Hawking<strong>and</strong> Unruh effects).Another complication needs to be addressed: Anomalies of the quantizedstress-energy tensor. It is not only that the value of (the expectation value of)the quantized stress-energy tensor isn’t fixed on curved spacetime due to theappearance of metric- or curvature-dependent renormalization ambiguities,but there are also curvature-induced anomalies. Recall that in quantum fieldtheory one generally says that a certain quantity is subject to an anomaly ifthe quantized/renormalized quantity fails to feature a property — mostly, asymmetry property — which is fulfilled for the counterpart of that quantityin classical field theory. In the case of the stress-energy tensor, this is thetrace anomaly or conformal anomaly. Suppose that φ is a C ∞ solution of theconformally coupled, massless linear scalar wave equation, i.e.(∇ a ∇ a + 1 6 R )φ =0on a globally hyperbolic spacetime M with scalar curvature R. Then theclassical stress-energy tensor of φ has vanishing trace, T a a = 0. This expressesthe conformal invariance of the equation of motion. However, as was shownby Wald [38], it is not possible to have the vanishing trace property for theexpectation value of the stress-energy tensor in the presence of curvature ifone insists on the stress-energy tensor having vanishing divergence: Underthese circumstances, one necessarily finds〈T a M a〉 ωM = −4Q M .Note that this is independent of the choice of Hadamard state ω M . The originof this trace-anomaly lies in having subtracted the divergence-causing termQg ab in the definition of the renormalized stress-energy tensor. In a sense,non-vanishing divergence of the renormalized stress-energy tensor could alsobe viewed as an anomaly, so in the case of the conformally coupled, masslessquantized linear scalar, one can trade the non-vanishing “divergence anomaly”of the renormalized stress-energy tensor for the trace anomaly. The conditionLCSE-4 assigns higher priority to vanishing divergence, whence onehas to put up with the trace anomaly.One may wonder why the features of the quantized/renormalized stressenergytensor have been discussed here at such an extent while our maintopic are local thermal equilibrium states. The reason is that the ambiguities<strong>and</strong> anomalies by which the definition of the stress-energy tensor for
236 R. Verchquantum fields in curved spacetime is plagued will also show up when tryingto generalize the concept of local thermal equilibrium in quantum field theoryin curved spacetime. Furthermore, the ambiguities <strong>and</strong> anomalies of thestress-energy tensor do play a role in semiclassical gravity (<strong>and</strong> semiclassicalcosmology), <strong>and</strong> we will come back to this point a bit later.3. Local thermal equilibrium3.1. LTE states on Minkowski spacetimeFollowing the idea of Buchholz, Ojima <strong>and</strong> Roos [7], local thermal equilibrium(LTE, for short) states are states of a quantum field for which local,intensive thermal quantities such as — most prominently — temperature <strong>and</strong>pressure can be defined <strong>and</strong> take the values they would assume for a thermalequlibribum state. Here “local” means, in fact, at a collection of spacetimepoints. We will soon be more specific about this.Although the approach of [7] covers also interacting fields, for the purposesof this contribution we will restrict attention to the quantized linearscalar field; furthermore, we start by introducing the concept of LTE stateson Minkowski spacetime. Consider the quantized massive linear field Φ 0 onMinkowski spacetime, in its usual vacuum representation, subject to the fieldequation (✷ + m 2 )Φ 0 = 0 (to be understood in the sense of operator-valueddistributions). Now, at each point x in Minkowski spacetime, one would liketo define a set of “thermal observables” Θ x formed by observables which aresensitive to intensive thermal quantities at x. If that quantity is, e.g., temperature,then the corresponding quantity in Θ x would be a thermometer“located” at spacetime point x. One may wonder if it makes physical senseto idealize a thermometer as being of pointlike “extension” in space <strong>and</strong> time,but as discussed in [7], this isn’t really a problem.What are typical elements of Θ x ? Let us look at a very particular example.Assume that m = 0, so we have the massless field. Moreover, fix someLorentzian frame consisting of a tetrad (e a 0,e a 1,e a 2,e a 3) of Minkowski vectorssuch that η ab e a μe b ν = η μν <strong>and</strong> with e a 0 future-directed. Relative to these data,let ω β,e0 denote the global thermal equilibrium state — i.e. KMS state —with respect to the time direction e 0 ≡ e a 0 at inverse temperature β>0 [19].Evaluating :Φ 2 0 :(x), the Wick square of Φ 0 at the spacetime point x, intheKMS state gives〈:Φ 2 0 :(x)〉 ωβ,e0 = 112β 2 .Recall that the Wick square is defined as:Φ 2 0 :(x) = lim Φ 0 (q x (ζ))Φ 0 (q x (−ζ)) −〈Φ 0 (q x (ζ))Φ 0 (q x (−ζ))〉 vac , (3.1)ζ→0where 〈 . 〉 vac is the vacuum state <strong>and</strong> we have setq x (ζ) =x + ζ (3.2)
Local Thermal Equilibrium <strong>and</strong> Cosmology 237for Minkowski space coordinate vectors x <strong>and</strong> ζ, tacitly assuming that ζisn’t zero or lightlike. Thus, :Φ 2 0 :(x) is an observable localized at x; strictlyspeaking, without smearing with test functions with respect to x, itisn’tan operator but a quadratic form. Evaluating :Φ 2 0 :(x) in the global thermalequilibrium state yields a monotonous function of the temperature. This isalso the case for mass m>0, only the monotonous function is a bit morecomplicated. So :Φ 2 0 :(x) can be taken as a “thermometer observable” at x.As in our simple model the KMS state for Φ 0 is homogeneous <strong>and</strong> isotropic,〈:Φ 2 0 :(x)〉 ωβ,e0 is, in fact, independent of x. Let us abbreviate the Wick-square“thermometer observable” byϑ(x) =:Φ 2 0 :(x) . (3.3)Starting from the Wick square one can, following [7], form many more elementsof Θ x . The guideline was that these elements should be sensitiveto intensive thermal quantitites at x. This can be achieved by forming thebalanced derivatives of the Wick square of order n, defined asð μ1 ...μ n:Φ 2 0 :(x))= lim ∂ ζ μ 1 ···∂ ζ μn(Φ 0 (q x (ζ))Φ 0 (q x (−ζ)) −〈Φ 0 (q x (ζ))Φ 0 (q x (−ζ))〉 vac .ζ→0(3.4)Prominent among these is the second balanced derivative becauseε μν (x) =− 1 4 ð μν :Φ 2 0 :(x) (3.5)is the thermal stress-energy tensor whichintheKMSstateω β,e0 takes on thevalues 1 ε μν (x) =〈ε μν (x)〉 ωβe0 = 1 ∫p μ p νd 3 p, (3.6)(2π) 3 R (e βp 3 0 − 1)p0where p μ = p a e a μ are the covariant coordinates of p with respect to the chosenLorentz frame, <strong>and</strong> p 0 = |p| with (p μ ) = (p 0 ,p). Again, this quantity isindependent of x in the unique KMS state of our quantum field model.Notice that the values of ε μν depend not only on the inverse temperature,but also on the time direction e 0 of the Lorentz frame with respect towhich ω β,e0 is a KMS state. Since the dependence of the thermal quantitieson β <strong>and</strong> e 0 in expectation values of ω β,e0 is always a function of the timelike,future-directed vector β = β · e 0 , it is hence useful to label the KMS statescorrespondingly as ω β , <strong>and</strong> to define LTE states with reference to β.Definition 3.1. Let ω be a state of the linear scalar field Φ 0 on Minkowskispacetime, <strong>and</strong> let N ∈ N.1 We caution the reader that in previous publications on LTE states, ϑ <strong>and</strong> ε μν are alwaysused to denote the expectation values of our ϑ <strong>and</strong> ε in LTE states. We hope that our useof bold print for the thermal observables is sufficient to distinguish the thermal observablesfrom their expectation values in LTE states.
238 R. Verch(i) Let D be a subset of Minkowski spacetime <strong>and</strong> let β : x ↦→ β(x) bea(smooth, if D is open) map assigning to each x ∈ D a future-directedtimelike vector β(x). Then we say that ω is a local thermal equilibriumstate of order N at sharp temperature for the temperature vector field β(for short, ω is a [D, β,N]-LTE state) if〈ð μ1···μ n:Φ 2 0 :(x)〉 ω = 〈ð μ1···μ n:Φ 2 0 :(0)〉 β(x) (3.7)holds for all x ∈ D <strong>and</strong> 0 ≤ n ≤ N. Here, we have written 〈 .〉 β(x) =ω β(x) ( . ) for the KMS state of Φ 0 defined with respect to the timelike,future-directed vector β(x). The balanced derivatives of the Wick squareon the right-h<strong>and</strong> side are evaluated at the spacetime point 0; since theKMS state on the right-h<strong>and</strong> side is homogeneous <strong>and</strong> isotropic, one hasthe freedom to make this choice.(ii) Let D be a subset of Minkowski spacetime <strong>and</strong> let m : x ↦→ ϱ x be a mapwhich assigns to each x ∈ D a probability measure compactly supportedon V + , the open set of all future-directed timelike Minkowski vectors. Itwill be assumed that the map is smooth if D is open. Then we say thatω is a local thermal equilibrium state of order N with mixed temperaturedistribution ϱ (for short, ω is a [D, ϱ, N]-LTE state) if〈ð μ1···μ n:Φ 2 0 :(x)〉 ω = 〈ð μ1···μ n:Φ 2 0 :(0)〉 ϱx (3.8)holds for all x ∈ D <strong>and</strong> 0 ≤ n ≤ N, where∫〈ð μ1···μ n:Φ 2 0 :(0)〉 ϱx = 〈ð μ1···μ n:Φ 2 0 :(0)〉 β ′ dϱ x (β ′ ) . (3.9)V +In this definition, Φ 0 can more generally also be taken as the linearscalar field with a finite mass different from zero. Let us discuss a couple offeatures of this definition <strong>and</strong> some results related to it.(A) The set of local thermal observables used for testing thermal propertiesof ω in (3.7) <strong>and</strong> (3.8) is Θ x(N) , formed by the Wick square <strong>and</strong> all of itsbalanced derivatives up to order N. One can generalize the condition tounlimited order of balanced derivatives, using as thermal observables the setΘ (∞)x = ⋃ ∞N=1 Θ(N) x .(B) The condition (3.7) dem<strong>and</strong>s that at each x in D, the expectation valuesof thermal observables in Θ (N)x evaluated in ω <strong>and</strong> in 〈 . 〉 β(x) coincide; inother words, with respect to these thermal observables at x, ω looks just likethe thermal equilibrium state 〈 . 〉 β(x) . Note that β(x) canvarywithx, soanLTE state can have a different temperature <strong>and</strong> a different “equilibrium restframe”, given by the direction of β(x), at each x.(C) With increasing order N, the sets Θ x(N) become larger; so the higher theorder N for which the LTE condition is fulfilled, the more can the state ω beregarded as coinciding with a thermal state at x. In this way, the maximumorder N for which (3.7) is fulfilled provides a measure of the deviation fromlocal thermal equilibrium (<strong>and</strong> similarly, for the condition (3.8)).
Local Thermal Equilibrium <strong>and</strong> Cosmology 239(D) Condition (3.7) dem<strong>and</strong>s that, on Θ (N)x , ω coincides with a thermalequilibrium state at sharp temperature, <strong>and</strong> sharp thermal rest-frame. Condition(3.8) is less restrictive, dem<strong>and</strong>ing only that ω coincides with a mixtureof thermal equilibrium states, described by the probability measure ϱ x ,onthe thermal observables Θ x . Of course, (3.7) is a special case of (3.8).(E) Clearly, each global thermal equilibrium state, or KMS state, ω β isan LTE state, with constant inverse temperature vector β. The interestingfeature of the definition of LTE states is that β(x) canvarywithx in spacetime,<strong>and</strong> the question of existence of LTE states which are not global KMSstates arises. In particular, fixing the order N <strong>and</strong> the spacetime region D,which functions β(x) orϱ x can possibly occur? They surely cannot be completelyarbitrary, particularly for open D, since the Wick square of Φ 0 <strong>and</strong> itsbalanced derivatives are subject to dynamical constraints which are a consequenceof the equation of motion for Φ 0 . We put on record some of theresults which have been established so far.The hot bang state [7, 6]A state of Φ 0 (zero mass case) which is an LTE state with a variable, sharpinverse temperature vector field on the open forward lightcone V + was constructedin [7] <strong>and</strong> further investigated in [6]. This state is called the hot bangstate ω HB , <strong>and</strong> for x, y in V + , its two-point function w HB has the formw HB (x, y) = 1 ∫(2π) 3 e −i(x−y)μ p μɛ(p 0 )δ(p μ 1p μ )R 1 − e 4 −a((x−y)μ p μ ) d4 p (3.10)where a>0 is some parameter <strong>and</strong> ɛ denotes the sign function. To compare,the two-point function of a KMS state with constant inverse temperaturevector β ′ is given byw β ′(x, y) = 1 ∫(2π) 3 e −i(x−y)μ p μɛ(p 0 )δ(p μ 1p μ )d 4 p. (3.11)R 4 1 − e −β′μ p μUpon comparison, one can see that ω HB has an inverse temperature vectorfieldβ(x) =2ax , x ∈ V + .Thus, the temperature diverges at the boundary of V + , with the thermal restframe tilting lightlike; moreover, the temperature decreases away from theboundary with increasing coordinate time x 0 . This behaviour is sketched inFigure 1.The hot bang state provides an example of a state where the thermalstress-energy tensor ε μν (x) =−(1/4)〈ð μν :Φ 2 0 :(x)〉 HB deviates from theexpectation values of the full stress-energy tensor 〈T μν (x)〉 HB . In fact, as discussedin [7], for the stress-energy tensor of the massless linear scalar field inMinkowski spacetime one finds generallyT μν (x) =ε μν + 1 12 (∂ μ∂ ν − η μν ✷):Φ 2 0 :(x) ,
240 R. VerchFigure 1. Sketch of temperature distribution of the hotbang state.<strong>and</strong> thus one obtains, for the hot bang state,whereasε (HB)μν (x) =〈ε μν (x)〉 HB = π21440a 4 4x μ x ν − η μν x λ x λ(x κ x κ ) 3 ,〈T μν (x)〉 HB = ε (HB)μν (x)+ 1440a2288π 2 ε(HB) μν (x) .The first term is due to the thermal stress-energy, while the second term is aconvection term which will dominate over the thermal stress-energy when theparameter a exceeds 1 by order of magnitude. Thus, for non-stationary LTEstates, the expectation value of the full stress-energy tensor can in generalnot be expected to coincide with the thermal stress-energy contribution dueto transport terms which are not seen in ε μν .Maximal spacetime domains for nontrivial LTE states [6]As mentioned before, the inverse temperature vector field for an LTE statecannot be arbitrary since the dynamical behaviour of the quantum field imposesconstraints. A related but in some sense stronger constraint whichappears not so immediate comes in form of the following result which wasestablished by Buchholz in [6]: Suppose that ω is a [D, β, ∞]-LTE state, i.e.an LTE state of infinite order with sharp inverse temperature vector field β.If D, the region on which ω has the said LTE property, contains a translateof V + ,thenforβ to be non-constant it is necessary that D is contained insome timelike simplicial cone, i.e. an intersection of characteristic half-spaces(which means, in particular, that D cannot contain any complete timelikeline).Existence of non-trivial mixed temperature LTE states [33]The hot bang state mentioned above has been constructed for the case of themassless linear scalar field. It turned out to be more difficult to construct nontrivialLTE states for the massive linear scalar field, <strong>and</strong> the examples knownto date for the massive case are mixed temperature LTE states. Recently,Solveen [33] proved a general result on the existence of non-trivial mixed
Local Thermal Equilibrium <strong>and</strong> Cosmology 241temperature LTE states: Given any compact subset D of Minkowski spacetime,there are non-constant probability measure-valued functions x ↦→ ϱ x ,x ∈ D, together with states ω which are [D, ϱ, ∞]-LTE states.LTE condition as generalization of the KMS condition [27]The choice of balanced derivatives of the Wick square as thermal observablesfixing the LTE property may appear, despite the motivation given in [7],as being somewhat arbitrary, so that one would invite other arguments fortheir prominent role in setting up the LTE condition. One attempt in thisdirection has been made by Schlemmer who pointed at a relation betweenan Unruh-like detector model <strong>and</strong> balanced derivatives of the Wick square[31]. On the other h<strong>and</strong>, recent work by Pinamonti <strong>and</strong> Verch shows thatthe LTE condition can be viewed as a generalization of the KMS condition.The underlying idea will be briefly sketched here, for full details see theforthcoming publication [27]. For a KMS state 〈 . 〉 β,e0 ,let(ϕ(τ) =ϕ x (τ) =〈Φ 0 q ( − 1 2 τe ) ) (0 Φ 0 q ( + 1 2 τe 0) )〉 .β,e 0Then the KMS condition implies that there is a functionf = f x : S β = {τ + iσ : τ ∈ R, 0
242 R. VerchStructure of the set of thermal observablesThe linearity of the conditions (3.7) <strong>and</strong> (3.8) implies that they are alsofulfilled for linear combinations of elements in Θ x(N) . This means that theLTE property of a state extends to elements of the vector space spanned byΘ x(N) . That is of some importance since one can show that span(Θ (N)x )isdense in the set of all thermal observables, its closure containing, e.g., theentropy flux density [7, 6].Furthermore, the equation of motion of Φ 0 not only provides constraintson the possible functions β(x) orϱ x which can occur for LTE states, but evendetermines evolution equations for these — <strong>and</strong> other — thermal observablesin LTE states [7, 6]. Let us indicate this briefly by way of an example. For anLTE state, it holds that the trace of the thermal stress-energy tensor vanishes,ε ν ν(x) = 0, due to the analogous property for KMS states. On the other h<strong>and</strong>,from relations between Wick products <strong>and</strong> their balanced derivatives oneobtains the equation ε ν ν(x) =∂ ν ∂ ν :Φ 2 0 :(x), <strong>and</strong> therefore, for any LTE statewhich fulfills the LTE condition on an open domain, the differential equation∂ ν ∂ ν ϑ(x) = 0 must hold, where ϑ(x) denotes the expectation values of ϑ(x),i.e. the Wick square, in the LTE state. This is a differential equation for (afunction of) β(x) orϱ x .3.2. LTE states in curved spacetimeThe concept of LTE states, describing situations which are locally approximatelyin thermal equilibrium, appears promising for quantum field theory incurved spacetime where global thermal equilibrium in general is not at h<strong>and</strong>.This applies in particular to early stages in cosmological scenarios wherethermodynamical considerations play a central role for estimating processeswhich determine the evolution <strong>and</strong> structures of the Universe in later epochs.However, usually in cosmology the thermal stress-energy tensor is identifiedwith the full stress-energy tensor, <strong>and</strong> it is also customary to use an “instantaneousequilibrium” description of the (thermal) stress-energy tensor. In viewof our previous discussion, this can at best be correct in some approximation.Additional difficulties arise from the ambiguities <strong>and</strong> anomalies affecting thequantized stress-energy tensor, <strong>and</strong> we will see that such difficulties are alsopresent upon defining LTE states in curved spacetime. Let us see how onemay proceed in trying to generalize the LTE concept to curved spacetime,<strong>and</strong> what problems are met in that attempt.We assume that we are given a local covariant quantum field Φ =(Φ M )with a local covariant stress-energy tensor; we also assume that Φ admitsglobal thermal equilibrium states for time symmetries on flat spacetime. Forconcreteness, we will limit our discussion to the case that our local covariantquantum field is a scalar field with curvature coupling ξ. The parameter ξcan be any real number, but the most important cases are ξ = 0 (minimalcoupling) or ξ =1/6 (conformal coupling). So Φ M obeys the field equation(∇ a ∇ a + ξR + m 2 )Φ M =0, (3.12)
Local Thermal Equilibrium <strong>and</strong> Cosmology 243where ∇ is the covariant derivative of M <strong>and</strong> R the associated scalar curvature,with mass m ≥ 0.The central objects in the definition of LTE states in flat spacetime werecontaining “thermalobservables” localized at x, <strong>and</strong> (ii) a set of global thermal equilibriumstates (KMS states) serving as “reference states” determining the thermalequilibrium values of elements in Θ x . In a generic curved spacetime one encountersthe problem that there aren’t any global thermal equilibrium states(unless the spacetime possesses suitable time symmetries) <strong>and</strong> thus there arein general no c<strong>and</strong>idates for the requisite reference states of (ii). To circumventthis problem, one can take advantage of having assumed that Φ is alocal covariant quantum field theory. Thus, let M be a globally hyperbolicspacetime, <strong>and</strong> let Φ M be the quantum field on M given by Φ. Thereisalsoa quantum field Φ 0 on Minkowski spacetime given by Φ. One can use theexponential map exp x in M at x to identify Φ M ◦ exp x <strong>and</strong> Φ 0 , <strong>and</strong> therebyone can push forward thermal equilibrium states ω β of Φ 0 to thermal referencestates on which to evaluate correlations of Φ M ◦ exp x infinitesimallyclose to x. This way of defining thermal reference states at each spacetimepoint x in M has been proposed by Buchholz <strong>and</strong> Schlemmer [8].As a next step, one needs a generalization of balanced derivatives of(i)ateachspacetimepointx a set (or family of sets) Θ (N)xthe Wick square of Φ M as elements of the Θ (N)M,x, where we used the labelM to indicate that the thermal observables are defined with respect to thespacetime M. So far, mostly the case N = 2 has been considered, whichis enough to study the thermal stress-energy tensor. In [32], the followingdefinition was adopted: Let ω M be a state of Φ M fulfilling the microlocalspectrum condition (in other words, ω M is a Hadamard state), <strong>and</strong> let w Mdenote the corresponding two-point function. We consider the smooth partu(x, y) =w M (x, y) − h M (x, y) obtained from the two-point function aftersubtracting the singular Hadamard parametrix h M (x, y) as in (2.6). Thenwe take its symmetric part u + (x, y) = 1 (u(x, y) +u(y, x)) <strong>and</strong> define the2expectation value of the Wick square as the coincidence limit〈:Φ 2 M :(x)〉 ωM = lim u + (x, y) . (3.13)y→xFurthermore, we define the second balanced derivative of the Wick square ofΦ M in terms of expectation values as()〈ð ab :Φ 2 M :(x)〉 ωM = limx ′ →x∇ a ∇ b −∇ a ∇ b ′ −∇ a ′∇ b + ∇ a ′∇ b ′ u + (x, x ′ ) ,(3.14)where on the right-h<strong>and</strong> side unprimed indices indicate covariant derivativeswith respect to x, whereas primed indices indicate covariant derivatives withrespect to x ′ . In [32] it is shown that (3.14) amounts to taking the secondderivatives of u + (exp x (ζ), exp x (−ζ)) with respect to ζ, <strong>and</strong> evaluatingat ζ = 0. Note that upon using the symmetric part u + of u in defining theWick product, its first balanced derivative vanishes, as it would on Minkowskispacetime for the normal ordering definition. The definitions (3.13) <strong>and</strong> (3.14)
244 R. Verchare referred to as symmetric Hadamard parametrix subtraction (SHP) prescription,as in [32]. Note, however, that for the massive case, m>0, the SHPprescription differs from the normal ordering prescription by m-dependentuniversal constants, a fact which must be taken into account when definingthe LTE condition; see [32] for details.By definition, Θ (2)M,xis then taken to consist of multiples of the unitoperator, :Φ 2 M :(x) <strong>and</strong>ð ab :Φ 2 M :(x), defined according to the SHP prescription.With these ingredients in place, one can attempt the definition of LTEstates on a curved, globally hyperbolic spacetime M.Definition 3.2. Let ω M be a state of Φ M fulfilling the microlocal spectrumcondition, <strong>and</strong> let D be a subset of the spacetime M.(i) We say that ω M is an LTE state with sharp temperature vector field(timelike, future-directed, <strong>and</strong> smooth if D is open) β : x ↦→ β(x)(x ∈ D) oforder2if〈:Φ 2 M :(x)〉 ωM = 〈:Φ 2 0 :(0)〉 β(x) <strong>and</strong> (3.15)〈ð μν :Φ 2 M :(x)〉 ωM = 〈ð μν :Φ 2 0 :(0)〉 β(x) (3.16)hold for all x ∈ D. On the right-h<strong>and</strong> side there appear the expectationvalues of the thermal equilibrium state 〈 . 〉 β(x) of Φ 0 on Minkowskispacetime, where the vector β(x) ∈ T x M on the left-h<strong>and</strong> side is identifiedwith a Minkowski space vector β(x) on the right-h<strong>and</strong> side via theexponential map exp x , using that exp x (0) = x, <strong>and</strong> the coordinates referto a choice of Lorentz frame at x. On the right-h<strong>and</strong> side, we have nowused the definition of Wick product <strong>and</strong> its second balanced derivativeaccording to the SHP prescription.(ii) Let ϱ : x ↦→ ϱ x (x ∈ D) beamapfromD to compactly supportedprobability measures in V + (assumed to be smooth if D is open). Wesay that ω M is an LTE state with mixed temperature distribution ϱ oforder 2 if〈:Φ 2 M :(x)〉 ωM = 〈:Φ 2 0 :(0)〉 ϱ(x) <strong>and</strong> (3.17)〈ð μν :Φ 2 M :(x)〉 ωM = 〈ð μν :Φ 2 0 :(0)〉 ϱ(x) (3.18)hold for all x ∈ D. The same conventions as in (i) regarding identificationof curved spacetime objects (left-h<strong>and</strong> sides) <strong>and</strong> Minkowski spaceobjects (right-h<strong>and</strong> sides) by exp x applies here as well. The definitionof the ϱ x -averaged objects is as in (3.9).Let us discuss some features of this definition <strong>and</strong> some first resultsrelated to it on generic spacetimes. We will present results pertaining to LTEstates on cosmological spacetimes in the next section.(α) The definition of thermal observables given here is local covariant sincethe Wick square <strong>and</strong> its covariant derivatives are local covariant quantumfields [22]. In particular, if A is the functor describing our local covariantquantum field Φ, then for any arrow M −→ ψN one has A (ψ)(Θ (2)M,x )=
Local Thermal Equilibrium <strong>and</strong> Cosmology 245Θ (2)N,ψ(x)(by a suitable extension of A , see [22]). Obviously, it is desirable todefine thermal observables in curved spacetimes in such a way as to be localcovariant. Otherwise, they would depend on some global properties of theparticular spacetimes, in contrast to their interpretation as local intensivequantities.(β) As is the case for the quantized stress-energy tensor, also the thermalobservables in Θ (2)M,x , i.e. the Wick square of Φ M <strong>and</strong> its second balancedderivative, are subject to renormalization ambiguities <strong>and</strong> anomalies. Supposingthat choices of :Φ 2[1]M :(x) <strong>and</strong>ð μν :Φ 2[1]M:(x) have been made, where[1] serves as label for the particular choice, one has, in principle, the freedomof redefining these observables by adding suitable quantities depending onlyon the local curvature of spacetime (so as to preserve local covariance), like:Φ 2[2]M:(x) =:Φ2[1]M:(x)+y[2][1] M (x) ,ð μν :Φ 2[2]M :(x) =ð μν :Φ 2[1][2][1]M:(x)+YMμν (x) .Therefore, on curved spacetime the thermal interpretation of Wick square<strong>and</strong> its balanced derivatives depends on the choice one makes here, <strong>and</strong> itis worth contemplating if there are preferred choices which may restrict theapparent arbitrariness affecting the LTE criterion.(γ) Similarly as observed towards the end of the previous section, thereare differential equations to be fulfilled in order that the LTE condition canbe consistent. For the case of the linear (minimally coupled, massless) scalarfield, Solveen [33] has noted that the condition of the thermal stress-energytensor having vanishing trace in LTE states leads to a differential equationof the form14 ∇a ∇ a ϑ(x)+ξR(x)ϑ(x)+U(x) =0,where ϑ is the expectation value of the Wick square in an LTE state, R is thescalar curvature of the underlying spacetime M, <strong>and</strong>U is another functiondetermined by the curvature of M. In view of the previous item (β), aredefinition of the Wick square will alter the function U(x) of the differentialequation that must be obeyed by ϑ(x). The consequences of that possibilityare yet to be determined. It is worth mentioning that for the Dirac field, whichwe don’t treat in these proceedings, an analogous consistency condition leadsto an equation which can only be fulfilled provided that — in this case —the first balanced derivative is defined appropriately, i.e. with addition of adistinct curvature term relative to the SHP definition of the Wick square [24].We admit that so far we do not fully underst<strong>and</strong> the interplay of the LTEcondition <strong>and</strong> the renormalization ambiguity which is present in the definitionof Wick products <strong>and</strong> their balanced derivatives in curved spacetime, buthope to address some aspects of that interplay in greater detail elsewhere[18].
246 R. Verch(δ) Finally we mention that one can prove so-called “quantum energy inequalities”,i.e. lower bounds on weighted integrals of the energy density inLTE states along timelike curves (see [14] for a review on quantum energyinequalities); the lower bounds depend on the maximal temperature an LTEstate attains along the curve. This holds for a wide range of curvature couplingsξ <strong>and</strong> all mass parameters m [32]. That is of interest since, whilestate-independent lower bounds on weighted integrals of the energy densityhave been established for all Hadamard states of the minimally coupled linearscalar field in generic spacetimes [13], such a result fails in this generality forthe non-minimally coupled scalar field [15]. We recommend that the readertakes a look at the references for further information on this circle of questions<strong>and</strong> their possible relevance regarding the occurrence of singularities insolutions to the semiclassical Einstein equations.4. LTE states on cosmological spacetimesAs mentioned previously, one of the domains where one can apply the conceptof LTE states <strong>and</strong> also examine its utility is early cosmology. Thus, we reviewthe steps which have been taken, or are currently being taken, in investigatingLTE states in cosmological scenarios.The central premise in st<strong>and</strong>ard cosmology is that one considers phenomenaat sufficiently large scales such that it is a good approximation toassume that, at each instant of time, the geometry of space (<strong>and</strong>, in orderto be consistent with Einstein’s equations, the distribution of matter <strong>and</strong> energy)is isotropic <strong>and</strong> homogeneous [42]. Making for simplicity the additionalassumption (for which there seems to be good observational motivation) thatthe geometry of space is flat at each time, one obtainsI × R 3 , ds 2 = dt 2 − a(t) 2( (dx 1 ) 2 +(dx 2 ) 2 +(dx 3 ) 2) (4.1)for the general form of spacetime manifold <strong>and</strong> metric, respectively, in st<strong>and</strong>ardcosmology. Here, I is an open interval hosting the time coordinate t,<strong>and</strong> a(t) is a smooth, strictly positive function called the scale factor. Withthe spacetime geometry of the general form (4.1), the only freedom is thetime function a(t), to be determined by Einstein’s equation together with amatter model <strong>and</strong> initial conditions.The time coordinate in (4.1) is called cosmological time. Under suitable(quite general) conditions on a(t) one can pass to a new time coordinate,called conformal time,∫ tdt ′η = η(t) =t 0a(t ′ ) dt′for some choice of t 0 . SettingΩ(η) =a(t(η)) ,
Local Thermal Equilibrium <strong>and</strong> Cosmology 247the metric (4.1) takes the formds 2 =Ω(η) 2 (dη 2 − (dx 1 ) 2 − (dx 2 ) 2 − (dx 3 ) 2 )(4.2)with respect to the conformal time coordinate, so it is conformally equivalentto flat Minkowski spacetime. 24.1. Existence of LTE states at fixed cosmological timeNow let Φ M be the linear, quantized scalar field on the cosmological spacetime(4.1) for some a(t), or equivalently on R 4 with metric (4.2); we assumearbitrary curvature coupling, as in (3.12). The first question one would liketo answer is if there are LTE states of order 2 for Φ M . As one might imagine,this is a very difficult problem, <strong>and</strong> it seems that the method employed bySolveen [33] to establish existence of non-trivial LTE states on bounded openregions of Minkowski spacetime cannot be used to obtain an analogous resultin curved spacetime. In view of the constraints on the temperature evolutionit seems a good starting point to see if there are any second-order LTE statesat some fixed cosmological time, i.e. on a Cauchy surface — this would postponethe problem of having to establish solutions to the evolution equationsof the temperature distribution. Moreover, to fit into the formalism, suchstates have to fulfill the microlocal spectrum condition, i.e. they have to beHadamard states. Allowing general a(t), this problem is much harder thanit seems, since if one took an “instantaneous KMS state” at some value ofcosmological time — such a state, defined in terms of the Cauchy data formulationof the quantized linear scalar field, appears as a natural c<strong>and</strong>idate foran LTE state at fixed time — then that state is in general not Hadamard ifa(t) is time-dependent. The highly non-trivial problem was solved by Schlemmerin his PhD thesis [30]. The result he established is as follows.Theorem 4.1. Let t 1 be a value of cosmological time in the interval I, <strong>and</strong>let e a 0 =(dt) a be the canonical time vector field of the spacetime (4.1). Thenthere is some β 1 > 0 (depending on m, ξ <strong>and</strong> the behaviour of a(t) near t 1 )such that, for each β
248 R. Verchsuch that ω β(x) (ϑ(x)) = θ(x). Likewise, one can look for second-order LTEstates ω βμν (x) at x with β μν (x) =β μν (x)e a 0 <strong>and</strong> ω βμν (x)(ε μν (x)) = ɛ μν (x)(note that there is no sum over the indices) with respect to a tetrad basisat x containing e a 0.Ifω is itself a second-order LTE state at sharp temperature,then all the numbers β(x) <strong>and</strong>β μν (x) must coincide, or rather theassociated absolute temperatures T(x) =1/kβ(x) <strong>and</strong>T μν (x) =1/kβ μν (x).Otherwise, the mutual deviation of these numbers can be taken as a measurefor the failure of ω to be a sharp temperature, second-order LTE state atx. Schlemmer has investigated such a case, choosing ξ =0.1 <strong>and</strong>m =1.5in natural units, <strong>and</strong> a(t) =e Ht with H =1.3; he constructed a spatiallyisotropic <strong>and</strong> homogeneous state ω which is second-order LTE at conformaltime η = η 1 = −1.0, <strong>and</strong> calculated, as just described, the “would-be LTE”comparison temperatures T(x) ≡ T(η) <strong>and</strong>T μν (x) ≡ T μν (η) numerically forearlier <strong>and</strong> later conformal times. The result is depicted in Figure 2.T8T from Θ7T from Ε 00T from Ε 116541.6 1.4 1.2 1.0 0.8 0.6 0.4ΗFigure 2. LTE comparison temperatures calculated fromθ(η) <strong>and</strong>ɛ μν (η) for a state constructed as second-order LTEstate at η = −1.0 [30]Figure 2 shows that the comparison temperatures deviate from eachother away from η = η 1 =1.0. However, they all drop with increasing ηas they should for an increasing scale factor. Both the absolute <strong>and</strong> relativetemperature deviations are larger at low temperatures than at high temperatures.This can be taken as an indication of a general effect, namely that theLTE property is more stable against perturbations (which drive the systemaway from LTE) at high temperature than at low temperature.4.2. LTE <strong>and</strong> metric scatteringLet us turn to a situation demonstrating that effect more clearly. We considerthe quantized, conformally coupled (ξ =1/6) linear scalar field Φ M on R 4with conformally flat metric of the form (4.2), where Ω(η) is chosen withΩ(η) 2 = λ + Δλ (1 + tanh(ρη)) , (4.3)2
Local Thermal Equilibrium <strong>and</strong> Cosmology 249where λ, Δλ <strong>and</strong> ρ are real positive constants. One can show that the quantumfield Φ M has asymptotic limits as η →±∞; in a slightly sloppy notation,{lim Φη ′ M (η + η ′ Φ in (η, x), x) =(η ∈ R, x =(x 1 ,x 2 ,x 3 ) ∈ R 3 ) ,→∓∞ Φ out (η, x)where Φ in <strong>and</strong> Φ out are copies of the quantized linear scalar field with massm on Minkowski spacetime, however with different (constant) scale factors.This means that Φ in <strong>and</strong> Φ out obey the field equations(λ✷ + m 2 )Φ in =0, ((λ +Δλ)✷ + m 2 )Φ out =0,where ✷ = ∂η 2 − ∂x 2 − ∂ 2 1 x− ∂ 2 2 x.Ifω 2 in is a state of Φ in , it induces a stateω out of Φ out by setting〈Φ out (η 1 , x 1 ) ···Φ out (η n , x n )〉 ωout = 〈Φ in (η 1 , x 1 ) ···Φ in (η n , x n )〉 ωin .Thus, ω out is the state obtained from ω in through the scattering processthe quantum field Φ M undergoes by propagating on a spacetime with theexp<strong>and</strong>ing conformal scale factor as in (4.3). This particular form of the scalefactor has the advantage that the scattering transformation taking Φ in intoΦ out can be calculated explicitly, <strong>and</strong> so can the relation between ω in <strong>and</strong>ω out [2]. It is known that if ω in is a Hadamard state, then so is ω out .In a forthcoming paper [25], we pose the following question: If we takeω in as a global thermal equilibrium state (with respect to the time coordinateη) ofΦ in ,isω out an LTE state of second order for Φ out ? The answers, whichwe will present in considerably more detail in [25], can be roughly summarizedas follows.(1) For a wide range of parameters m, λ, Δλ <strong>and</strong> ρ, <strong>and</strong> inverse temperatureβ in of ω in , ω out is a mixed temperature LTE state (with respect to thetime direction of η) of second order.(2) Numerical calculations of the comparison temperatures T <strong>and</strong> T μν fromθ <strong>and</strong> ɛ μν for ω out show that they mutually deviate (so ω out is no sharptemperature LTE state). Numerically one can show that the deviationsdecrease with increasing temperature T in of ω in (depending, of course,on the other parameters m, λ, Δλ <strong>and</strong> ρ).This shows that scattering of the quantum field by the exp<strong>and</strong>ing spacetimemetric tends to drag the initially global thermal state away from equilibrium.However, this effect is the smaller the higher the initial temperature,as one would expect intuitively. As the initial temperature goes to zero, ω inapproaches the vacuum state. In this case, ω out is a non-vacuum state dueto quantum particle creation induced by the time-varying spacetime metricwhich is non-thermal [41]. That may appear surprising in view of the closeanalogy of the particle-creation-by-metric-scattering effect to the Hawkingeffect. However, it should be noted that the notion of temperature in thecontext of the Hawking effect is different form the temperature definitionentering the LTE condition. In the framework of quantum fields on de Sitterspacetime, Buchholz <strong>and</strong> Schlemmer [8] noted that the KMS temperature of
250 R. Vercha state with respect to a Killing flow differs, in general, from the LTE temperature.To give a basic example, we note that the vacuum state of a quantumfield theory in Minkowski spacetime is a KMS state at non-zero temperaturewith respect to the Killing flow of the Lorentz boosts along a fixed spatialdirection (that’s the main assertion of the Bisognano-Wichmann theorem,see [19] <strong>and</strong> references cited there). On the other h<strong>and</strong>, the vacuum state isalso an LTE state of infinite order, but at zero temperature. It appears thatthe question how to modify the LTE condition such that it might becomesensitive to the Hawking temperature has not been discussed so far.4.3. LTE temperature in the Dappiaggi-Fredenhagen-Pinamonti cosmologicalapproachIn this last part of our article, we turn to an application of the LTE concept inan approach to cosmology which takes as its starting point the semiclassicalEinstein equationG Mab (x) =8πG〈T Mab (x)〉 ωM . (4.4)On the left-h<strong>and</strong> side, we have the Einstein tensor of the spacetime geometryM, on the right-h<strong>and</strong> side the expectation value of the stress-energy tensorof Φ M which is part of a local covariant quantum field M in a state ω M .Adopting the st<strong>and</strong>ing assumptions of st<strong>and</strong>ard cosmology, it seems a fairansatz to assume that a solution can be obtained for an M of the spatially flatFriedmann-Robertson-Walker form (4.1) while taking for Φ M the quantizedlinear conformally coupled scalar field whose field equation is)(∇ a ∇ a + 1 6 R + m2 Φ M =0. (4.5)And, in fact, in a formidable work, Pinamonti [26] has recently shown thatthis assumption is justified: Adopting the metric ansatz (4.1) for M, hecouldestablish that, for any given m ≥ 0, there are an open interval of cosmologicaltime, a scale factor a(t) <strong>and</strong> a Hadamard state ω M such that (4.4) holds forthe t-values in the said interval (identifying x =(t, x 1 ,x 2 ,x 3 )).However, we will simplify things here considerably by setting m =0from now on, so that we have a conformally covariant quantized linear scalarfield with field equation(∇ a ∇ a + 1 6 R )Φ M =0. (4.6)As pointed out above in Sec. 2, in the case of conformal covariant Φ M onehas to face the trace anomalyaT M a (x) =−4Q M (x)1 , (4.7)where Q M is the divergence-compensating term of (2.7), a quantity which isdetermined by the geometry of M <strong>and</strong> which hence is state-independent, so(4.7) is to be read as an equation at the level of operators. However, equation(4.7) is not entirely complete as it st<strong>and</strong>s, since it doesn’t make explicit thatT Mab (x) is subject to a renormalization ambiguity which lies in the freedomof adding divergence-free, local covariant tensor fields C Mab (x). Therefore,
Local Thermal Equilibrium <strong>and</strong> Cosmology 251also the trace of the renormalized stress-energy tensor is subject to such anambiguity. Let us now be a bit more specific about this point. Suppose wedefine, in a first step, a renormalized ˜T Mab (x) by the SHP renormalizationprescription. Then defineT Mab (x) =T [0]Mab (x) =˜T Mab (x) − Q M (x)g ab (x) ,where Q M (x) is the divergence-compensating term defined with respect to˜T Mab (x). Now, if T Mab (x) is re-defined asT [C]Mab(x) =T[0]Mab (x)+C Mab(x)with C Mab (x) local covariant <strong>and</strong> divergence-free, one obtains for the traceT [C] aM a (x) = ( − 4Q M (x)+CMa(x) a ) 1 . (4.8)For generic choice of C Mab , the right-h<strong>and</strong> side of (4.8) contains derivativesof the spacetime metric of higher than second order. However, Dappiaggi,Fredenhagen <strong>and</strong> Pinamonti [9] have pointed out that one can make a specificchoice of C Mab such that the right-h<strong>and</strong> side of (4.8) contains only derivativesof the metric up to second order. Considering again the semiclassical Einsteinequation (4.4) <strong>and</strong> taking traces on both sides, one getsR M (x) =8πG ( − 4Q M (x)+CMa(x) a ) , (4.9)which (in the case of a conformally covariant Φ M considered here) determinesthe spacetime geometry independent of the choice of a state ω M (after supplyingalso initial conditions). With our metric ansatz (4.1), equation (4.9) isequivalent to a non-linear differential equation for a(t), of the general formF ( a (n) (t),a (n−1) (t),...,a (1) (t),a(t) ) =0, (4.10)with a (j) (t) =(d j /dt j )a(t) <strong>and</strong>withn denoting the highest derivative orderin the differential equation for a(t). For generic choices of C Mab , n turnsout to be greater than 2. On the other h<strong>and</strong>, if C Mab is specifically chosenso that the trace T [C]M a a contains only derivatives up to second order of thespacetime metric, then n = 2. Making or not making this choice has drasticconsequences for the behaviour of solutions a(t) to (4.10). The case ofgeneric C Mab , leading to n ≥ 3 in (4.10), was investigated in a famous articleby Starobinski [34]. He showed that, in this case, the differential equation(4.10) for a(t) has solutions with an inflationary, or accelerating (a (2) (t) > 0),phase which is unstable at small time scales after the initial conditions. Thisprovided a natural argument why the inflationary phase of early cosmologywould end after a very short timespan. However, recent astronomical observationshave shown accelerating phases of the Universe over a large timescale atlate cosmic times. The popular explanation for this phenomenon postulatesan exotic form of energy to be present in the Universe, termed “Dark Energy”[42]. In contrast, Dappiaggi, Fredenhagen <strong>and</strong> Pinamonti have shownthat the specific choice of C Mab leads to a differential equation (4.10) fora(t) withn = 2 which admits stable solutions with a long-term acceleratingphase at late cosmological times [9]. In a recent work [10], it was investigated
252 R. Verchif such solutions could account for the observations of the recently observedaccelerated cosmic expansion. Although that issue remains so far undecided,also in view of the fact that linear quantized fields are certainly too simpleto describe the full physics of quantum effects in cosmology, it bears theinteresting possibility that “cosmic acceleration” <strong>and</strong> “Dark Energy” couldactually be traced back to the renormalization ambiguities <strong>and</strong> anomaliesarising in the definition of the stress-energy tensor of quantum fields in thepresence of spacetime curvature. In that light, one can take the point ofview that the renormalization ambiguity of T Mab is ultimately constrained(<strong>and</strong> maybe fixed) by the behaviour of solutions to the semiclassical Einsteinequation: One would be inclined to prefer such C Mab whichleadtostablesolutions, in spite of the argument of [34]. At any rate, only those C Mab canbe considered which lead to solutions to the semiclassical Einstein equationscompatible with observational data. After all, the renormalization freedomin quantum field theoretic models of elementary particle physics is fixed in avery similar way.The differential equation with stable solutions a(t) derived in [9] canbe expressed in terms of the following differential equation for the Hubblefunction H(t) =a (1) (t)/a(t),Ḣ(H 2 − H 0 )=−H 4 +2H 0 H 2 , (4.11)where H 0 is some universal positive constant <strong>and</strong> the dot means differentiationwith respect to t. There are two constant solutions to (4.11) (obviouslyH(t) = 0 is a solution) as well as non-constant solutions depending on initialconditions. For the non-constant solutions, one finds an asymptotic behaviouras follows [9, 17]: For early cosmological times,H(t) ≈ 1 , a(t) ≈ Γ(t − t 0 ) , (4.12)t − t 0with some constants Γ > 0<strong>and</strong>realt 0 ,fort>t 0 . On the other h<strong>and</strong>, for latecosmological times:H(t) ≈ √ 2H 0 coth ( 2 √ 2H 0 t − 1 ) . (4.13)Therefore, H(t) <strong>and</strong>a(t) have, for early cosmological times, a singularity ast → t 0 , but different from the behaviour of a Universe filled with classicalradiation, which is known to yield [39, 42]H rad (t) ≈12(t − t 0 ) , a rad(t) ≈ Γ ′√ t − t 0with some positive constant Γ ′ . For the temperature behaviour of radiationclose to the singularity at t → t 0 one then obtainsκ ′T rad (t) ≈ √ t − t0with another constant κ ′ .Now we wish to compare this to the temperature behaviour, as t → t 0 ,of any second-order, sharp temperature LTE state ω M of Φ M fulfilling the
Local Thermal Equilibrium <strong>and</strong> Cosmology 253semiclassical Einstein equation in the Dappiaggi-Fredenhagen-Pinamonti approachfor t → t 0 . Of course, the existence of such LTE states is an assumption,<strong>and</strong> in view of the results of Subsection 4.1, cf. Figure 2, this assumptionis certainly an over-idealization. On the other h<strong>and</strong>, Figure 2 can also be interpretedas saying that, at early cosmic times, while sharp temperature LTEstates might possibly not exist, it is still meaningful to attribute an approximatetemperature behaviour to states as t → t 0 .In order to determine the temperature behaviour of the assumed secondorderLTE state ω M , one observes first that [32]〈[C] Tab (x)〉 = εω ab (x)+ 1M 12 ∇a ∇ b ϑ(x)+− 1 3 g ab(x)ε c c(x) (4.14)+ q M (x)ϑ(x)+K [C]Mab (x)with some state-independent tensor K [C]Mab which depends on C Mab <strong>and</strong> whichhas local covariant dependence of the spacetime geometry M; likewise sohas q M . Both K [C]Mab <strong>and</strong> q M can be explicitly calculated once C Mab <strong>and</strong>,consequently, a(t) (respectively, H(t)) are specified. Then, ε ab (x)<strong>and</strong>ϑ(x)arefunctions of β(t) (making the usual assumption that ω M is homogeneous <strong>and</strong>isotropic). To determine β(t), relation (4.14) is plugged into the vanishingof-divergenceequation∇ a〈 T [C]ab (x)〉 =0,ω Mthus yielding a non-linear differential equation involving β(t) <strong>and</strong>a(t). Insertingany a(t) coming from the non-constant Dappiaggi-Fredenhagen-Pinamontisolutions, one can derive the behaviour, as t → t 0 ,ofβ(t), <strong>and</strong> it turnsout [17] that β(t) ≈ γa(t) with another constant γ>0. This resembles thebehaviour of classical radiation. But in view of the different behaviour of a(t)as compared to early cosmology of a radiation-dominated Universe, one nowobtains a temperature behaviourκT(t) ≈ ,t − t 0with yet another constant κ>0; this is more singular than T rad (t) in the limitt → t 0 . We will present a considerably more detailed analysis of the temperaturebehaviour of LTE states in the context of the Dappiaggi-Fredenhagen-Pinamonti cosmological model elsewhere [18].5. Summary <strong>and</strong> outlookThe LTE concept allows it to describe situations in quantum field theorywhere states are no longer in global thermal equilibrium, but still possess,locally, thermodynamic parameters. In curved spacetime, this concept is intriguinglyinterlaced with local covariance <strong>and</strong> the renormalization ambiguitieswhich enter into its very definition via Wick products <strong>and</strong> their balancedderivatives. Particularly when considering the semiclassical Einstein equationsin a cosmological context this plays a role, <strong>and</strong> the thermodynamic
254 R. Verchproperties of quantum fields in the very early stages of cosmology can turnout to be different from what is usually assumed in considerations based on,e.g., modelling matter as classical radiation. The implications of these possibilitiesfor theories of cosmology remain yet to be explored — so far we haveonly scratched the tip of an iceberg, or so it seems.There are several related developments concerning the thermodynamicbehaviour of quantum fields in curved spacetime, <strong>and</strong> in cosmological spacetimesin particular, which we haven’t touched upon in the main body of thetext. Worth mentioning in this context is the work by Holl<strong>and</strong>s <strong>and</strong> Leileron a derivation of the Boltzmann equation in quantum field theory [21]. Itshould be very interesting to try <strong>and</strong> explore relations between their approach<strong>and</strong> the LTE concept. There are also other concepts of approximatethermal equilibrium states [11], <strong>and</strong> again, the relation to the LTE conceptshould render interesting new insights. In all, the new light that these recentdevelopments shed on quantum field theory in early cosmology is clearlyconceptually fruitful <strong>and</strong> challenging.References[1] C. Bär, N. Ginoux <strong>and</strong> F. Pfäffle, Wave equations on Lorentzian manifolds <strong>and</strong>quantization, European<strong>Mathematical</strong>Society,Zürich, 2007[2] N.D. Birrell <strong>and</strong> P.C.W. Davies, <strong>Quantum</strong> fields in curved space, CambridgeUniversity Press, Cambridge, 1982[3] N.N. Bogoliubov <strong>and</strong> D.V. Shirkov, Introduction to the theory of quantizedfields, Wiley-Interscience, New York, 1959[4] R. Brunetti, K. Fredenhagen <strong>and</strong> M. Köhler, The microlocal spectrum condition<strong>and</strong> Wick polynomials of free fields on curved space-times, Commun. Math.Phys. 180 (1996) 633[5] R. Brunetti, K. Fredenhagen <strong>and</strong> R. Verch, The generally covariant localityprinciple — a new paradigm for local quantum field theory, Commun. Math.Phys. 237 (2003) 31[6] D. Buchholz, On hot bangs <strong>and</strong> the arrow of time in relativistic quantum fieldtheory, Commun. Math. Phys. 237 (2003) 271[7] D. Buchholz, I. Ojima <strong>and</strong> H.-J. Roos, Thermodynamic properties of nonequilibriumstates in quantum field theory, Annals Phys. 297 (2002) 219[8] D. Buchholz <strong>and</strong> J. Schlemmer, Local temperature in curved spacetime, Class.Quant. Grav. 24 (2007) F25[9] C. Dappiaggi, K. Fredenhagen <strong>and</strong> N. Pinamonti, Stable cosmological modelsdriven by a free quantum scalar field, Phys. Rev. D77 (2008) 104015[10] C. Dappiaggi, T.-P. Hack, J. Möller <strong>and</strong> N. Pinamonti, Dark energy from quantummatter, e-Print: arXiv:1007.5009 [astro-ph.CO][11] C. Dappiaggi, T.-P. Hack <strong>and</strong> N. Pinamonti, Approximate KMS states forscalar <strong>and</strong> spinor fields in Friedmann-Robertson-Walker spacetimes, e-Print:arXiv:1009.5179 [gr-qc][12] C.J. Fewster, On the notion of “the same physics in all spacetimes”, intheseproceedings, e-Print: arXiv:1105.6202 [math-ph]
Local Thermal Equilibrium <strong>and</strong> Cosmology 255[13] C.J. Fewster, A general worldline quantum inequality, Class. <strong>Quantum</strong> Grav.17 (2000) 1897[14] C.J. Fewster, Energy inequalities in quantum field theory, in: J. Zambrini (ed.),Proceedings of the XIV International Congress on <strong>Mathematical</strong> Physics, Lisbon,2003, World Scientific, Singapore, 2005. e-Print: arXiv:math-ph/0501073[15] C.J. Fewster <strong>and</strong> L. Osterbrink, <strong>Quantum</strong> energy inequalities for the nonminimallycoupled scalar field, J. Phys. A41(2008) 025402[16] C.J. Fewster <strong>and</strong> C.J. Smith, Absolute quantum energy inequalities in curvedspacetime, Ann. Henri Poincaré 9 (2008) 425[17] M. Gransee, Diploma Thesis, Institute of Theoretical Physics, University ofLeipzig, 2010[18] A. Knospe, M. Gransee <strong>and</strong> R. Verch, in preparation[19] R. Haag, Local quantum physics, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York, 1996[20] S. Holl<strong>and</strong>s, Renormalized quantum Yang-Mills fields in curved spacetime, Rev.Math. Phys. 20 (2008) 1033[21] S. Holl<strong>and</strong>s <strong>and</strong> G. Leiler, On the derivation of the Boltzmann equation inquantum field theory: flat spacetime, e-Print: arXiv:1003.1621 [cond-mat.statmech][22] S. Holl<strong>and</strong>s <strong>and</strong> R.M. Wald, Local Wick polynomials <strong>and</strong> time ordered productsof quantum fields in curved space-time, Commun. Math. Phys. 223 (2001) 289[23] Kay, B.S. <strong>and</strong> Wald, R.M., Theorems on the uniqueness <strong>and</strong> thermal propertiesof stationary, nonsingular, quasifree states on space-times with a bifurcateKilling horizon, Phys. Rep. 207 (1991) 49[24] A. Knospe, Diploma Thesis, Institute of Theoretical Physics, University ofLeipzig, 2010[25] F. Lindner <strong>and</strong> R. Verch, to be published[26] N. Pinamonti, On the initial conditions <strong>and</strong> solutions of the semiclassical Einsteinequations in a cosmological scenario, e-Print: arXiv:1001.0864 [gr-qc][27] N. Pinamonti <strong>and</strong> R. Verch, in preparation[28] M.J. Radzikowski, Micro-local approach to the Hadamard condition in quantumfield theory on curved space-time, Commun. Math. Phys. 179 (1996) 529[29] K. S<strong>and</strong>ers, Equivalence of the (generalised) Hadamard <strong>and</strong> microlocal spectrumcondition for (generalised) free fields in curved spacetime, Commun. Math.Phys. 295 (2010) 485[30] J. Schlemmer, PhD Thesis, Faculty of Physics, University of Leipzig, 2010[31] J. Schlemmer, Local thermal equilibrium states <strong>and</strong> Unruh detectors in quantumfield theory, e-Print: hep-th/0702096 [hep-th][32] J. Schlemmer <strong>and</strong> R. Verch, Local thermal equilibrium states <strong>and</strong> quantumenergy inequalities, Ann. Henri Poincaré 9 (2008) 945[33] C. Solveen, Local thermal equilibrium in quantum field theory on flat <strong>and</strong> curvedspacetimes, Class. Quant. Grav. 27 (2010) 235002[34] A. Starobinski, A new type of isotropic cosmological models without singularity,Phys. Lett. B91 (1980) 99
256 R. Verch[35] A. Strohmaier, R. Verch <strong>and</strong> M. Wollenberg, Microlocal analysis of quantumfields on curved spacetimes: analytic wavefront sets <strong>and</strong> Reeh-Schlieder theorems,J. Math. Phys. 43 (2002) 5514[36] R. Verch, A spin-statistics theorem for quantum fields on curved spacetimemanifolds in a generally covariant framework, Commun. Math. Phys. 223(2001) 261[37] R.M. Wald, The back reaction effect in particle creation in curved spacetime,Commun. Math. Phys. 54 (1977) 1[38] R.M. Wald, Trace anomaly of a conformally invariant quantum field in curvedspace-time, Phys. Rev. D17 (1978) 1477[39] R.M. Wald, General relativity, University of Chicago Press, Chicago, 1984[40] R.M. Wald, <strong>Quantum</strong> field theory in curved spacetime <strong>and</strong> black hole thermodynamics,University of Chicago Press, Chicago, 1994[41] R.M. Wald, Existence of the S matrix in quantum field theory in curved spacetime,Ann. Physics (N.Y.) 118 (1979) 490[42] S. Weinberg, Cosmology, Oxford University Press, Oxford, 2008Rainer VerchInstitut für Theoretische PhysikUniversität LeipzigVor dem Hospitaltore 1D-04103 LeipzigGermanye-mail: verch@itp.uni-leipzig.de
Shape Dynamics. An IntroductionJulian BarbourAbstract. Shape dynamics is a completely background-independent universalframework of dynamical theories from which all absolute elementshave been eliminated. For particles, only the variables that describe theshapes of the instantaneous particle configurations are dynamical. Inthe case of Riemannian three-geometries, the only dynamical variablesare the parts of the metric that determine angles. The local scale factorplays no role. This leads to a shape-dynamic theory of gravity inwhich the four-dimensional diffeomorphism invariance of general relativityis replaced by three-dimensional diffeomorphism invariance <strong>and</strong>three-dimensional conformal invariance. Despite this difference of symmetrygroups, it is remarkable that the predictions of the two theories –shape dynamics <strong>and</strong> general relativity – agree on spacetime foliations byhypersurfaces of constant mean extrinsic curvature. However, the twotheories are distinct, with shape dynamics having a much more restrictiveset of solutions. There are indications that the symmetry group ofshape dynamics makes it more amenable to quantization <strong>and</strong> thus to thecreation of quantum gravity. This introduction presents in simple termsthe arguments for shape dynamics, its implementation techniques, <strong>and</strong>a survey of existing results.Mathematics Subject Classification (2010). 70G75, 70H45, 83C05, 83C45.Keywords. <strong>Gravity</strong> theory, conformal invariance, Mach’s principle.1. IntroductionOne of Einstein’s main aims in creating general relativity was to implementMach’s idea [1, 2] that dynamics should use only relative quantities <strong>and</strong> thatinertial motion as expressed in Newton’s first law should arise, not as aneffect of a background absolute space, but from the dynamical effect of theuniverse as a whole. Einstein called this Mach’s principle [3]. However, ashe explained later [4, 5] (p. 186), he found it impractical to realize Mach’sprinciple directly <strong>and</strong> was forced to use coordinate systems. This has obscuredthe extent to which <strong>and</strong> how general relativity is a background-independentF. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_13,© Springer Basel AG 2012257
258 J. Barbourtheory. My aim in this paper is to present a universal framework for the direct<strong>and</strong> explicit creation of completely background-independent theories.I shall show that this leads to a theory of gravity, shape dynamics, thatis distinct from general relativity because it is based on a different symmetrygroup, according to which only the local shapes of Riemannian 3-geometriesare dynamical. Nevertheless, it is remarkable that the two theories have anontrivial ‘intersection’, agreeing exactly in spatially closed universes whenever<strong>and</strong> wherever Einsteinian spacetimes admit foliation by hypersurfaces ofconstant mean extrinsic curvature. However, many solutions of general relativitythat appear manifestly unphysical, such as those with closed timelikecurves, are not allowed in shape dynamics. In addition, it appears that thestructure of shape dynamics makes it significantly more amenable to quantizationthan general relativity.This is not the only reason why I hope the reader will take an interest inshape dynamics. The question of whether motion is absolute or relative hasa venerable history [6, 7], going back to long before Newton made it famouswhen he formulated dynamics in terms of absolute space <strong>and</strong> time [8]. Whatis ultimately at stake is the definition of position <strong>and</strong>, above all, velocity. Thishas abiding relevance in our restless universe. I shall show that it is possibleto eliminate every vestige of Newtonian absolutes except for just one. Butthis solitary remnant is hugely important: it allows the universe to exp<strong>and</strong>.Shape dynamics highlights this remarkable fact.This introduction will be to a large degree heuristic <strong>and</strong> based on Lagrangianformalism. A more rigorous Hamiltonian formulation of shape dynamicsbetter suited to calculations <strong>and</strong> quantum-gravity applications was recentlydiscovered in [9] (a simplified treatment is in [10]). Several more papersdeveloping the Hamiltonian formulation in directions that appear promisingfrom the quantum-gravity perspective are in preparation. A dedicated website(shapedynamics.org) is under construction; further background informationcan be found at platonia.com.The contents list (see the start of this volume) obviates any further introduction,but a word on terminology will help. Two distinct meanings ofrelative are often confused. Mach regarded inter-particle separations as relativequantities; in Einstein’s theories, the division of spacetime into space <strong>and</strong>time is made relative to an observer’s coordinate system. To avoid confusion,Iuserelational in lieu of Mach’s notion of relative.2. The relational critique of Newton’s dynamics2.1. Elimination of redundant structureNewton’s First Law states: “Every body continues in its state of rest oruniform motion in a right line unless it is compelled to change that state byforces impressed on it.” Since the (absolute) space in which the body’s motionis said to be straight <strong>and</strong> the (absolute) time that measures its uniformity areboth invisible, this law as stated is clearly problematic. Newton knew this <strong>and</strong>
Shape Dynamics. An Introduction 259argued in his Scholium in the Principia [8] that his invisible absolute motionscould be deduced from visible relative motions. This can be done but requiresmore relative data than one would expect if only directly observable initialdata governed the dynamics. As we shall see, this fact, which is not widelyknown, indicates how mechanics can be reformulated with less kinematicstructure than Newton assumed <strong>and</strong> simultaneously be made more predictive.It is possible to create a framework that fully resolves the debate about thenature of motion. In this framework, the fewest possible observable initial datadetermine the observable evolution. 1I show first that all c<strong>and</strong>idate relational configurations 2 of the universehave structures determined by a Lie group, which may be termed their structuregroup. The existence of such a group is decisive. It leads directly to anatural way to achieve the aim just formulated <strong>and</strong> to a characteristic universalstructure of dynamics applicable to a large class of systems. It is presentin modern gauge theories <strong>and</strong>, in its most perfect form, in general relativity.However, the relational core of these theories is largely hidden because theirformulation retains redundant kinematic structure.To identify the mismatch that shape dynamics aims to eliminate, thefirst step is to establish the essential structure that Newtonian dynamicsemploys. It will be sufficient to consider N, N ≥ 3, point particles interactingthrough Newtonian gravity. In an assumed inertial frame of reference, eachparticle a, a =1, ..., N, has coordinates x i a(t), i = x, y, z, that depend on t,the Newtonian time. The x i a’s <strong>and</strong> t are all assumed to be observable. Theparticles, assumed individually identifiable, also have constant masses m a .For the purposes of our discussion, they can be assumed known.Let us now eliminate potentially redundant structure. Newton grantedthat only the inter-particle separations r ab , assumed to be ‘seen’ all at once,are observable. In fact, this presupposes an external (absolute) ruler. Closerto empirical reality are the dimensionless ratios˜r ab :=r √ab∑, R rmh := rab 2 R , (1)rmha
260 J. Barbourrelative to each other. All that we have are the sets {˜r ab }. The totality ofsuch sets is shape space Q N ss , which only exists for N ≥ 3.3 The number ofdimensions of Q N ss is 3N − 7: from the 3N Cartesian coordinates, six aresubtracted because Euclidean translations <strong>and</strong> rotations do not change ther ab ’s <strong>and</strong> the seventh because the {˜r ab }’s are scale invariant.Shape space is our key concept. <strong>Mathematical</strong>ly, we reach it through asuccession of spaces, the first being the 3N-dimensional Cartesian configurationspace Q N . In it, all configurations that are carried into each other bytranslations t in T, the group of Euclidean translations, belong to a commonorbit of T. Thus,T decomposes Q N into its group orbits, which are definedto be the points of the (3N − 3)-dimensional quotient space T N := Q N /T.This first quotienting to T N is relatively trivial. More significant is the furtherquotienting by the rotation group R to the (3N −6)-dimensional relativeconfiguration space Q N rcs := QN /TR [11]. The final quotienting by the dilatation(scaling) group S leads to shape space Q N ss := QN /TRS [12]. The groupsT <strong>and</strong> R together form the Euclidean group, while the inclusion of S yieldsthe similarity group. The orbit of a group is a space with as many dimensionsas the number of elements that specify a group element. The orbits of S thushave seven dimensions (Fig. 1).The groups T, R, S are groups of motion, orLie groups (groups that aresimultaneously manifolds, i.e., their elements are parametrized by continuousparameters). If we have a configuration q of N particles in Euclidean space,q ∈ Q N , we can ‘move it around’ with T or R or ‘change its size’ with S.This intuition was the basis of Lie’s work. It formalizes the fundamentalgeometrical notions of congruence <strong>and</strong> similarity. Two figures are congruentif they can be brought to exact overlap by a combination of translations <strong>and</strong>rotations <strong>and</strong> similar if dilatations are allowed as well.Relational particle dynamics can be formulated in any of the quotientspaces just considered. Intuition suggests that the dynamics of an ‘isl<strong>and</strong>universe’ in Euclidean space should deal solely with its possible shapes. Thesimilarity group is then the fundamental structure group. 4 This leads to particleshape dynamics <strong>and</strong> by analogy to the conformal geometrodynamicsthat will be considered in the second part of the paper.Lie groups <strong>and</strong> their infinite-dimensional generalizations are fundamentalin modern mathematics <strong>and</strong> theoretical physics. They play a dual role inshape dynamics, first in indicating how potentially redundant structure canbe pared away <strong>and</strong>, second, in providing the tool to create theories that arerelationally perfect, i.e., free of the mismatch noted above. Moreover, because3 A single point is not a shape, <strong>and</strong> the distance between two particles can be scaled to anyvalue, so nothing dimensionless remains to define a shape. Also the configuration in whichall particles coincide is not a shape <strong>and</strong> does not belong to shape space.4 One might want to go further <strong>and</strong> consider the general linear group, under which anglesare no longer invariant. I will consider this possibility later.
Shape Dynamics. An Introduction 261Figure 1. Shape space for the 3-body problem is obtainedby decomposing the Newtonian configuration space Q 3 intoorbits of the similarity group S. The points on any givenvertical line (any group orbit) correspond to all possible representationsin Euclidean space of one of the possible shapesof the triangle formed by the three particles. Each such shapeis represented below its orbit as a point in shape space. Eachorbit is actually a seven-dimensional space. The effects of rotation<strong>and</strong> scaling are shown.Lie groups, as groups that are simultaneously manifolds, have a common underlyingstructure <strong>and</strong> are ubiquitous, they permit essentially identical methodsto be applied in many different situations. This is why shape dynamicsis a universal framework.2.2. Newtonian dynamics in shape spaceWe now identify the role that absolute space <strong>and</strong> time play in Newtoni<strong>and</strong>ynamics by projection to shape space. We have removed structure from theq’s in Q N , reducing them to points s ∈ Q N ss . This is projection of q’s. Wecan also project complete Newtonian histories q(t). To include time at thestart, we adjoin to Q N the space T of absolute times t, obtaining the spaceQ N T. Newtonian histories are then (monotonically rising) continuous curvesin Q N T. However, clocks are parts of the universe; there is no external clockavailable to provide the reading for the T axis. All the objective informationis carried by the successive configurations of the universe. We must thereforeremove the T axis <strong>and</strong>, in the first projection, label the points representingthe configurations in Q N by an arbitrary increasing parameter λ <strong>and</strong> thenmake the further projection to the shape space Q N ss. The history becomes s(λ)
262 J. BarbourFigure 2. In Newtonian dynamics, the history of a systemis a monotonically rising curve q(t) inQ N T or a curve q(λ)in Q N labelled by a monotonic λ. The objective observablehistory is the projected curve s(λ) inshapespaceQ N ss .(Fig. 2). A history is the next most fundamental concept in shape dynamics.There is no ‘moving now’ in this concept. History is not a spot moving alongs(λ), lighting up ‘nows’ as it goes. It is the curve; λ merely labels its points.Newtonian dynamics being time-reversal invariant, there is no past-to-futuredirection on curves in Q N ss .Given a history of shapes s, we can define a shape velocity. Supposefirst that in fact by some means we can define a distinguished parameterp, orindependent variable, along a suitably continuous curve in Q N ss .Thenat each point along the curve we have a shape s <strong>and</strong> its (multi-component)velocity ds/dp. Thisisatangent vector to the curve. If we have no p butonly an arbitrary λ, we can still define shape velocities ds/dλ = s ′ , but allwe really have is the direction d (in Q N ss )inwhichs is changing. The differencebetween tangent vectors <strong>and</strong> directions associated with curves in shape spacewill be important later.We can now identify the mismatch that, when eliminated, leads to theshape-dynamic ideal. To this end, we recall Laplacian determinism in Newtoni<strong>and</strong>ynamics: given q <strong>and</strong> ˙q at some instant, the evolution of the system isuniquely determined (the particle masses <strong>and</strong> the force law assumed known).The question is this: given the corresponding shape projections s <strong>and</strong> d, isthe evolution in shape space Q N ss uniquely determined? The answer is no fora purely geometrical reason. The fact is that certain initial velocities whichare objectively significant in Newtonian dynamics can be generated by purely
Shape Dynamics. An Introduction 263bac1δx i atδx i arδx i adδx i aFigure 3. The two triangles of slightly different shapesformed by three particles a, b, c define a point s <strong>and</strong> directiond uniquely in shape space, but changes to an originalplacing (1) in Euclidean space of the dashed triangle relativeto the grey one generated by translations (t), rotations (r),<strong>and</strong> dilatations (d) give rise to different Newtonian initialvelocities dx i a/dt.group actions. To be precise, different Newtonian velocities can be generatedfrom identical data in Q N ss . This is illustrated for the 3-body problem (in twodimensions) in Fig. 3.I will not go into the details of the proof (see [13]), but in a Newtonian N-body system the velocities at any given instant can be uniquely decomposedinto parts due to an intrinsic change of shape <strong>and</strong> three further parts due tothe three different group actions – translations, rotations, <strong>and</strong> dilatations –applied as in Fig. 3. These actions are obviously ‘invisible’ in the shape-spaces <strong>and</strong> d, which define only the shape <strong>and</strong> the way it is changing.By Galilean relativity, translations of the system have no effect in Q N ss .We can ignore them but not rotations <strong>and</strong> dilatations. Four dimensionlessdynamically effective quantities are associated with them. First, two anglesdetermine the direction in space of a rotation axis. Second, from the kineticenergies associated with rotation, T r , dilatation T d , <strong>and</strong> change of shape, T s ,we can form two dimensionless ratios, which it is natural to take to be T r /T s<strong>and</strong> T d /T s (since change of shape, represented by T s , is our ‘gold st<strong>and</strong>ard’).Thus, the kinematic action of the Lie groups generates four parameters thataffect the histories in shape space without changing the initial s <strong>and</strong> d. Thisis already so for pure inertial motion. If forces are present, there is a fifthparameter, the ratio T/V of the system’s kinetic energy T to its potentialenergy V , that is dynamically significant but is also invisible in the s <strong>and</strong> din shape space.We now see that although Newtonian dynamics seems wonderfully rational<strong>and</strong> transparent when expressed in an inertial frame of reference, itdoes not possess perfect Laplacian determinism in shape space. This failureappears especially odd if N is large. Choose some coordinates s i ,i =1, 2, ..., 3N −7, in shape space <strong>and</strong> take one of them, call it τ, as a surrogate forNewton’s t. If only shapes had dynamical effect, then by analogy with inertialframeNewtonian dynamics, the initial values of τ,s i , ds i /dτ,i =1..., 3N − 8would fix the evolution. They do not.
264 J. BarbourFive more data are needed <strong>and</strong> must be taken from among the secondderivatives d 2 s i /dτ 2 . Moreover, no matter how large N, say a million as ina globular cluster, we always need just five. 5 They make no sense from theshape-dynamic perspective. Poincaré, writing as a philosopher deeply committedto relationalism, found the need for them repugnant [14, 15]. But, inthe face of the manifest presence of angular momentum in the solar system,he resigned himself to the fact that there is more to dynamics than, literally,meets the eye. 6 In fact, the extra d 2 s i /dτ 2 ’s are explained by Newton’s assumptionof an all-controlling but invisible frame for dynamics. They are theevidence, <strong>and</strong> the sole evidence at that, for absolute space.For some reason, Poincaré did not consider Mach’s suggestion [1, 2] thatthe universe in its totality might somehow determine the structure of thedynamics observed locally. Indeed, the universe exhibits evidence for angularmomentum in innumerable localized systems but none overall. This suggeststhat, regarded as a closed dynamical system, it has no angular momentum<strong>and</strong> meets the Poincaré principle: either a point <strong>and</strong> direction (strong form)or a point <strong>and</strong> a tangent vector (weak form) in the universe’s shape spacedetermine its evolution. The stronger form of the principle will hold if theuniverse satisfies a geodesic principle in Q N ss, since a point <strong>and</strong> a direction arethe initial conditions for a geodesic. The need for the two options, either ofwhich may serve as the definition of Mach’s principle [16], will be clarified inthe next section.To summarize: on the basis of Poincaré’s analysis <strong>and</strong> intuition, wewould like the universe to satisfy Laplacian determinism in its shape space<strong>and</strong> not merely in a special frame of reference that employs kinematic structurenot present in shape space.3. The universal structure of shape dynamics3.1. The elimination of timeIn st<strong>and</strong>ard dynamical theory, the time t is an independent variable suppliedby an external clock. But any clock is a mechanical system. If we wish totreat the universe as a single system, the issue of what clock, if any, to usebecomes critical. In fact, it is not necessary to use any clock.This can be demonstrated already in Q N . We simply proceed withouta clock. Histories of the system are then just curves in Q N , <strong>and</strong> we seek alaw that determines them. An obvious possibility is to define a metric on Q N<strong>and</strong> require histories to be geodesics with respect to it.5 If N = 3 or 4, there are insufficient d 2 s i /dτ 2 ’s <strong>and</strong> we need higher derivatives too.6 Poincaré’s penetrating analysis, on which this subsection is based, only takes into accountthe role of angular momentum in the ‘failure’ of Newtonian dynamics when expressed inrelational quantities. Despite its precision <strong>and</strong> clarity, it has been almost totally ignoredin the discussion of the absolute vs relative debate in dynamics.
Shape Dynamics. An Introduction 265A metric is readily found because the Euclidean geometry of space thatdefines Q N in the first place also defines a natural metric on Q N :√ ∑ds kin =am a2 dx a · dx a . (3)This is called the kinetic metric [17]; division of dx a by an external dt transformsthe radic<strong>and</strong> into the Newtonian kinetic energy. We may call (3) asupermetric. We shall see how it enables us to exploit structure defined atthe level of Q N at the shape-space level.We can generate further such supermetrics from (3) by multiplying itsradic<strong>and</strong> by a function on Q N , for example ∑ a
266 J. Barboursignal is generated using (7). The reason is important. Suppose we used onlythe numerators in (7) to measure time; then subsystems without interactionswould generate time signals that march in step, but with interactions onesystem may be sinking into its potential well as another is rising out of its.Then the ‘time’ generated by the former will pass faster than the latter’s.However the denominators in (7) correct this automatically since E − Vincreases or decreases with T . Time must be measured by some motion, butfor generic systems only the time label that ensures conservation of the energycan meet the marching-in-step criterion. Duration is defined as uniquely asentropy is through the logarithm of probability.In textbooks, (4) is derived as Jacobi’s principle [17] <strong>and</strong> used to determinethe dynamical orbit of systems in Q N (as, for example, a planet’sorbit, which is not to be confused with a group orbit). The speed in orbitis then determined from (6) regarded as the energy theorem. The derivationabove provides the deeper interpretation of (6) in a closed system. It is thedefinition of time. Note that time is eliminated from the initial kinematicsby a square root in the Lagrangian. This pattern will be repeated in more refinedrelational settings below, in which we can address the question of whatpotentials V are allowed in relational dynamics.A final comment. Time has always appeared elusive. It is representedin dynamics as the real line R 1 . Instants are mere points on the line, eachidentical to the other. This violates the principle that things can be distinguishedonly by differences. There must be variety. In relational dynamics R 1is redundant <strong>and</strong> there are only configurations, but they double as instantsof time. The need for variety is met.3.2. Best matchingThe next step is to determine curves in shape space Q N ss that satisfy thestrong or weak form of the Poincaré principle. As already noted, the strongform, with which we begin, will be satisfied by geodesics with respect toa metric defined on Q N ss. For this, given two nearly identical shapes, s 1 ,s 2 ,i.e., neighbouring points in Q N ss, we need to define a ‘distance’ between thembased on their difference <strong>and</strong> nothing else. Once again we use the Euclideangeometry that underlies both Q N <strong>and</strong> Q N ss.Shape s 1 in Q N ss has infinitely many representations in Q N : all of thepoints on its group orbit in Q N . Pick one with coordinates x 1 a.Pickanearbypoint on the orbit of s 2 with coordinates x 2 a. In Newtonian dynamics, thecoordinate differences dx a = x 2 a −x 1 a are physical displacements, but in shapedynamics they mix physical difference of shape with spurious difference dueto the arbitrary positioning of s 1 <strong>and</strong> s 2 on their orbits. To obtain a measureof the shape difference, hold s 1 fixed in Q N <strong>and</strong> move s 2 around in its orbit, 7for the moment using only Euclidean translations <strong>and</strong> rotations. This changes7 Recall that a group orbit is generically a multi-dimensional space.
Shape Dynamics. An Introduction 267dx a = x 2 a − x 1 a <strong>and</strong> simultaneously√ds trial := (E − V ) ∑ am a2 dx a · dx a . (8)Since (8) is positive definite <strong>and</strong> defines a nonsingular metric on Q N ,it will be possible to move shape s 2 into the unique position in its orbit atwhich (8) is minimized (for given position of s 1 ). This unique position canbe characterized in two equivalent ways: 1) Shape s 2 has been moved tothe position in which it most closely ‘covers’ s 1 , i.e., the two shapes, whichare incongruent, have been brought as close as possible to congruence, asmeasured by (8). This is the best-matched position. 2) The 3N-dimensionalvector joining s 1 <strong>and</strong> s 2 in their orbits in Q N is orthogonal to the orbits. Thisis true in the first place for the kinetic metric, for which E − V = 1, but alsofor all choices of E − V . For each, the best-matched position is the same butthere is a different best-matched ‘distance’ between s 1 <strong>and</strong> s 2 :√ds bm := min of (E − V ) ∑ m a2 dx a · dx a between orbits (9)aBecause orthogonality of two vectors can only be established if all componentsof both vectors are known, best matching introduces a further degreeof holism into relational physics. The two ways of conceptualizing best matchingare shown in Fig. 4 for the 3-body problem in two dimensions.It is important that the orthogonal separation (9) is the same at allpoints within the orbits of either s 1 or s 2 . This is because the metric (8) onQ N is equivariant: if the same group transformations are applied to the configurationsin Q N that represent s 1 <strong>and</strong> s 2 , the value of (8) is unchanged. Indifferential-geometric terms, equivariance is present because the translation<strong>and</strong> rotation group orbits are Killing vectors of the kinetic metric in Q N .Theequivariance property only holds if E − V satisfies definite conditions, whichI have tacitly assumed so far but shall spell out soon.In fact, it is already lost if we attempt to include dilational best matchingwith respect to the kinetic metric in order to determine a ‘distance’ betweenshapes rather than only relative configurations as hitherto. For suppose werepresent two shapes by configurations of given sizes in Q N <strong>and</strong> find theirbest-matched separation d bm using Euclidean translations <strong>and</strong> rotations. Weobtain some value for d bm . If we now change the scale of one of the shapes,d bm must change because the kinetic metric has dimensions m 1/2 l <strong>and</strong> scalestoo. To correct for this in a natural way, we can divide the kinetic metric bythesquarerootofI cms , the centre-of-mass moment of inertia (2), <strong>and</strong> thenbest match to get the inter-shape distance√∑ds sbm := min of Icms−1 m a dx a · dx a between orbits (10)aAs it must be, ds sbm is dimensionless <strong>and</strong> defines a metric on Q N ss. Itispreciselysuch a metric that we need in order to implement Poincaré’s principle.
268 J. Barbouraba b cFigure 4. a) An arbitrary placing of the dashed triangle relativeto the undashed triangle; b) the best-matched placingreached by translational <strong>and</strong> rotational minimization of (8);c) the two positions of the triangle configurations on theirgroup orbits in Q N . The connecting ‘strut’ is orthogonal withrespect to the supermetric on Q N in the best-matched position.Best matching brings the centres of mass to coincidence<strong>and</strong> reduces the net rotation to zero.Terminologically, it will be convenient to call directions in Q N that lieentirely in group orbits vertical <strong>and</strong> the best-matched orthogonal directionshorizontal. Readers familiar with fibre bundles will recognize this terminology.A paper presenting best-matching theory ab initio in terms of fibre bundlesis in preparation.3.3. The best-matched action principleWe can now implement the strong Poincaré principle. We calculate in Q N ,but the reality unfolds in Q N ss . The task is this: given two shapes s a <strong>and</strong>s b in Q N ss, find the geodesic that joins them. The distance along the trialcurves between s a <strong>and</strong> s b is to be calculated using the best-matched metric(10) found in Q N <strong>and</strong> then ‘projected’ down to Q N ss. The projected metric isunique because the best-matching metric in Q N is equivariant.The action principle in Q N has the form∫δI bm =0, I bm =2 dλ √ WT bm , T bm = 1 ∑ dx bma2 dλ · dxbm adλ , (11)a
where dx bmaShape Dynamics. An Introduction 269/dλ is the limit of δx bma /δλ as dλ → 0, <strong>and</strong> the potential-type termW must be such that equivariance holds. In writing the action in this way, Ihave taken a short cut. Expressed properly [11], I bm contains the generatorsof the various group transformations, <strong>and</strong> the variation with respect to them/dλ. It is assumed in (11) that thisvariation has already been done.The action (11) is interpreted as follows. One first fixes a trial curve inQ N ss between s a <strong>and</strong> s b <strong>and</strong> represents it by a trial curve in Q N through theorbits of the shapes in the Q N ss trialcurve.TheQN trial curve must never‘run vertically’. It may run orthogonally to the orbits, <strong>and</strong> this is just whatwe want. For if it does, the δx a that connect the orbits are best matched. Itleads to the best-matched velocities dx bma, dependent only on the shape differences, that are to determinethe action.To make the trial curve in Q N orthogonal, we divide it into infinitesimalsegments between adjacent orbits 1, 2, 3, ..., m (orbits 1 <strong>and</strong> m are s a <strong>and</strong> s b ,respectively). We hold the initial point of segment 1–2 fixed <strong>and</strong> move theother end into the horizontal best-matched position on orbit 2. We then movethe original 2–3 segment into the horizontal with its end 2 coincident with theend of the adjusted 1–2 segment. We do this all the way to the s b orbit. Makingthe segment lengths tend to zero, we obtain a smooth horizontal curve.Because the Q N metric is equivariant, this curve is not unique – its initialpoint can be moved ‘vertically’ to any other point on the initial orbit; all theother points on the curve are then moved vertically by the same amount. IfM is the dimension of the best-matching group (M = 7 for the similaritygroup), we obtain an M-parameter family of horizontal best-matched curvesthat all yield a common unique value for the action along the trial curve inis these δx bmaQ N ss. This is illustrated in Fig. 5.In this way we obtain the action for all trial curves in shape spacebetween s a <strong>and</strong> s b . The best-matching construction ensures that the actiondepends only on the shapes that are explored by the trial curves <strong>and</strong> nothingelse. It remains to find which trial curve yields the shortest distance betweens a <strong>and</strong> s b . This requires us to vary the trial curve in Q N ss, which of coursechanges the associated trial curves in Q N ,which,whenbestmatched,givedifferent values for the best-matched action. When we find the (in generalunique) curve for which the shape-space action is stationary, we have foundthe solution that satisfies the strong Poincaré principle. Theories satisfyingonly the weak principle arise when the equivariance condition imposed on Win (11) is somewhat relaxed, as we shall now see.3.4. Best-matching constraints <strong>and</strong> consistencyTo obtain a definite representation in the above picture of best matching, wemust refer the initial shape s 1 to a particular Cartesian coordinate systemwith a definite choice of scale. This ‘places’ shape s 1 at some position on itsgroup orbit in Q N . If we now place the next, nearly identical shape s 2 onits orbit close to the position chosen for s 1 on its orbit but not in the bestmatchedposition, we obtain certain coordinate differences δx a = x 2 a − x 1 a.
270 J. BarbourFigure 5. The action associated with each trial curve betweenshapes s a <strong>and</strong> s b in shape space is calculated byfinding a best-matched curve in Q N that runs through thegroup orbits ‘above’ the trial curve. The best-matched curveis determined uniquely apart from ‘vertical lifting’ by thesame amount in each orbit, which does not change the bestmatchedaction I bm . The trial curve in shape space for whichI bm is extremal is the desired curve in shape space.Dividing these by a nominal δt, we obtain velocities from which, in Newtonianterms, we can calculate a total momentum P = ∑ a m aẋ a , angular momentumL = ∑ a m ax a × ẋ a <strong>and</strong> rate of change of I: I ˙ = D =2∑a m ax a · ẋ a .Wecan change their values by acting on s 2 with translations, rotations, <strong>and</strong>dilatations respectively. Indeed, it is intuitively obvious that by choosingthese group transformations appropriately we can ensure thatP =0, (12)L =0, (13)D =0. (14)It is also intuitively obvious that the fulfilment of these conditions is preciselythe indication that the best-matching position has been reached.Let us now st<strong>and</strong> back <strong>and</strong> take an overall view. The reality in shapedynamics is simply a curve in Q N ss , which we can imagine traversed in eitherdirection. There is no rate of change of shapes, just their succession. Theonly convenient way to represent this succession is in Q N . However, any onecurve in Q N ss, denote it C ss , is represented by an infinite set {CssQN } of curvesin Q N . They all pass through the orbits of the shapes in C ss , within which
Shape Dynamics. An Introduction 271the {CssQN } curves can run anywhere. Prior to the introduction of the bestmatchingdynamics, all the curves {CssQN } are equivalent representations ofC ss <strong>and</strong> no curve parametrization is privileged.Best matching changes this by singling out curves in the set {Css QN }that ‘run horizontally’. They are distinguished representations, uniquely determinedby the best matching up to a seven-parameter freedom of position inone nominally chosen initial shape in its orbit. There is also a distinguishedcurve parametrization (Sec. 3.1), uniquely fixed up to its origin <strong>and</strong> unit.When speaking of the distinguished representation, I shall henceforth meanthat the curves in Q N <strong>and</strong> their parametrization have both been chosen inthe distinguished form (modulo the residual freedoms).Let us now consider how the dynamics that actually unfolds in Q N ssis seen to unfold in the distinguished representation. From the form of theaction (11), knowing that Newton’s second law can be recovered from Jacobi’sprinciple by choosing the distinguished curve parameter using (7), we see thatwe shall recover Newton’s second law exactly. We derive not only Newton’sdynamics but also the frame <strong>and</strong> time in which it holds (Fig. 6). Thereis a further bonus, for the best-matching dynamics is more predictive: theconditions (12), (13) <strong>and</strong> (14) must hold at any initial point that we choose<strong>and</strong> be maintained subsequently. Such conditions that depend only on theinitial data (but not accelerations) <strong>and</strong> must be maintained (propagated) arecalled constraints. This is the important topic treated by Dirac [18].Since the dynamics in the distinguished representation is governed byNewton’s second law, we need to establish the conditions under which itwill propagate the constraints (12), (13) <strong>and</strong> (14). In fact, we have to imposeconditions on the potential term W in (11). If (12) is to propagate, W must bea function of the coordinate differences x a −x b ; if (13) is to propagate, W c<strong>and</strong>epend on only the inter-particle separations r ab . These are both st<strong>and</strong>ardconditions in Newtonian dynamics, in which they are usually attributed tothe homogeneity <strong>and</strong> isotropy of space. Here they ensure consistency of bestmatching wrt the Euclidean group. Propagation of (14) introduces a novelelement. It requires W to be homogeneous with length dimension l −2 .Thisrequirement is immediately obvious in (11) from the length dimension l 2 ofthe kinetic term, which the potential must balance out. Note that in this casea constant E, corresponding to a nonzero energy of the system, cannot appearin W . The system must, in Newtonian terms, have total energy zero. However,potentials with dimension l −2 are virtually never considered in Newtoni<strong>and</strong>ynamics because they do not appear to be realized in nature. 8 I shall discussthis issue in the next subsection after some general remarks.8 It is in fact possible to recover Newtonian gravitational <strong>and</strong> electrostatic forces exactlyfrom l −2 potentials by dividing the l −1 Newtonian potentials by the square root of themoment of inertia I cms . This is because I cms is dynamically conserved <strong>and</strong> is effectivelyabsorbed into the gravitational constant G <strong>and</strong> charge values. However, the presence ofI cms in the action leads to an additional force that has the form of a time-dependent‘cosmological constant’ <strong>and</strong> ensures that I cms remains constant. See [12] for details.
272 J. BarbourFigure 6. The distinguished representation of bestmatchedshape dynamics for the 3-body problem. For theinitial shape, one chooses an arbitrary position in Euclideanspace. Each successive shape is placed on its predecessor inthe best-matched position (‘horizontal stacking’). The ‘vertical’separation is chosen in accordance with the distinguishedcurve parameter t determined by the condition (7). In theframework thus created, the particles behave exactly as Newtonianparticles in an inertial frame of reference with totalmomentum <strong>and</strong> angular momentum zero.Best matching is a process that determines a metric on Q N ss . For this,three things are needed: a supermetric on Q N , best matching to find theorthogonal inter-orbit separations determined by it, <strong>and</strong> the equivarianceproperty that ensures identity of them at all positions on the orbits. Naturegives us the metric of Euclidean space, <strong>and</strong> hence the supermetric on Q N ;thesecond <strong>and</strong> third requirements arise from the desire to implement Poincarétype dynamics in Q N ss. The orbit orthogonality, leading to the constraints (12),(13), <strong>and</strong> (14), distinguishes best-matched dynamics from Newtonian theory,which imposes no such requirements. Moreover, the constraint propagationneeded for consistency of best matching enforces symmetries of the potentialthat in Newtonian theory have to be taken as facts additional to the basicstructure of the theory.It is important that the constraints (12), (13), <strong>and</strong> (14) apply only to the‘isl<strong>and</strong> universe’ of the complete N-body system. Subsystems within it thatare isolated from each other, i.e., exert negligible forces on each other, canperfectly well have nonvanishing values of P, L,D. It is merely necessary thattheir values for all of the subsystems add up to zero. However, the consistency
Shape Dynamics. An Introduction 273conditions imposed on the form of the potential must be maintained at thelevel of the subsystems.We see that shape dynamics has several advantages over Newtoni<strong>and</strong>ynamics. The two forms of dynamics have Euclidean space in common,but shape dynamics derives all of Newton’s additional kinematic structure:absolute space (inertial frame of reference), the metric of time (duration),<strong>and</strong> the symmetries of the potential. Besides these qualitative advantages,shape dynamics is more powerful: fewer initial data predict the evolution.This subsection has primarily been concerned with the problem of definingchange of position. Newton clearly understood that this requires one toknow when one can say a body is at the same place at different instantsof time. Formally at least he solved this problem by the notion of absolutespace. Best matching is the relational alternative to absolute space. Forwhenone configuration has been placed relative to another in the best-matchedposition, every position in one configuration is uniquely paired with a positionin the other. If a body is at these paired positions at the two instants,one can say it is at the same place. The two positions are equilocal. Theimageof ‘placing’ one configuration on another in the best-matched position isclearly more intuitive than the notion of inter-orbit orthogonality. It makesthe achievement of relational equilocality manifest. It is also worth notingthat the very thing that creates the problem of defining change of position –the action of the similarity group – is used to resolve it in best matching.3.5. Two forms of scale invarianceWe now return to the reasons for the failure of Laplacian determinism ofNewtonian dynamics when considered in shape space. This will explain whyit is desirable to keep open the option of the weaker form of the Poincaréprinciple. It will be helpful to consider Newtonian dynamics once more in theform of Jacobi’s principle:∫δI =0, I =2 dλ √ (E − V (q))T kin , T kin := 1 ∑ dx am a2 dλ · dx adλ . (15)aTypically V (q) is a sum of terms with different, usually integer homogeneitydegrees: gravitational <strong>and</strong> electrostatic potentials have l −1 ,harmonicoscillatorpotentials are l 2 . Moreover, since (15) is timeless <strong>and</strong> only the dimensionlessmass ratios have objective meaning, length is the sole significantdimension. Because all terms in V must have the same dimension, dimensionfulcoupling constants must appear. One can be set to unity because anoverall factor multiplying the action has no effect on its extremals. If we takeG=1, a fairly general action will haveW = E − V = E + ∑ m i m j− g i V i , (16)ri
274 J. Barbourbe expressed through an initial point <strong>and</strong> direction in Q N ss , which cannotencode dimensionful information. Thus, each such E <strong>and</strong> g i present in (16)adds a one-parameter degree of uncertainty into the evolution from an initialpoint <strong>and</strong> direction in Q N ss . If the strong Poincaré principle holds, this unpredictabilityis eliminated. There may still be several different terms in V butthey must all have the same homogeneity degree −2 <strong>and</strong> dimensionless couplingconstants; in particular, the constant E cannot be present. Note alsothat any best matching enhances predictability <strong>and</strong> eliminates potentiallyredundant structure. But other factors may count. Nature may have reasonsnot to best match with respect to all conceivable symmetries.Indeed, the foundation of particle shape dynamics on the similaritygroup precluded consideration of the larger general linear group. I suspectthat this group would leave too little structure to construct dynamics at alleasily <strong>and</strong> that angles are the irreducible minimum needed. Another factor,possibly more relevant, is the difference between velocities (<strong>and</strong> momenta),which are vectors, <strong>and</strong> directions, which are not (since multiplication of themby a number is meaningless). Vectors <strong>and</strong> vector spaces have mathematicallydesirable properties. In quantum mechanics, the vector nature of momentaensures that the momentum <strong>and</strong> configuration spaces have the same numberof dimensions, which is important for the equivalence of the position <strong>and</strong> momentumrepresentations (transformation theory). If we insist on the strongPoincaré principle, the equivalence will be lost for a closed system regarded asan isl<strong>and</strong> universe. There are then two possibilities: either equivalence is lost,<strong>and</strong> transformation theory only arises for subsystems (just as inertial framesof reference arise from shape dynamics), or the strong Poincaré principle isrelaxed just enough to maintain equivalence.There is an interesting way to do this. In the generic N-body problemthe energy E <strong>and</strong> angular momentum L, as dimensionful quantities, are notscale invariant. But they are if E = L = 0. Then the behaviour is scaleinvariant. Further [12], there is a famous qualitative result in the N-bodyproblem, first proved by Lagrange, which is that Ï>0ifE ≥ 0. Then thecurve of I as a function of time is concave upwards <strong>and</strong> its time derivative,which is 2D (defined just before (14)), is strictly monotonic, increasing from−∞ to ∞ (if the evolution is taken nominally to begin at D = −∞).Now suppose that, as I conjecture, in its classical limit the quantummechanicsoftheuniversedoesrequiretheretobevelocities(<strong>and</strong>withthemmomenta) in shape space <strong>and</strong> its geometrodynamical generalization, to whichwe come soon. Then there must at the least be a one-parameter family ofsolutions that emanate from a point <strong>and</strong> a direction in Q N ss. We will certainlywant rotational best matching to enforce L = 0. We will then have to relaxdilatational best matching in such a way that a one-parameter freedom isintroduced. In the N-body problem we can do this, without having a bestmatchingsymmetry argument that enforces it, by requiring E = 0. Thecorresponding one-parameter freedom in effect converts a direction in Q N ssinto a vector. The interesting thing is now that, by Lagrange’s result, D is
Shape Dynamics. An Introduction 275monotonic when E = 0. This means that the shape-space dynamics can bemonotonically parametrized by the dimensionless ratio D c /D 0 ,whereD 0 isan initial value of D <strong>and</strong> D c is the current value. Thus, D c /D 0 provides anobjective ‘time’ difference between shapes s 1 <strong>and</strong> s 2 . The scare quotes areused because it does not march in step with the time defined by (6).An alternative dimensionless parametrization of the shape-space curvesin this case is by means of the (not necessarily monotonic) ratio I c /I 0 .Becausethe moment of inertia measures the ‘size’ of the universe, this ratiomeasures ‘the expansion of the universe’ from an initial size to its currentsize. One might question whether in this case one should say that the dynamicsunfolds on shape space. Size still has some meaning, though not atany one instant but only as a ratio at two instants. Moreover, on shape spacethis ratio plays the role of ‘time’ or ‘independent variable’. It does not appearas a dependent dynamical variable. This is related to the cosmological puzzlethat I highlighted at the end of the introduction: from the shape-dynamicperspective, the expansion of the universe seems to be made possible by alast vestige of Newton’s absolute space. I shall return to this after presentingthe dynamics of geometry in terms of best matching.To conclude the particle dynamics, the strong form of the Poincaréprinciple does almost everything that one could ask. It cannot entirely fixthe potential term W but does require all of its terms to be homogeneous ofdegree l −2 with dimensionless coefficients, one of which can always be set tounity. If the strong Poincaré principle fails, the most interesting way the weakformcanholdintheN-body problem is if E = 0. In this case a one-parameterfreedom in the shape-space initial data for given s <strong>and</strong> d is associated withthe ratio T s /T in Q N .4. Conformal geometrodynamicsAlthough limited to particle dynamics, the previous section has identified thetwo universal elements of shape dynamics: derivation of time from difference<strong>and</strong> best matching to obviate the introduction of absolute (nondynamical)structure. However, nothing can come of nothing. The bedrock on whichdynamics has been derived is the geometrical structure of individual configurationsof the universe. We began with configurations in Euclidean space <strong>and</strong>removed from them more <strong>and</strong> more structure by group quotienting. We leftopen the question of how far such quotienting should be taken, noting thatnature must decide that. In this section, we shall see that, with two significantadditions, the two basic principles of shape dynamics can be directlyapplied to the dynamics of geometry, or geometrodynamics. This will leadto a novel derivation of, first, general relativity, then special relativity (<strong>and</strong>gauge theory) <strong>and</strong> after that to the remarkable possibility that gravitationaltheory introduces a dynamical st<strong>and</strong>ard of rest in a closed universe.
276 J. BarbourIn this connection, let me address a likely worry of the reader, anticipatedin footnote 2, about the fundamental role given to instantaneous configurationsof the universe. Does this not flagrantly contradict the relativityof simultaneity, which is confirmed by countless experiments? In response, letme mention some possibly relevant facts.When Einstein <strong>and</strong> Minkowski created special relativity, they did notask how it is that inertial frames of reference come into existence. They tookthem as given. Even when creating general relativity, Einstein did not directlyaddress the origin of local inertial frames of reference. Moreover, althoughhe gave a definition of simultaneity at spatially separated points, he neverasked how temporally separated durations are to be compared. What does itmean to say that a second today is the same as a second yesterday? Shapedynamics directly addresses both of these omissions of Einstein, to whichmay be added his adoption of length as fundamental, which Weyl questionedin 1918 [19, 20]. Finally, it is a pure historical accident that Einstein, as hehimself said, created general relativity so early, a decade before quantummechanics was discovered. Now it is an architectonic feature of quantummechanics that the Schrödinger wave function is defined on configurationspace, not (much to Einstein’s dismay) on spacetime.This all suggests that instantaneous spatial configurations of the universecould at the least be considered as the building blocks of gravitationaltheory. Indeed, they are in the Hamiltonian dynamical form of general relativityintroduced by Dirac [21] <strong>and</strong> Arnowitt, Deser <strong>and</strong> Misner [22]. However,many relativists regard that formulation as less fundamental than Einstein’soriginal one. In contrast, I shall argue that the shape-dynamical approachmight be more fundamental <strong>and</strong> that the geometrical theory of gravity couldhave been found rather naturally using it. I ask the reader to keep an openmind.4.1. Superspace <strong>and</strong> conformal superspaceDifferential geometry begins with the idea of continuity, encapsulated in thenotion of a manifold, the rigorous definition of which takes much care. Iassume that the reader is familiar with the essentials <strong>and</strong> also with diffeomorphisms;if not, [23] is an excellent introduction. To model a closed universe,we need to consider closed manifolds. The simplest possibility thatmatches our direct experience of space is S 3 ,whichcanbepicturedasthethree-dimensional surface of a four-dimensional sphere.Now suppose that on S 3 we define a Riemannian 3-metric g ij (x). As a3×3 symmetric matrix at each space point, it can always be transformed at agiven point to diagonal form with 1, 1, 1 on the diagonal. Such a metric doesthree things. First, it defines the length ds of the line element dx i connectingneighbouring points of the manifold: ds = √ g ij dx i dx j .Thisiswellknown.However, for shape dynamics it is more important that g ij (x) determinesangles. Lettwocurvesatx be tangent to the line elements dx i <strong>and</strong> dy i <strong>and</strong>
Shape Dynamics. An Introduction 277θ be the angle between them. Thencos θ =g ij dx i dy j√gkl dx k dx l g mn dy m dy n . (17)The third thing that the metric does (implicitly) is give information aboutthe coordinates employed to express the metric relations.We see here an immediate analogy between a 3-metric <strong>and</strong> an N-bodyconfiguration of particles in Euclidean space. Coordinate information is mixedup with geometrical information, which itself comes in two different forms:distances <strong>and</strong> angles. Let us take this analogy further <strong>and</strong> introduce correspondingspaces <strong>and</strong> structure groups.Riem(S 3 ) is the (infinite-dimensional) space of all suitably continuousRiemannian 3-metrics g ij on S 3 (henceforth omitted). Thus, each point inRiem is a 3-metric. However, many of these 3-metrics express identical distancerelationships on the manifold that are simply expressed by means ofdifferent coordinates, or labels. They can therefore be carried into each otherby three-dimensional diffeomorphisms without these distance relations beingchanged. They form a diffeomorphism equivalence class {g ij } diff ,<strong>and</strong>the3-diffeomorphisms form a structure group that will play a role analogous to theEuclidean group in particle dynamics. Each such equivalence class is an orbitof the 3-diffeomorphism group in Riem <strong>and</strong> is defined as a three-geometry. Allsuch 3-geometries form superspace. This is a familiar concept in geometrodynamics[24]. Less known is conformal superspace, which is obtained fromsuperspace by the further quotienting by conformal transformations:g ij (x) → φ(x) 4 g ij (x), φ(x) > 0. (18)Here, the fourth power of the position-dependent function φ is chosen forconvenience, since it makes the transformation of the scalar curvature Rsimple (in four dimensions, the corresponding power is 2); the condition φ>0is imposed to stop the metric being transformed to the zero matrix.The transformations (18) change the distance relations on the manifoldbut not the angles between curves. Moreover, distances are not directly observable.To measure an interval, we must lay a ruler adjacent to it. If theinterval <strong>and</strong> the ruler subtend the same angle at our eye, we say that theyhave the same length. This is one reason for thinking that angles are morefundamental than distances; another is that they are dimensionless. We alsohave the intuition that shape is more basic than size; we generally speak ofthe, not an, equilateral triangle. It is therefore natural to make the combinationof the group of 3-diffeomorphisms <strong>and</strong> the conformal transformations(18) the structure group of conformal geometrodynamics.Before continuing, I want to mention the subgroup of the transformations(18) that simply multiply the 3-metric by a constant C:g ij (x) → Cg ij (x), C > 0. (19)One can say that the transformations (19) either ‘change the size ofthe universe’ or change the unit of distance. Like similarity transformations,
278 J. Barbourthey leave all length ratios unchanged, <strong>and</strong> are conceptually distinct from thegeneral transformations (18), which change the ratios of the geodesic lengthsd(a, b) <strong>and</strong>d(c, d) between point pairs a, b <strong>and</strong> c, d. As a result, general conformaltransformations open up a vastly richer field for study than similaritytransformations. Another subgroup consists of the volume-preserving conformaltransformations (18). They leave the total volume V = ∫ √ gd 3 x of theuniverse unchanged. We shall see that these transformations play an importantrole in cosmology. The seemingly minor restriction of the transformations(18) to be volume preserving is the mysterious last vestige of absolute spacethat I mentioned in the introduction.The idea of geometrodynamics is nearly 150 years old. Clifford, thetranslator of Riemann’s 1854 paper on the foundations of geometry, conjecturedin 1870 that material bodies in motion might be nothing more thanregions of empty but differently curved three-dimensional space moving relativeto each other [24], p. 1202. This idea is realized in Einstein’s generalrelativity in the vacuum (matter-free) case in the geometrodynamic interpretationadvocated by Wheeler [24]. I shall briefly describe his superspace-basedpicture, before taking it further to conformal superspace.Consider a matter-free spacetime that is globally hyperbolic. This meansthat one can slice it by nowhere intersecting spacelike hypersurfaces identifiedby a monotonic time label t (Fig. 7). Each hypersurface carries a 3-geometry,which can be represented by many different 3-metrics g ij .Atanypointx onone hypersurface labelled by t one can move in spacetime orthogonally tothe t + δt hypersurface, reaching it after the proper time δτ = Nδt,whereN is called the lapse. If the time labelling is changed, N is rescaled in suchawaythatNδt is invariant. In general, the coordinates on successive 3-geometries will be chosen arbitrarily, so that the point with coordinate x onhypersurface t + δt will not lie at the point at which the normal erected atpoint x on hypersurface t pierces hypersurface t + δt. There will be a lateraldisplacement of magnitude δx i = N i δt. ThevectorN i is called the shift.The lapse <strong>and</strong> shift encode the g 00 <strong>and</strong> g 0i components respectively of the4-metric: g 00 = N i N i − N 2 , g 0i = N i .Each 3-metric g ij on the successive hypersurfaces is a point in Riem, <strong>and</strong>the one-parameter family of successive g ij ’s is represented as a curve in Riemparametrized by t. This is just one representation of the spacetime. First, onecan change the time label freely on the curve (respecting monotonicity). Thisleaves the curve in Riem unchanged <strong>and</strong> merely changes its parametrization.Second, by changing the spatial coordinates on each hypersurface one canchange the successive 3-metrics <strong>and</strong> move the curve around to a considerabledegree in Riem. However, each of these curves corresponds to one <strong>and</strong>thesamecurveinsuperspace.But,third,one<strong>and</strong>thesamespacetimecanbe sliced in many different ways because the definition of simultaneity ingeneral relativity is to a high degree arbitrary (Fig. 8). Thus, an infinity ofcurves in superspace, <strong>and</strong> an even greater infinity of curves in Riem, representthe same spacetime. In addition, they can all carry infinitely many different
Shape Dynamics. An Introduction 279Figure 7. The 3 + 1 decomposition of spacetime as explainedin the text.parametrizations by time labels. This huge freedom corresponds to the possibilityof making arbitrary four-dimensional coordinate transformations, orequivalently 4-diffeomorphisms, on spacetime.As long as one insists on the equal status of all different slicings byspacelike hypersurfaces – on slicing or foliation invariance – it is not possibleto represent the evolution of 3-geometry by a unique curve in a geometricalconfiguration space. This is the widely accepted view of virtually all relativists.Shape dynamics questions this. I shall now sketch the argument.Purely geometrically, distinguished foliations in spacetime do exist. Theflat intrinsic (two-dimensional) geometry of a sheet of paper is unchangedwhen it is rolled into a tube <strong>and</strong> acquires extrinsic curvature. Byanalogy,just as a 3-metric g ij describes intrinsic geometry, a second fundamentalform, also a 3 × 3 symmetric tensor K ij , describes extrinsic curvature. Itstrace K = g ij K ij is the mean extrinsic curvature. Aconstant-mean-curvature(CMC) hypersurface is one embedded in spacetime in such a way that K iseverywhere constant. In three-dimensional Euclidean space two-dimensionalsoap bubbles have CMC surfaces. Such surfaces are extremal <strong>and</strong> are thereforeassociated with ‘good’ mathematics. At least geometrically, they are clearlydistinguished.A complete underst<strong>and</strong>ing of the possibilities for slicing a spatiallyclosed vacuum Einsteinian spacetime, i.e., one that satisfies Einstein’s fieldequations G μν =0,byCMChypersurfacesdoesnotyetexist. 9 However, as9 There certainly exist spacetimes that satisfy Einstein’s field equations <strong>and</strong> do not admitCMC foliation. However, shape dynamics does not have to yield all solutions allowed by
280 J. BarbourSlicings of spacetimeCurves insuperspaceFigure 8. Because there is no distinguished definition ofsimultaneity in general relativity, a spacetime can be slicedin many different ways. This slicing, or foliation, freedomleads to many different representations of the spacetime bycurves in superspace. Two slicings <strong>and</strong> corresponding curvesin superspace are shown.we shall see, there exists a very effective <strong>and</strong> reliable way to generate ‘patches’of CMC-foliated Einsteinian spacetimes. In such a patch CMC-foliated spacetimeexists in an open neighbourhood either side of some CMC hypersurfacelabelled by t = 0. A noteworthy property of CMC foliations is that K, whichis necessarily a spatial constant on each hypersurface, must change monotonicallyin the spatially closed case. Moreover, K measures the rate of changeof the spatial volume V = ∫ √ g d 3 x in unit proper time. 10 In both theserespects, K is closely analogous to the quantity D (14) in particle mechanics.We recall that it is the rate of increase of the moment of inertia I, which,like V , characterizes the size of the universe.Let us now suppose that we do have a vacuum Einsteinian spacetimethat is CMC foliated either in its entirety or in some patch. On each leaf (slice)of the foliation there will be some 3-geometry <strong>and</strong> a uniquely determined conformal3-geometry, i.e., that part of the 3-geometry that relates only to anglemeasurements. We can take the successive conformal 3-geometries <strong>and</strong> plotthem as a curve in conformal superspace (CS). Having done this, we couldchange the slicing in the spacetime, obtaining a different curve of 3-geometriesin superspace. They too would have associated conformal 3-geometries, <strong>and</strong>each different curve of 3-geometries in superspace would generate a differentcurve of conformal 3-geometries in CS. According to the st<strong>and</strong>ard interpretationof general relativity, all these different curves in superspace <strong>and</strong> in CSgeneral relativity but only those relevant for the description of the universe. The ability toreproduce nature, not general relativity, is what counts.10 The value of K is only defined up to its sign.
Shape Dynamics. An Introduction 281are to be regarded as physically equivalent. I believe there are grounds to atleast question this.If we go back to Clifford’s original inspiration, note that only anglesare observable, <strong>and</strong> insist on either the strong or weak form of the Poincaréprinciple, we are led naturally to the desire to create a dynamical theoryof conformal geometry in which either a point <strong>and</strong> a direction in CS or apoint <strong>and</strong> a tangent vector in CS suffice to determine a unique evolutionin CS. This will be exactly analogous to the aim of particle shape dynamics<strong>and</strong> will be implemented in the following subsections. What makes thisshape-dynamic approach interesting is that the successions of conformal 3-geometries generated in the weak case correspond exactly to the successionsof conformal 3-geometries obtained on CMC foliated Einsteinian spacetimes.Moreover, the best matching by which the dynamic curves in CS are obtainedsimultaneously generates the complete spacetime as the distinguishedrepresentation of the conformal dynamics. Once this spacetime has been generatedin the CMC foliation, one can go over to an arbitrary foliation withinit <strong>and</strong> recover all of the familiar results of general relativity. Three distinctingredients create conformal dynamics. I shall present them one by one.4.2. The elimination of timeIt is easy (Sec. 3.1) to remove time from the kinematics of particle dynamics<strong>and</strong> recover it as a distinguished parameter from geodesic dynamics. It willhelp now to look at the structure of the canonical momenta in relationalparticle dynamics. Given a Lagrangian L(q a ,q a) ′ that depends on dynamicalvariables q a <strong>and</strong> their velocities q a ′ =dq a /dλ, the canonical momentum of q ais p a := ∂L/∂q a. ′ For the best-matched action (11),√p a := ∂Lbm W dx bma∂x ′ = m aa T bm dλ , Lbm = √ WT bm . (20)The distinguished time label t is obtained by choosing λ such that alwaysW = T bm so that the cofactor of m a dx bma /dλ is unity. The definition is holisticfor two reasons. First, the dx bma /dλ are obtained by global best matching <strong>and</strong>are therefore determined by all the changes of the relative separations of theparticles. Second, the denominator of the factor √ W/T bm is a sum overthe displacements of all the particles in the universe. This is seen explicitlyin the expression (6). The p a have a further important property: they arereparametrization invariant. If one rescales λ, λ → ¯λ(λ), the velocities q a′scale, but because velocities occur linearly in the numerator <strong>and</strong> denominatorof (20) there is no change in p a , which is in essence a direction. (It is in facta direction cosine wrt to the conformal metric obtained by multiplying thekinetic metric by W .)However, one could take the view that, at any instant, one should obtaina local measure of time derived from purely local differences. This would stillyield an holistic notion of time if the local differences were obtained by bestmatching. However, in the case of particle dynamics, a local derived timeof this kind cannot be obtained for the simple reason that particles are, by
282 J. Barbourdefinition, structureless. The situation is quite different in field dynamicsbecause fields have several components at each space point. This opens upthe possibility of a local measure of time, as I shall now show (deferring theconceptually distinct issue of best matching until later).Let the action on Riem, the configuration space in which calculationsare of necessity made, have the form∫∫I = dλL, L = d 3 x √ gWT, (21)where g = √ det g ij is introduced explicitly to make the integr<strong>and</strong> a tensordensity, the scalar W is a local functional of g ij (that is, it depends on g ij<strong>and</strong> its spatial derivatives up to some finite order, which will in fact be thesecond), <strong>and</strong> T depends quadratically on the metric velocities g ′ ij := dg ij/dλ<strong>and</strong> also quadratically on g ij . It will actually have the formT = G ijklA g′ ijg ′ kl, G ijklA= g ik g jl + Ag ij g kl . (22)Here, G ijklA,inwhichA is an as yet arbitrary constant, is a supermetric (cf(3)) <strong>and</strong> appears because one needs to construct from the velocities g ij ′ aquantity that is a scalar under 3-diffeomorphisms. Because g ij′ is, like the3-metric, a symmetric tensor, there are only two independent scalars thatone can form from it by contraction using the inverse metric. 11The key thing about (21) is that one first forms a quantity quadratic inthe velocities at each space point, takes the square root at each space point,<strong>and</strong> only then integrates over space. This is a local square root <strong>and</strong> can bejustified as follows. First, a square root must be introduced in some wayto create a theory without any external time variable. Next, there are twoways in which this can be done. The first is by direct analogy with Jacobi’sprinciple (4) or (15), <strong>and</strong> would lead to an action with ‘global’ square rootsof the form∫ ∫ ∫I = dλ√d 3 x gW√ √ d 3 x √ gT. (23)Besides being a direct generalization, the action (23) is on the face of itmathematically more correct than (21) since it defines a proper metric onRiem, which is not the case if the square root is taken, as in (21), before theintegration over space. Nevertheless, it turns out that an action of the form(21) does lead to a consistent theory. This will be shown below, but we canalready see that in such a case we obtain a theory with a local emergent time.For this, we merely need to calculate the form of the canonical momenta of11 The most general supermetric formed from g ij <strong>and</strong> acting on a general tensor has threeterms. For A = −1, we obtain the DeWitt supermetric, which will appear later. In principle,one could also consider supermetrics formed with spatial derivatives of g ij , but these wouldlead to very complicated theories. As Einstein always recommended, it is advisable to lookfirst for the simplest nontrivial realizations of an idea.
Shape Dynamics. An Introduction 283the 3-metric g ij that follow from (21):p ij := ∂L√W∂g ij′ =T Gijkl g kl. ′ (24)The similarity of the p ij to the particle canonical momenta p a (20) is obvious.First, under λ → ¯λ(λ), the momenta p ij are, like x a , unchanged. Second, thecomplex√of bare velocities G ijkl gkl ′ is multiplied by the Jacobi-type factorW/T. However, the key difference is that this factor is no longer a globalbut a position-dependent local quantity. I will not go into further detailsyet except to say that when the theory is fully worked out it leads to theappearance of a local increment of proper time given by δτ = Nδλ,whereN = √ T/4R can be identified with the lapse in general relativity.Whereas the elimination of time in Jacobi’s principle <strong>and</strong> for an actionlike (23) with global square roots is, at the classical level at least, 12 atrivial matter with no impact on the best matching (<strong>and</strong> vice versa), theelimination of external time by the local square root has a huge effect <strong>and</strong>its consequences become intimately interconnected with those of the bestmatching. Perhaps the most important effect is that it drastically reducesthe number of consistent actions that one can construct. This was first recognizedby my collaborator Niall Ó Murchadha, <strong>and</strong> its consequences wereexplored in [28], about which I shall say something after the description ofgeometrodynamic best matching.4.3. Geometrodynamic best matchingThe basic idea of geometrodynamic best matching is exactly as for particlesbut leads to a vastly richer theory because a 3-geometry, either Riemannianor conformal, is infinitely more structured than a configuration of particlesin Euclidean space. However, the core idea is the same: to ‘minimize the incongruence’of two intrinsically distinct configurations. This is done by usingthe spatial structure groups of the configurations to bring one configurationinto the position in which it most closely overlaps the other.Let us first consider 3-diffeomorphisms. If we make an infinitesimal coordinatetransformation on a given 3-metric g ij (x), obtaining new functionsof new coordinates, g ij (x) → ḡ ij (¯x),<strong>and</strong>thenconsiderḡ ij (¯x) attheoldx values,the resulting 3-metric ḡ ij (x) is what one obtains by a 3-diffeomorphismgenerated by some 3-vector field ξ i (x): ḡ ij (x) = g ij (x) +ξ (i;j) (the semicolondenotes the covariant derivative wrt to g ij <strong>and</strong> the round parenthesessymmetrization). The two 3-metrics g ij (x) <strong>and</strong>ḡ ij (x) are diffeomorphicallyrelated representations of one <strong>and</strong> the same 3-geometry. This is analogous tochanging the Cartesian coordinates of a particle configuration.Now suppose that g ij (x) +δg ij (x) represents a 3-geometry genuinelydistinct from g ij (x), i.e., δg ij cannot be represented in the form ξ (i;j) ,which12 In quantum mechanics, the effect is dramatic, since the quantization of Jacobi’s principleleads to a time-independent, <strong>and</strong> not time-dependent Schrödinger equation. This is oneaspect of the famous ‘problem of time’ in canonical quantum gravity [25, 26, 27].
284 J. Barbourwould indicate a spurious diffeomorphically-induced change. The difficultythat we now face is that, because we are considering intrinsically different3-geometries, mere identity of the coordinate values x i does not mean thatthey specify ‘the same point’ in the two different 3-geometries. In fact, theproblem has nothing to do with coordinates. Given an apple <strong>and</strong> a pear, theredoes not appear to be any way to establish a 1-to-1 pairing of all the pointson the apple’s surface with all those on the pear’s. However, best matchingdoes just that if the compared objects differ only infinitesimally. To applythe technique rigourously, one must use rates of change rather than finitedifferences.<strong>Mathematical</strong>ly, we can always specify a 3-metric g ij (x) <strong>and</strong> its velocityg ij ′ =dg ij/dλ. The problem is that g ij ′ =dg ij/dλ mixes information aboutthe intrinsic change of the described 3-geometry with arbitrary informationabout the way in which the coordinates are laid down as the 3-geometrychanges. There is an equivalence class of velocities {g ij ′ − ξ′ (i;j)} that all representthe same intrinsic change. The task of best matching is to select aunique one among them that can be said to measure the true change.We note first that there is no objection to fixing coordinates on theoriginal 3-geometry, giving g ij (x), just as we chose an initial Cartesian representationfor the particle configurations. To fix the way the coordinates arethen laid down, we consider the effect of λ-dependent diffeomorphisms on(21). It becomes∫∫I = dλL, L = d 3 x √ gWT,T = G ijklA (g′ ij − ξ (i;j) ′ )(g′ kl − ξ (k;l) ′ ). (25)The possibility of constructing consistent geometrodynamical theoriesis considered in [28], 13 to which I refer the reader for details, since I only wishto indicate what the results are.The basic theoretical structure obtained in geometrodynamics is broadlythe same as in particle dynamics. One obtains constraints <strong>and</strong> conditionsunder which they propagate consistently. These conditions strongly restrictthe set of consistent theories. I shall first identify the constraints <strong>and</strong> thenindicate how they act as ‘theory selectors’.First, there are constraints because of the local square root in (25).Before giving them, I need to draw attention to a similar constraint, or ratheridentity, in the particle model. It follows from the form (20) of the canonicalmomenta p a that∑ p a · p a≡ W. (26)2ma aThis is a square-root identity, since it follows directly from the square root inthe Lagrangian <strong>and</strong> means that the p a are in essence direction cosines. In theHamiltonian formalism, (26) becomes a constraint <strong>and</strong> is a single global relation.In contrast, the geometrodynamic action contains an infinity of square13 Some of the conclusions reached in [28] are too strong, being based on tacit simplicityassumptions that Anderson identified [29, 30]. I shall report here the most interestingresults that are obtained when the suitable caveats are made.
Shape Dynamics. An Introduction 285roots, one at each space point. Correspondingly, the canonical momenta satisfyinfinitely many identities (or Hamiltonian constraints):p ij p ij −2A3A − 1 p2 ≡ gW, p = g ij p ij . (27)Second, constraints arise through the best matching wrt diffeomorphisms.This is implemented by variation of (25) with respect to ξ i ′,treatedas a Lagrange multiplier. This also leads to a constraint at each space point:p ij ;j=0. (28)These linear constraints are closely analogous to the linear constraints (12)<strong>and</strong> (13) in particle dynamics. For the form (25) of the action, they propagateautomatically <strong>and</strong> do not lead to restrictions. This is because the action(21) was chosen in advance in a form invariant under λ-independent3-diffeomorphisms, which in turn ensured that (25) is invariant under λ-dependent 3-diffeomorphisms. Had we chosen a more general functional ofg ij <strong>and</strong> its spatial <strong>and</strong> λ derivatives, propagation of the constraints (28)would have forced us to specialize the general form to (25). This is anothermanifestation of the power of combining the structure group of the 3-metrics(the 3-diffeomorphism group) with the Poincaré requirement.There is no analogous control over the quadratic constaints (27) thatarise from the local square root in (21) <strong>and</strong> (25). As is shown in [28], the onlyaction that consistently propagates both the quadratic <strong>and</strong> linear constraintshas the form∫ ∫I BSW = dλ d 3 x √ (Λ + dR)T A=−1 , (29)where the subscript A = −1 ofT indicates that the undetermined coefficientin the supermetric is forced to take the DeWitt value. More impressive isthe drastic restriction on the possible form of the potential term W ,whichis restricted to be Λ + dR, d =0or ± 1. The action (29) is in fact theBaierlein–Sharp–Wheeler action [31], which is dynamically equivalent to theEinstein–Hilbert action for globally hyperbolic spacetimes. The only freedomis in the choice of the constant Λ, which corresponds to the cosmologicalconstant, <strong>and</strong> the three options for d. The case d = 0 yields so-called stronggravity <strong>and</strong> is analogous to pure inertial motion for particles. The case dcorresponds to a Lorentzian spacetime <strong>and</strong> hence to the st<strong>and</strong>ard form ofgeneral relativity, while d = −1 gives Euclidean general relativity.When translated into spacetime terms, 14 the constraints (27) <strong>and</strong> (28)are respectively the 00 <strong>and</strong> 0i, i =1, 2, 3, Einstein field equations G μν =0,μ,ν =0, 1, 2, 3. Whereas the particle dynamics associated with the globalEuclidean group leads to global relations, implementation of the Poincaréprinciple in geometrodynamics by the local elimination of time <strong>and</strong> bestmatching wrt local 3-diffeomorphisms leads to local constraints, the propagationof which directly determines the simplest nontrivial realization of the14 The ‘construction of spacetime’ will be described later.
286 J. Barbourwhole idea: general relativity. Of course, the immense power of local symmetryrequirements was one of the great discoveries of 20th-century physics. Itfirst became apparent with Einstein’s creation of general relativity. If shapedynamics has value, it is not so much in the locality of the symmetries as intheir choice <strong>and</strong> in the treatment of time. I shall compare the shape-dynamicapproach with Einstein’s at the end of the paper. Here I want to continuewith the results of [28].So far, we have considered pure geometrodynamics. The assumptionthat the structure of spacetime always reduces locally to the Minkowskispaceform of special relativity (a key element in Einstein’s approach) playedno role in the derivation. The manner in which special relativity arises in [28]is striking. In field theory, the essence of special relativity is a universal lightcone: all fields must have the same limiting signal propagation velocity. Nowvacuum general relativity has a ‘light cone’. What happens if we attempt,as the simplest possibility, to couple a scalar field ϕ to vacuum geometrodynamicsdescribed by the action (25)?The propagation speed of such a field, with action containing the fieldvelocities dϕ/dλ <strong>and</strong> first spatial derivatives ∂ i ϕ quadratically, is determinedby a single coefficient C, which fixes the ratio of the contributions of dϕ/dλ<strong>and</strong> ∂ i ϕ to the action. When the scalar field is added to the action (for detailssee [28]), the constraints (27) <strong>and</strong> (28) acquire additional terms, <strong>and</strong> onemust verify that the modified constraints are propagated by the equationsof motion. It is shown in [28] that propagation of the modified quadraticconstraint fixes the coefficient C to be exactly such that it shares the geometrodynamiclight cone. Otherwise, the scalar field can have a term in itsaction corresponding to a mass <strong>and</strong> other self-interactions.The effect of attempting to couple a single 3-vector field A to the geometryis even more remarkable. In this case, there are three possible terms thatcan be formed from the first spatial (necessarily covariant) derivatives of A.Each may enter in principle with an arbitrary coefficient. The requirementthat the modified quadratic constraint propagate not only fixes all three coefficientsin such a way that the 3-vector field has the same light cone as thegeometry but also imposes the requirement that the canonical momenta P ofA satisfy the constraint div P = 0. In fact, the resulting field is none otherthan the Maxwell field interacting with gravity. The constraint div P =0isthe famous Gauss constraint. This can be taken further [32]. If one attemptsto construct a theory of several 3-vector fields that interact with gravity <strong>and</strong>with each other, they have to be Yang–Mills gauge fields. Unlike the scalarfield, all the gauge fields must be massless.To conclude this subsection, let us indulge in some ‘what-might-havebeen’history. Clifford’s ‘dream’ of explaining all motion <strong>and</strong> matter in termsof dynamical Riemannian 3-geometry was in essence a proposal for a newontology of the world. The history of science shows that new, reasonablyclearly defined ontologies almost always precede major advances. A good exampleis Descartes’s formulation of the mechanical world view; it led within
Shape Dynamics. An Introduction 287a few decades to Newton’s dynamics ([7], Chaps. 8–10). Clifford died tragicallyyoung; he could have lived to interact with both Mach <strong>and</strong> Poincaré.Between them, they had the ideas <strong>and</strong> ability needed to create a relationaltheory of dynamical geometry (<strong>and</strong> other fields) along the lines describedabove. In this way, well before 1905, they could have discovered, first, generalrelativity in the form of the Baierlein–Sharp–Wheeler action (29), next specialrelativity through a universal light cone, <strong>and</strong> even, third, gauge theory.All of this could have happened as part of a programme to realize Clifford’soriginal inspiration in the simplest nontrivial way.I want to emphasize the role that the concept of time would have playedin such a scenario. In 1905, Einstein transformed physics by insisting thatthe description of motion has no meaning “unless we are quite clear as towhat we underst<strong>and</strong> by ‘time’ ” [33]. He had in mind the problem of definingsimultaneity at spatially separated points. Resolution of this issue in 1905was perhaps the single most important thing that then led on to generalrelativity. However, in 1898, Poincaré [34, 35] had noted the existence of twofundamental problems related to time: the definition of simultaneity <strong>and</strong> theolder problem of defining duration: What does it mean to say that a secondtoday is the same as a second tomorrow? Even earlier, in 1883, Mach hadsaid: “It is utterly beyond our power to measure the changes of things bytime. Quite the contrary, time is an abstraction at which we arrive by meansof the changes of things.” Both Mach <strong>and</strong> Poincaré had clearly recognizedthe need for a theory of duration along the lines of Sec. 3.1. 15There is now an intriguing fact. The structure of dynamics so far presentedin this paper has been based on two things: best matching <strong>and</strong> a theoryof duration. Both were initially realized globally, after which a local treatmentwas introduced. Moreover, entirely different schemes were used to achieve thedesired aims of a relational treatment of displacement <strong>and</strong> of duration (bestmatching <strong>and</strong> a square root in the action respectively). Remarkably, Einstein’stheory of simultaneity appeared as a consequence of these relationalinputs. In line with my comments at the end of Sec. 3.1, I believe that theconcept of duration as a measure of difference is more fundamental than thedefinition of simultaneity, so it is reassuring that Einstein’s well confirmedresults can be recovered starting from what may be deeper foundations. Inthis connection, there is another factor to consider. In the st<strong>and</strong>ard representationof general relativity, spacetime is a four-dimensional block. One is notsupposed to think that the Riemannian 3-geometry on the leaves of a 3+1foliation is more fundamental than the lapse <strong>and</strong> shift, which tell one howthe 3-geometries on the leaves ‘fit together’ (Fig. 7). The lapse is particularlyimportant: it tells you the orthogonal separation (in spacetime) between the3-geometries that are the leaves of a 3+1 foliation. However, the G 00 Einsteinfield equation enables one to solve algebraically for the lapse in terms of the15 Despite a careful search through his papers, I have been unable to find any evidence thatEinstein ever seriously considered the definition of duration. As we saw in Sec. 3.1, this isintimately related to the theory of clocks, which Einstein did grant had not been properlyincluded in general relativity. He called the omission a ‘sin’ [36].
288 J. Barbourother variables. It is precisely this step that led Baierlein, Sharp <strong>and</strong> Wheelerto the BSW action (29). It contains no lapse, but, as we have seen, is exactlythe kind of action that one would write down to implement (locally) Mach’srequirement that time (duration) be derived from differences. Thus, there isan exactly right theory of duration at the heart of general relativity, but it ishidden in the st<strong>and</strong>ard representation.However, this is not the end of the story. Quite apart from the implicationsof the two aspects of time – duration <strong>and</strong> simultaneity – for thequantum theory of the universe, there is also what Weyl [37] called the “disturbingquestion of length”: Why does nature seem to violate the principlethat size should be relative? We shall now see that a possible answer to thisquestion may add yet another twist to the theory of time.4.4. Conformal best matchingIn best matching wrt 3-diffeomorphisms, we are in effect looking at all possibleways in which all points on one 3-geometry can be mapped bijectively tothe points of an intrinsically different 3-geometry <strong>and</strong> selecting the bijectionthat extremalizes 16 the quantity chosen to measure the incongruence of thetwo. So far, we have not considered changing the local scale factor of the3-metrics in accordance with the conformal transformations (18). But giventhat only angles are directly observable, we have good grounds for supposingthat lengths should not occur as genuine dynamical degrees of freedom in thedynamics of geometry. If we best match wrt conformal transformations, onlythe angle-determining part of 3-metrics can play a dynamical role. Moreover,we have already noted (Sec. 4.1) the possibility that we might wish to bestmatch only wrt volume-preserving conformal transformations.At this point, it is helpful to recall the geometrical description of bestmatching in the particle model. It relies on a supermetric on the ‘large’ configurationspace Q N , which is foliated by the orbits of whatever group one isconsidering. Each orbit represents the intrinsic physical configuration of thesystem. Hitherto the supermetric chosen on the ‘large’ space (Q N or Riem)has been equivariant, so that the orthogonal separation ds between neighbouringorbits is the same at all points on the orbits. This made it possible tocalculate the orthogonal ds anywhere between the orbits <strong>and</strong>, knowing thatthe same value would always be obtained, project any such ds down to thephysical quotient space. This met the key aim – to define a metric on thephysical space.Now there is in principle a different way in which this aim can be met.It arises if the orthogonal separation between the orbits is not constant but16 We have to extremalize rather than minimize because the DeWitt supermetric (G ijklA=−1in (22)) is indefinite. Einsteinian gravity is unique among all known physical fields in thatits kinetic energy is not positive definite. The part associated with expansion of space –the second term in (25) – enters with the opposite sign to the part associated with thechange of the conformal part of the 3-metric, i.e., its shape.
Shape Dynamics. An Introduction 289has a unique extremum at some point between any two considered orbits. 17This unique extremal value can then be taken to define the required distanceon the physical quotient space. I shall now indicate how this possibility canbe implemented. Since the equations become rather complicated, I shall notattempt to give them in detail but merely outline what happens.We start with the BSW action (29), since our choices have already beenrestricted to it by the local square root <strong>and</strong> the diffeomorphism best matching(neither of which we wish to sacrifice, though we will set Λ = 0 for simplicity).As just anticipated, we immediately encounter a significant difference fromthe best-matching wrt to 3-diffeomorphisms, for which we noted that (21)is invariant under λ-independent diffeomorphisms. In the language of gaugetheory, (21) has a global (wrt λ) symmetry that is subsequently gauged byreplacing the bare velocity g ij′ by the corrected velocity g′ ij − ξ′ (i;j) .Itisthe global symmetry which ensures that the inter-orbit separation in Riemis everywhere constant (equivariance). In contrast to the invariance of (21)under λ-independent diffeomorphisms, there is no invariance of (21) underλ-independent conformal transformations of the form (18). The kinetic termby itself is invariant, but √ gR is not. Indeed,√√ √ gR → gφ4R − 8 ∇2 φφ . (30)It should however be stressed that when (21) is ‘conformalized’ in accordancewith (18) the resulting action is invariant under the combined gaugetypetransformationg ij → ω 4 g ij , φ → φ ω , (31)where ω = ω(x, λ) is an arbitrary function. This exactly matches the invarianceof (25) under 3-diffeomorphisms that arises because the transformationof g ij ′ is offset by a compensating transformation of the best-matching correctionξ(i;j) ′ . The only difference is that under the diffeomorphisms the velocitiesalone are transformed because of the prior choice of an action that is invariantunder λ-independent transformations, whereas (31) generates transformationsof both the dynamical variables <strong>and</strong> their velocities.We now note that if we best match (25) wrt unrestricted conformaltransformations, we run into a problem since we can make the action eversmaller by taking the value of φ ever smaller. Thus, we have no chance offinding an extremum of the action. There are two ways in which this difficultycan be resolved. The first mimics what we did in particle dynamics in orderto implement the strong Poincaré principle on shape space, namely use aLagrangian that overall has length dimension zero.In the particle model we did this by dividing the kinetic metric dsby √ I cms ,whereI cms is the cms moment of inertia. The analog of I cms in17 To the best of my knowledge, this possibility (which certainly does not occur in gaugetheory) was first considered by Ó Murchadha, who suggested it as a way to implementconformal best matching in [38].
290 J. Barbourgeometrodynamics is V , the total volume of the universe, <strong>and</strong> division ofthe Lagrangian in (21) by V 2/3 achieves the desired result. This route isexplored in [39]. It leads to a theory on conformal superspace that satisfiesthe strong Poincaré principle <strong>and</strong> is very similar to general relativity, exceptfor an epoch-dependent emergent cosmological constant. This has the effect ofenforcing V = constant, with the consequence that the theory is incapable ofexplaining the diverse cosmological phenomena that are all so well explainedby the theory of the exp<strong>and</strong>ing universe. The theory is not viable.An alternative is to satisfy the weak Poincaré principle by restrictingthe conformal transformations (18) to be such that they leave the total volumeunchanged. At the end of Sec. 4.1, I briefly described the consequences.Let me now give more details; for the full theory, see [40]. The physical spaceis initially chosen to be conformal superspace (CS), to which the space Vof possible volumes V of the universe is adjoined, giving the space CS+V.One obtains a theory that in principle yields a unique curve between any twopoints in CS+V. These two points are specified by giving two conformal geometriesc 1 <strong>and</strong> c 2 , i.e., two points in CS, <strong>and</strong> associated volumes V 1 <strong>and</strong> V 2 .However, there are two caveats. First, one cannot guarantee monotonicity ofV . This difficulty can be avoided by passing from V to its canonically conjugatevariable; in spacetime terms, this turns out to be K, the constant meancurvature of CMC hypersurfaces. Second, both V <strong>and</strong> K have dimensions<strong>and</strong> as such have no direct physical significance. Only the curves projectedfrom CS+V to CS correspond to objective reality. In fact, a two-parameterfamily of curves in CS+V projects to a single-parameter family of curves inCS labelled by the dimensionless values of V 2 /V 1 or, better, the monotonicK 2 /K 1 . 18A comparison with the st<strong>and</strong>ard variational principle for the N-bodyproblem is here helpful. In it one specifies initial <strong>and</strong> final configurations inQ N , i.e., 2 × 3N numbers, together with a time difference t 2 − t 1 .Thus,the variational problem is defined by 6N + 1 numbers. However, the initialvalue problem requires only 6N numbers: a point in Q N <strong>and</strong> the 3N numbersrequired to specify the (unconstrained) velocities at that point. In a geodesicproblem, one requires respectively 6N <strong>and</strong> 6N −1 numbers in the two differentbut essentially equivalent formulations. In the conformal theory, we thus havesomething very like a monotonic ‘time’, but it does not enter as a differencet 2 − t 1 but as the ratio K 2 /K 1 . This result seems to me highly significantbecause it shows (as just noted in the footnote) that in the shape-dynamicdescription of gravity one can interpret the local shapes of space as the truedegrees of freedom <strong>and</strong> K 2 /K 1 as an independent variable. As K 2 /K 1 varies,the shapes interact with each other. This mirrors the interaction of particlepositions in Newtonian dynamics as time, or, as we saw earlier, T/V changes.18 This fact escaped notice in [40]. Its detection led to [41], which shows that a point <strong>and</strong>a tangent vector in CS are sufficient to determine the evolution in CS. In turn, this meansthat the evolution is determined by exactly four local Hamiltonian shape degrees of freedomper space point. The paper [40] was written in the mistaken belief that one extra globaldegree of freedom, the value of V , also plays a true dynamical role.
Shape Dynamics. An Introduction 291However, the closer analogy in Newtonian dynamics is with the system’schange of the shape as D 2 /D 1 changes.The only input data in this form of the shape-dynamic conformal theoryare the initial point <strong>and</strong> tangent vector in CS. There is no trace of localinertial frames of reference, local proper time, or local proper distance. Inthe st<strong>and</strong>ard derivation of general relativity these are all presupposed inthe requirement that locally spacetime can be approximated in a sufficientlysmall region by Minkowski space. 19 In contrast, in the conformal approach,this entire structure emerges from specification of a point <strong>and</strong> direction inconformal superspace.Oneortwopointsmaybemadeinthisconnection.First,asthereadercan see in [40], the manner in which the theory selects a distinguished 3-geometry in a theory in which only conformal 3-geometry is presupposedrelies on intimate interplay of the theory’s ingredients. These are the localsquare root <strong>and</strong> the two different best matchings: wrt to diffeomorphisms<strong>and</strong> conformal transformations. Second, the construction of spacetime in aCMC foliation is fixed to the minutest detail from input that can in no waybe reduced. Expressed in terms of two infinitesimally differing conformal 3-geometries C 1 <strong>and</strong> C 2 , the outcome of the best matchings fixes the localscale factor √ g on C 1 <strong>and</strong> C 2 , making them into 3-geometries G 1 <strong>and</strong> G 2 .Thus, it takes one to a definite position in the conformal orbits. This is thebig difference from the best matching with respect to diffeomorphisms alone<strong>and</strong> what happens in the particle model <strong>and</strong> gauge theory. In these cases theposition in the orbit is not fixed. Next, the best matching procedure pairs eachpoint on G 1 with a unique point on G 2 <strong>and</strong> determines a duration betweenthem. In the spacetime that the theory ‘constructs’ the paired points areconnected by spacetime vectors orthogonal to G 1 <strong>and</strong> G 2 <strong>and</strong> with lengthsequal to definite (position-dependent) proper times. These are determined onthe basis of the expression (24) for the canonical momenta, in which W = gR.The lapse N is N = √ T/4R <strong>and</strong> the amount of proper time δτ between thepaired points is δτ = Nδλ. It is obvious that δτ is the outcome of a hugeholistic process: the two best matchings together determine not only whichpoints are to be paired but also the values at the paired points of all thequantities that occur in the expression N = √ T/4R.We can now see that there are two very different ways of interpretinggeneral relativity. In the st<strong>and</strong>ard picture, spacetime is assumed from thebeginning <strong>and</strong> it must locally have precisely the structure of Minkowski space.From the structural point of view, this is almost identical to an amalgamof Newton’s absolute space <strong>and</strong> time. This near identity is reflected in theessential identity locally of Newton’s first law <strong>and</strong> Einstein’s geodesic lawfor the motion of an idealized point particle. In both cases, it must move in19 The 4-metric g μν has 10 components, of which four correspond to coordinate freedom.If one takes the view, dictated by general covariance, that all the remaining six are equallyphysical, then the entire theory rests on Minkowski space. One merely allows it to be bent,as is captured in the ‘comma goes to semicolon’ rule.
292 J. Barboura straight line at a uniform speed. As I already mentioned, this very rigidinitial structure is barely changed by Einstein’s theory in its st<strong>and</strong>ard form. InWheeler’s aphorism [24], “Space tells matter how to move, matter tells spacehow to bend.” But what we find at the heart of this picture is Newton’s firstlaw barely changed. No explanation for the law of inertia is given: it is a – oneis tempted to say the – first principle of the theory. The wonderful structureof Einstein’s theory as he constructed it rests upon it as a pedestal. I hopethat the reader will at least see that there is another way of looking at the lawof inertia: it is not the point of departure but the destination reached after ajourney that takes into account all possible ways in which the configurationof the universe could change.This bears on the debate about reductionism vs holism. I believe that thest<strong>and</strong>ard spacetime representation of general relativity helps to maintain theplausibility of a reductionist approach. Because Minkowski’s spacetime seemsto be left essentially intact in local regions, I think many people (includingthose working in quantum field theory in external spacetimes) unconsciouslyassume that the effect of the rest of the universe can be ignored. Well, for somethings it largely can. However, I feel strongly that the creation of quantumgravity will force us to grasp the nettle. What happens locally is the outcomeof everything in the universe. We already have a strong hint of this fromthe classical theory, which shows that the ‘reassuring’ local Minkowskianframework is determined – through elliptic equations in fact – by every laststructural detail in the remotest part of the universe.4.5. Shape dynamics or general relativity?There is no question that general relativity has been a wonderful success <strong>and</strong>as yet has passed every experimental test. The fact that it predicts singularitiesis not so much a failure of the theory as an indication that quantumgravity must at some stage come into play <strong>and</strong> ‘take over’. A more seriouscriticism often made of general relativity is that its field equations G μν = T μνallow innumerable solutions that strike one as manifestly unphysical, for example,the ones containing closed timelike curves. There is a good case forseeking a way to limit the number of solutions. Perhaps the least controversialis the route chosen by Dirac [21] <strong>and</strong> Arnowitt, Deser, <strong>and</strong> Misner (ADM)[22]. The main justification for their 3+1 dynamical approach is the assumptionthat gravity can be described in the Hamiltonian framework, which isknown to be extremely effective in other branches of physics <strong>and</strong> especiallyin quantum mechanics.If a Hamiltonian framework is adopted, it then becomes especially attractiveto assume that the universe is spatially closed. This obviates the needfor arbitrary boundary conditions, <strong>and</strong>, as Einstein put it when discussingMach’s principle ([42], p. 62), “the series of causes of mechanical phenomena[is] closed”.The main difficulty in suggesting that the spacetime picture should bereplaced by the more restrictive Hamiltonian framework arises from the relativityprinciple, i.e., the denial of any distinguished definition of simultaneity.
Shape Dynamics. An Introduction 293In the ideal form of Hamiltonian theory, one seeks to have the dynamics representedby a unique curve in a phase space of true Hamiltonian degreesof freedom. This is equivalent to having a unique curve in a correspondingconfiguration space of true geometrical degrees of freedom even if mathematicaltractability means that the calculations must always be made in Riem.Dirac <strong>and</strong> ADM showed that dynamics in Riem could be interpreted in superspace,thereby reducing the six degrees of freedom per space point in a3-metric to the three in a 3-geometry. But the slicing freedom within spacetimemeans that a single spacetime still corresponds to an infinity of curves insuperspace. The Hamiltonian ideal is not achieved. The failure is tantalizing,because much evidence suggests that gravity has only two degrees of freedomper space point, hinting at a configuration space smaller than superspace.As long as relativity of simultaneity is held to be sacrosanct, there isno way forward to the Hamiltonian ideal. York <strong>and</strong> Wheeler came close tosuggesting that it was to be found in conformal superspace, but ultimatelybalked at jettisoning the relativity principle. 20 In this connection, it is worthpointing out that Einstein’s route to general relativity occurred at a particularpoint in history <strong>and</strong> things could have been approached differently. I thinkit entirely possible that Einstein’s discovery of his theory of gravitation inspacetime form could be seen as a glorious historical accident. In particular,Einstein could easily have looked differently at certain fundamental issuesrelated to the nature of space, time, <strong>and</strong> motion. Let me end this introductionto shape dynamics with some related observations on each.Space. Riemann based his generalization of Euclidean geometry on lengthas fundamental. It was only in 1918, three years after the creation of generalrelativity, that Weyl [19, 20] challenged this <strong>and</strong> identified – in a fourdimensionalcontext – angles as more fundamental. I will argue elsewherethat Weyl’s attempt to generalize general relativity to eliminate the correctlyperceived weakness of Riemann’s foundations failed because it wasnot sufficiently radical – instead of eliminating length completely from thefoundations, Weyl retained it in a less questionable form.Time. As I noted earlier, in 1898 Poincaré [34, 35] identified two equallyfundamental problems related to time: how is one to define duration <strong>and</strong> howis one to define simultaneity at spatially separated points? Einstein attackedthe second problem brilliantly but made no attempt to put a solution to thesecond into the foundations of general relativity.Motion. In the critique of Newtonian mechanics that was such a stimulusto general relativity, Mach argued that only relative velocities should20 York’s highly important work on the initial-value problem of general relativity [43, 44]is intimately related to the shape-dynamic programme <strong>and</strong> was one of its inspirations.For a discussion of the connections, see [40]. One of the arguments for the shape-dynamicapproach is that it provides a first-principles derivation of York’s method, which in itsoriginal form was found by trial <strong>and</strong> error. It may also be noted here that York’s methods,which were initially developed for the vacuum (matter-free) case, can be extended to includematter [45, 46]. This suggests that the principles of shape dynamics will extend to the casein which matter is present.
294 J. Barbouroccur in dynamics. Einstein accepted this aspiration, but did not attemptto put it directly into the foundations of general relativity, arguing that itwas impractical ([3, 5], p. 186). Instead it was necessary to use coordinatesystems <strong>and</strong> achieve Mach’s ideal by putting them all on an equal footing(general covariance).All three alternatives in approach listed above are put directly into shapedynamics. I think that this has been made adequately clear with regard tothe treatment of space <strong>and</strong> motion. I wish to conclude with a comment onthe treatment of time, which is rather more subtle.It is well known that Einstein regarded special relativity as a principlestheory like thermodynamics, which was based on human experience: heat energynever flows spontaneously from a cold to a hot body. Similarly, uniformmotion was always found to be indistinguishable – within a closed system– from rest. Einstein took this fact as the basis of relativity <strong>and</strong> never attemptedto explain effects like time dilatation at a microscopic level in theway Maxwell <strong>and</strong> Boltzmann developed the atomic statistical theory of thermodynamics.Since rods <strong>and</strong> clocks are ultimately quantum objects, I do notthinksuchaprogrammecanbeattemptedbeforewehaveabetterideaofthe basic structure of quantum gravity. However, I find it interesting <strong>and</strong>encouraging that a microscopic theory of duration is built in at a very basiclevel in shape dynamics. This is achieved in particle dynamics using Jacobi’sprinciple, which leads to a global definition of duration, <strong>and</strong> in conformaldynamics using the local-square-root action (21). I also find it striking that,as already noted, the simple device of eliminating the lapse from Einstein’sspacetime theory immediately transforms his theory from one created withoutany thought of a microscopic theory of duration into one (based on theBaierlein–Sharp–Wheeler action (29)) that has such a theory at its heart.A theory of duration was there all along. It merely had to be uncovered byremoving some of the structure that Einstein originally employed – truly acase of less is more.The effect of the local square root is remarkable. At the level of theorycreation in superspace, in which length is taken as fundamental, the localsquare root acts as an extremely powerful selector of consistent theories <strong>and</strong>,as we have seen, enforces the appearance of the slicing freedom, universalityof the light cone, <strong>and</strong> gauge fields as the simplest bosonic fields that couple todynamic geometry. As I have just noted, it also leads to a microscopic theoryof local duration (local proper time). Thus, the mere inclusion of the localsquare root goes a long way to establishing a constructive theory of specialrelativisticeffects. It is not the whole way, because quantum mechanics mustultimately explain why physically realized clocks measure the local propertime created by the local theory of duration.The effect of the local square root is even more striking when appliedin theory creation in conformal superspace. It still enforces universality ofthe light cone <strong>and</strong> the appearance of gauge fields but now does two furtherthings. First, it leads to a microscopic theory of length. For the conformal
Shape Dynamics. An Introduction 295best matching, in conjunction with the constraints that follow from the localsquare root, fixes a distinguished scale factor of the 3-metric. Second, it introducesthe distinguished CMC foliation within spacetime without changingany of the classical predictions of general relativity. It leads to a theory ofsimultaneity.Thus, the conformal approach to geometrodynamics suggests that thereare two c<strong>and</strong>idate theories of gravity that can be derived from differentfirst principles. Einstein’s general relativity is based on the idea that spacetimeis the basic ontology; its symmetry group is four-dimensional diffeomorphisminvariance. But there is also an alternative dual theory based onthree-dimensional diffeomorphism invariance <strong>and</strong> conformal best matching[9, 10, 40, 41]. The set of allowed solutions of the conformal theory is significantlysmaller than the general relativity set. In principle, this is a goodfeature, since it makes the conformal theory more predictive, but it cannot beruled out that, being tied to CMC foliations, the conformal theory will be unableto describe physically observable situations that are correctly describedby general relativity.I will end with two comments. First, shape dynamics in conformal superspaceis a new <strong>and</strong> mathematically well-defined framework of dynamics.Second, its physical applications are most likely to be in quantum gravity.AcknowledgementsThe work reported in this paper grew out of collaboration with (in chronologicalorder) Bruno Bertotti, Niall Ó Murchadha, Brendan Foster, EdwardAnderson, Bryan Kelleher, Henrique Gomes, Sean Gryb, <strong>and</strong> Tim Koslowski.Discussions over many years with Karel Kuchař <strong>and</strong> Lee Smolin were also veryhelpful, as were extended discussions with Jimmy York in 1992. I am muchindebted to them all. Thanks also to Boris Barbour for creating the figures.This work was funded by a grant from the Foundational Questions Institute(FQXi) Fund, a donor advised fund of the Silicon Valley CommunityFoundation on the basis of proposal FQXi-RFP2-08-05 to the FoundationalQuestions Institute.References[1] E Mach. Die Mechanik in ihrer Entwickelung historisch-kritisch dargestellt.1883.[2] E Mach. The Science of Mechanics. Open Court, 1960.[3] A Einstein. Prinzipielles zur allgemeinen Relativitätstheorie. Annalen derPhysik, 55:241–244, 1918.[4] A Einstein. Dialog über Einwände gegen die Relativitätstheorie. Die Naturwissenschaften,6:697–702, 1918.[5] J Barbour <strong>and</strong> H Pfister, editors. Mach’s Principle: From Newton’s Bucket to<strong>Quantum</strong> <strong>Gravity</strong>, volume 6 of Einstein Studies. Birkhäuser, Boston, 1995.[6] J Barbour. Absolute or Relative Motion? Volume 1. The Discovery of Dynamics.Cambridge University Press, 1989.
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Shape Dynamics. An Introduction 297[29] E Anderson. On the recovery of geometrodynamics from two different setsof first principles. Stud. Hist. Philos. Mod. Phys., 38:15, 2007. arXiv:grqc/0511070.[30] E Anderson. Does relationalism alone control geometrodynamics with sources?2007. arXiv:0711.0285.[31] R Baierlein, D Sharp, <strong>and</strong> J Wheeler. Three-dimensional geometry as a carrierof information about time. Phys. Rev., 126:1864–1865, 1962.[32] E Anderson <strong>and</strong> J Barbour. Interacting vector fields in relativity without relativity.Class. <strong>Quantum</strong> Grav., 19:3249–3262, 2002, gr-qc/0201092.[33] A Einstein. Zur Elektrodynamik bewegter Körper. Ann. Phys., 17:891–921,1905.[34] H Poincaré. La mesure du temps. Rev. Métaphys. Morale, 6:1, 1898.[35] H Poincaré. The measure of time. In The Value of Science. 1904.[36] A Einstein. Autobiographical notes. In P Schilpp, editor, Albert Einstein:Philosopher–Scientist. Harper <strong>and</strong> Row, New York, 1949.[37] H Weyl. Symmetry. Princeton University Press, 1952.[38] J Barbour <strong>and</strong> N Ó Murchadha. Classical <strong>and</strong> quantum gravity on conformalsuperspace. 1999, gr-qc/9911071.[39] E Anderson, J Barbour, B Z Foster, <strong>and</strong> N Ó Murchadha. Scale-invariant gravity:geometrodynamics. Class. <strong>Quantum</strong> Grav., 20:1571, 2003, gr-qc/0211022.[40] E Anderson, J Barbour, B Z Foster, B Kelleher, <strong>and</strong> N Ó Murchadha. The physicalgravitational degrees of freedom. Class. <strong>Quantum</strong> Grav., 22:1795–1802,2005, gr-qc/0407104.[41] J Barbour <strong>and</strong> N Ó Murchadha. Conformal Superspace: the configuration spaceof general relativity, arXiv:1009.3559.[42] A Einstein. The Meaning of Relativity. Methuen <strong>and</strong> Co Ltd, London, 1922.[43] J W York. Gravitational degrees of freedom <strong>and</strong> the initial-value problem.Phys. Rev. Letters, 26:1656–1658, 1971.[44] J W York. The role of conformal 3-geometry in the dynamics of gravitation.Phys. Rev. Letters, 28:1082–1085, 1972.[45] J Isenberg, N Ó Murchadha, <strong>and</strong> J W York. Initial-value problem of generalrelativity. III. Phys. Rev. D, 12:1532–1537, 1976.[46] J Isenberg <strong>and</strong> J Nester. Extension of the York field decomposition to generalgravitationally coupled fields. Ann. Phys., 108:368–386, 1977.Julian BarbourCollege Farm, South NewingtonBanbury, Oxon, OX15 4JGUKe-mail: Julian.Barbour@physics.ox.ac.ukjulian@platonia.com
On the Motion of Point Defects inRelativistic <strong>Field</strong>sMichael K.-H. KiesslingAbstract. We inquire into classical <strong>and</strong> quantum laws of motion for thepoint charge sources in the nonlinear Maxwell–Born–Infeld field equationsof classical electromagnetism in flat <strong>and</strong> curved Einstein spacetimes.Mathematics Subject Classification (2010). 78A02, 78A25, 83A05, 83C10,83C50, 83C75.Keywords. Electromagnetism, point charges, Maxwell–Born–Infeld fieldequations, laws of motion.1. IntroductionShortly after it was realized that there was a problem, quantumphysics was invented. Since then we have been doing quantumphysics, <strong>and</strong> the problem was forgotten. But it is still with us!Detlef Dürr <strong>and</strong> Sergio Albeverio, late 1980sThe “forgotten problem” that Dürr <strong>and</strong> Albeverio were talking aboutsome 20+ years ago is the construction of a consistent classical theory for thejoint evolution of electromagnetic fields <strong>and</strong> their point charge sources. Ofcourse the problem was not completely forgotten, but it certainly has becomea backwater of mainstream physics with its fundamental focus on quantumtheory: first quantum field theory <strong>and</strong> quantum gravity, then string, <strong>and</strong> inrecent years now M-theory. Unfortunately, more than a century of researchinto quantum physics has not yet produced a consistent quantum field theoryof the electromagnetic interactions without artificial irremovable mathematicalregularizers; the incorporation of the weak <strong>and</strong> strong interactions in thest<strong>and</strong>ard model has not improved on this deficiency. The consistent theoryof quantum gravity has proved even more elusive, <strong>and</strong> nobody (presumably)knows whether M-theory is ever going to see the light of the day. So it mayyet turn out that the “forgotten classical problem” will be solved first.F. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_14,© Springer Basel AG 2012299
300 M. KiesslingIn the following, I will report on some exciting recent developments towardsthe solution of the “forgotten classical problem” in general-relativisticspacetimes, in terms of the coupling of the Einstein–Maxwell–Born–Infeldtheory for an electromagnetic spacetime with point defects (caused by pointcharge sources) to a Hamilton–Jacobi theory of motion for these point defects.Mostly I will talk about the special-relativistic spacetime limit, though. SinceI want to emphasize the evolutionary aspects of the theory, I will work in aspace+time splitting of spacetime rather than using the compact formalismof spacetime geometry. Furthermore I will argue that this putative solutionto the classical problem also teaches us something new about the elusiveconsistent quantum theory of electromagnetism with point sources, <strong>and</strong> itscoupling to gravity. Namely, while the spacetime structure <strong>and</strong> the electromagneticfields will still be treated at the classical level, replacing our classicalHamilton–Jacobi law of motion for the electromagnetic point defects by a deBroglie–Bohm–Dirac quantum law of motion for the point defects yields areasonable “first quantization with spin” of the classical theory of motion ofpoint defects in the fields, which has the additional advantage that it doesn’tsuffer from the infamous measurement problem. In all of these approaches,the structure of spacetime is classical. I will have to leave comments on thepursuit of the photon <strong>and</strong> the graviton, <strong>and</strong> quantum spacetimes to a futurecontribution.I now begin by recalling the “forgotten classical problem.”2. Lorentz electrodynamics with point chargesI briefly explain why the formal equations of classical Lorentz electrodynamicswith point charges fail to yield a well-defined classical theory of electromagnetism.12.1. Maxwell’s field equationsI prepare the stage by recalling Maxwell’s field equations of electromagnetismin Minkowski spacetime, written with respect to any convenient flat foliation(a.k.a. Lorentz frame) into space points s ∈ R 3 at time t ∈ R. Supposea relativistic theory of matter has supplied an electric charge density ρ(t, s)<strong>and</strong> an electric current vector-density j(t, s), satisfying the local law of chargeconservation,∂ρ(t, s)+∇ · j(t, s) =0. (2.1)∂tMaxwell’s electromagnetic field equations comprise the two evolution equations1 ∂c1 ∂c∂t B m (t, s) =−∇ ×E m (t, s) , (2.2)∂t D m (t, s) =+∇ ×H m (t, s) − 4π 1 cj(t, s) , (2.3)1 I hope that this also dispels the perennial myth in the plasma physics literature that theseill-defined equations were “the fundamental equations of a classical plasma.”
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 301<strong>and</strong> the two constraint equations∇ ·B m (t, s) =0, (2.4)∇ ·D m (t, s) =4πρ(t, s) . (2.5)These field equations need to be supplemented by a relativistic “constitutivelaw” which expresses the electric <strong>and</strong> magnetic fields E m <strong>and</strong> H m in terms ofthe magnetic induction field B m <strong>and</strong> the electric displacement field D m .Theconstitutive law reflects the “constitution of matter” <strong>and</strong> would have to besupplied by the theory of matter carrying ρ <strong>and</strong> j. (Later we will adopt adifferent point of view.)For matter-free space Maxwell proposedH m (t, s) =B m (t, s) , (2.6)E m (t, s) =D m (t, s) . (2.7)The system of Maxwell field equations (2.2), (2.3), (2.4), (2.5) with ρ ≡ 0<strong>and</strong> j ≡ 0, supplemented by Maxwell’s “law of the pure aether” (2.6) <strong>and</strong>(2.7), will be called the Maxwell–Maxwell field equations. They feature a largenumber of conserved quantities [AnTh2005], including the field energy, thefield momentum, <strong>and</strong> the field angular momentum, given by, respectively (cf.[Abr1905], [Jac1975]),E f = 1 ∫(|Emm (t, s)| 2 + |B mm (t, s)| 2) d 3 s, (2.8)8π R 3P f = 1 ∫E mm (t, s) ×B mm (t, s)d 3 s, (2.9)4πc R 3L f = 1 ∫s × (E mm (t, s) ×B mm (t, s)) d 3 s. (2.10)4πc R 3These integrals will retain their meanings also in the presence of point sources.2.2. The Maxwell–Lorentz field equationsAlthough Maxwell pondered atomism — think of Maxwell’s velocity distribution<strong>and</strong> the Maxwell–Boltzmann equation in the kinetic theory of gases —,it seems that he did not try to implement atomistic notions of matter into hiselectromagnetic field equations. This step had to wait until the electron wasdiscovered, by Wiechert [Wie1897] <strong>and</strong> Thomson [Tho1897] (see [Pip1997]<strong>and</strong> [Jos2002]). Assuming the electron to be a point particle with charge −e,the Maxwell field equations for a single electron embedded in Maxwell’s “pureaether” at Q(t) ∈ R 3 at time t become the Maxwell–Lorentz field equations(for a single electron),1 ∂c∂t B ml (t, s) =−∇ ×E ml (t, s) , (2.11)1 ∂c ∂t E ml (t, s) =+∇ ×B ml (t, s)+4πeδ Q(t) (s) 1 ˙Q(t) c, (2.12)∇ ·B ml (t, s) =0, (2.13)∇ ·E ml (t, s) =−4πeδ Q(t) (s) , (2.14)
302 M. Kiesslingwhere “δ( · )” is Dirac’s delta function, <strong>and</strong> ˙Q(t) the velocity of the pointelectron. Note that the point charge “density” ρ(t, s) ≡−eδ Q(t) (s) <strong>and</strong>currentvector-“density” j(t, s) ≡−eδ Q(t) (s) ˙Q(t) jointly satisfy the continuityequation (2.1) in the sense of distributions; of course, the Maxwell–Lorentzfield equations have to be interpreted in the sense of distributions, too. Asis well-known, the Maxwell–Lorentz field equations are covariant under thePoincaré group; of course, the position <strong>and</strong> velocity of all the point chargesare transformed accordingly as well.Given any twice continuously differentiable, subluminal (| ˙Q(t)| < c)motion t ↦→ Q(t), the Maxwell–Lorentz field equations are solved byE retlw (t, s) =−e 1( n − ˙Q/c(1 − n · ˙Q/c) 3 γ 2 r 2 + n × [(n − ˙Q/c) × ¨Q/c 2 ])∣∣∣retr(2.15)B retlw (t, s) =n| retret×E lw (t, s) , (2.16)where n =(s − Q(t))/r with r = |s − Q(t)|, γ 2 =1/(1 −|˙Q(t)| 2 /c 2 ), <strong>and</strong>where“ret”meansthatthet-dependent functions Q(t), ˙Q(t), <strong>and</strong> ¨Q(t) areto be evaluated at the retarded time t ret defined implicitly by c(t − t ret )=|s − Q(t ret )|; see [Lié1898], [Wie1900]. By linearity, the general solution ofthe Maxwell–Lorentz equations with many point sources is now obtained byadding all their pertinent Liénard–Wiechert fields to the general solution tothe Maxwell–Maxwell equations.As the Maxwell–Lorentz field equations are consistent with any smoothsubluminal motion t ↦→ Q(t), to determine the physical motions a law ofmotion for the point electron has to be supplied. For the physicist of the late19th <strong>and</strong> early 20th centuries this meant a Newtonian law of motion.2.3. The Lorentz force2.3.1. The test charge approximation: all is well! Lorentz [Lor1904], Poincaré[Poi1905/6], <strong>and</strong> Einstein [Ein1905a] showed that at the level of the testparticle approximation, Newton’s law for the rate of change of its mechanicalmomentum, equipped with the Lorentz force [Lor1892]dP (t)[= −e E(t, Q(t)) + 1dt˙Q(t)]c×B(t, Q(t)) , (2.17)combined with the relativistic law between mechanical momentum <strong>and</strong> velocity,1 dQ(t) P (t)= √c dt m2 e c 2 + |P (t)| , (2.18)2provides an empirically highly accurate law of motion for the point electron;in (2.17), E = E mm <strong>and</strong> B = B mm are solutions of the Maxwell–Maxwellfield equations, <strong>and</strong> the m e in (2.18) is the empirical inert rest mass of thephysical electron, which is defined through this test particle law! This law oftest particle motion is globally well-posed as a Cauchy problem for the map
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 303t ↦→ (P (t), Q(t)) given any Lipschitz continuous so-called external fields E mm<strong>and</strong> B mm .2.3.2. The Lorentz self-force: infinite in all directions! Clearly, fundamentallythere is no such thing as a “test charge” in “external fields;” onlythe total fields, the solutions to the Maxwell–Lorentz field equations withthe point charge as source, can be fundamental in this theory. However,the law of motion (2.17), (2.18) is a priori undefined when formally coupledwith (2.11), (2.12), (2.13), (2.14), so that E = E ml ≡E retlw + E mm <strong>and</strong>B = B ml ≡B retlw + B mm in (2.17); indeed, inspection of (2.15), (2.16) makes itplain that “E retlw (t, Q(t))” <strong>and</strong> “Bret lw (t, Q(t))” are “infinite in all directions,”by which I mean that for any limit s → Q(t) ofE retlw (t, s) <strong>and</strong>Bret lw (t, s), thefield magnitudes diverge to infinity while their limiting directions, wheneversuch exist, depend on how the limit is taken.Lorentz <strong>and</strong> his peers interpreted the infinities to mean that the electronis not really a point particle. To uncover its structure by computing details ofthe motion which depend on its structure became the goal of “classical electrontheory” [Lor1895, Wie1896, Abr1903, Lor1904, Abr1905, Lor1915]. Thisstory is interesting in its own right, see [Roh1990, Yag1992], but would lead ustoo far astray from our pursuit of a well-defined theory of electromagnetismwith point charges. 22.3.3. Regularization <strong>and</strong> renormalization? Precisely because “E retlw (t, Q(t))”<strong>and</strong> “B retlw(t, Q(t))” are “infinite in all directions,” it is tempting to inquire intothe possibility of defining r.h.s.(2.17) for solutions of the Maxwell–Lorentzfield equations through a point limit ϱ(s|Q(t)) → δ Q(t) (s) ofaLorentzforcefield E ml (t, s)+ 1 ˙Q(t) c×B ml (t, s) averaged over some normalized regularizingdensity ϱ(s|Q(t)). Of course, for the Maxwell–Maxwell part of the Maxwell–Lorentz fields this procedure yields E mm (t, Q(t)) + 1 ˙Q(t) c×B mm (t, Q(t)). Forthe Liénard–Wiechert fields on the other h<strong>and</strong> such a definition via regularizationcannot be expected to be unique (if it leads to a finite result atall); indeed, one should expect that one can obtain any limiting averagedforce vector by choosing a suitable family of averages around the location of2 However, I do take the opportunity to advertise a little-known but very important fact. Bypostulating energy-momentum (<strong>and</strong> angular momentum) conservation of the fields alone,Abraham <strong>and</strong> Lorentz derived an effective Newtonian test particle law of motion fromtheir “purely electromagnetic models” in which both the external Lorentz force <strong>and</strong> theparticle’s inertial mass emerged through an expansion in a small parameter.Yet, in [ApKi2001] Appel <strong>and</strong> myself showed that the Abraham–Lorentz proposal ismathematically inconsistent: generically their “fundamental law of motion” does not admita solution at all! But how could Abraham <strong>and</strong> Lorentz arrive at the correct approximate lawof motion in the leading order of their expansion? The answer is: if you take an inconsistentnonlinear equation <strong>and</strong> assume that it has a solution which provides a small parameter,then formally exp<strong>and</strong> the equation <strong>and</strong> its hypothetical solution in a power series w.r.t.this parameter, then truncate the expansion <strong>and</strong> treat the retained expansion coefficientsas free parameters to be matched to empirical data, then youmayverywellendupwithan accurate equation — all the inconsistencies are hidden in the pruned-off part of theexpansion! But you haven’t derived anything, seriously.
304 M. Kiesslingthe point charge, so that such a definition of the Lorentz force is quite arbitrary.In the best case one could hope to find some physical principle whichselects a unique way of averaging the Lorentz force field <strong>and</strong> of taking thepoint limit of it. Meanwhile, in the absence of any such principle, one mayargue that anything else but taking the limit R → 0 of a uniform average ofE retlw (t, s)+ 1 ˙Q(t)×B retclw (t, s) over a sphere of radius R centered at Q(t) in theinstantaneous rest frame of the point charge, followed by a boost back intothe original Lorentz frame, would be perverse.Unfortunately, work by Dirac [Dir1938] has revealed that the pointlimit of such a “spherical averaging” does not produce a finite r.h.s.(2.17)with E retlw + E mm <strong>and</strong> B retlw + B mm in place of E <strong>and</strong> B. Instead, to leadingorder in powers of R the spherical average of the Lorentz “self-force” fieldE retlw (t, s) + 1 ˙Q(t) c×B retlw (t, s) isgivenby3 − e2 R −1 ∂ 2c 2 ∂t (γ(t) ˙Q(t)). Unless theacceleration vanishes at time t, this term diverges ↑∞in magnitude whenR ↓ 0. Incidentally, the fact that this term vanishes when the point particle isunaccelerated shows that the infinities of the Lorentz self-force in stationarymotion have been removed by spherical averaging.In addition to the divergence problems of the Lorentz self-force on apoint charge, the field energy integral (2.8) diverges for the Liénard–Wiechertfields because of their local singularity at s = Q(t) (a “classical UV divergence”).4 This confronts us also with the problem that by Einstein’s E = mc 2[Ein1905b] the electromagnetic field of a point charge should attach an infiniteinert mass to the point charge. In a sense, this is precisely what the in leadingorder divergent Lorentz self-force term “lim R↓0 − e2 R −1 ∂ (γ(t) ˙Q(t))”2c 2 ∂texpresses. But how does that fit in with the finite m e in (2.18)? There is noeasy way out of this dilemma.In [Dir1938] Dirac proposed to assign an R-dependent bare mass m b (R)to the averaging sphere of radius R, such that m b (R)+ e22cR −1 → m 2 e as R ↓ 0,where m e is the empirical rest mass of the electron (see above). It seemsthat this was the first time such a radical step was proposed, a precursorto what eventually became “renormalization theory;” cf. [GKZ1998]. Dirac’sprocedure in effect removes the divergent “self-force” from r.h.s.(2.17), withE retlw + E mm <strong>and</strong> B retlw + B mm in place of E <strong>and</strong> B, <strong>and</strong> produces the so-called3 The fact that e 2 /2R coincides with the electrostatic field energy of a surface-chargedsphere of radius R is a consequence of Newton’s theorem. The regularization of the Lorentzforce field of a point charge by “spherical averaging” should not be confused with settingup a dynamical Lorentz model of a “surface-charged sphere,” e.g. [ApKi2001], [ApKi2002].4 The field energy integral (2.8) for the Liénard–Wiechert fields diverges also because oftheir slow decay as s → ∞, but this “classical IR divergence” can be avoided by adding asuitable solution of the Maxwell–Maxwell field equations.
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 305Abraham–Lorentz–Dirac equation, viz. 5dP (t)[= −e E mm (t, Q(t)) + 1dt˙Q(t)]c×B mm (t, Q(t))[+ 2e23cI + γ 2 1 ˙Q 3 c⊗ ˙Q] [· 3γ 4 12c( ˙Q · ¨Q) ¨Q + γ 2... ]Q (t)2(2.19)coupled with (2.18); here I is the identity operator. The occurrence in (2.19)of the third time derivative of Q(t) signals that our troubles are not overyet. Viewed as a Cauchy problem, ¨Q(0) has now to be prescribed in additionto the familiar initial data Q(0) <strong>and</strong> ˙Q(0), <strong>and</strong> most choices will leadto self-accelerating run-away solutions. To eliminate these pathological solutionsamongst all solutions with the same initial data Q(0) <strong>and</strong> ˙Q(0), Dirac[Dir1938] integrated (2.19) once with the help of an integrating factor, obtainingan integro-differential equation which is of second order in the timederivative of Q(t) but now with the applied force integrated over the wholefuture of the trajectory. This implies that the remaining solutions are “preaccelerated,”6 <strong>and</strong> they can be non-unique [CDGS1995].Note that Dirac’s ad hoc procedure actually means a step in the directionof taking a renormalized point-particle limit in a family of “extendedelectron” models, though Dirac made no attempt (at this point) to come upwith a consistent dynamical model of an extended electron. If he had, hewould have found that for a family of dynamically consistent models of a“relativistically spinning Lorentz sphere” one cannot take the renormalizedpoint charge limit, cf. [ApKi2001].2.4. The effective equation of motion of L<strong>and</strong>au–LifshitzDespite the conceptual problems with its third-order derivative, the Abraham–Lorentz–Diracequation has served L<strong>and</strong>au <strong>and</strong> Lifshitz [LaLi1962] (<strong>and</strong>Peierls) as point of departure to arrive at an effective second-order equation ofmotion which is free of runaway solutions <strong>and</strong> pre-acceleration. They arguedthat whenever the familiar second-order equation of test charge motion isempirically successful, the second line at r.h.s.(2.19) should be treated asa tiny correction term to its first line. As a rule of thumb this should betrue for test particle motions with trajectories whose curvature κ is muchsmaller than m e c 2 /e 2 , the reciprocal of the “classical electron radius.” Butthen, in leading order of a formal expansion in powers of the small parameterκe 2 /m e c 2 , the third time derivative of Q(t) in (2.19) can be expressed interms of Q(t) <strong>and</strong> ˙Q(t), obtained from the equations of test particle motion5 The complicated correction term to the “external Lorentz force” at r.h.s.(2.19) is simplythe space part of the Laue [Lau1909] four-vector (2e 2 /3c 3 )(g + u ⊗ u) · ◦◦ u , divided byγ(t). Here, u is the dimensionless four-velocity of the point charge, ◦ means derivativew.r.t. “c×proper time,” <strong>and</strong> g is the metric tensor for Minkowski spacetime with signature(−, +, +, +). Note that g + u ⊗ u projects onto the subspace four-orthogonal to u.6 “It is to be hoped that some day the real solution of the problem of the charge-fieldinteraction will look different, <strong>and</strong> the equations describing nature will not be so highlyunstable that the balancing act can only succeed by having the system correctly preparedahead of time by a convenient coincidence.” Walter Thirring, p. 379 in [Thi1997].
306 M. Kiessling(2.17) <strong>and</strong> (2.18) as follows: invert (2.18) to obtain P (t) =m e γ(t) ˙Q(t), thentake the time derivative of this equation <strong>and</strong> solve for ¨Q(t) toget[¨Q(t) = 1m e γ(t)I − 1 ˙Q(t) c⊗ ˙Q(t)]·Ṗ (t). (2.20)2Now substitute the first line at r.h.s.(2.19) for Ṗ (t) at r.h.s.(2.20), then takeanother time derivative of the so rewritten equation (2.20) to obtain Q(t)...in terms of Q, ˙Q, ¨Q; for the ¨Q terms in this expression, now resubstitute...(2.20), with the first line at r.h.s.(2.19) substituted for Ṗ (t), to get Q(t)in terms of Q, ˙Q. Inserting this final expression for Q(t) ...into the secondline at r.h.s.(2.19) yields the so-called L<strong>and</strong>au–Lifshitz equation of motion ofthe physical electron, 7 coupled with (2.18). The error made by doing so isof higher order in the small expansion parameter than the retained terms. Irefrain from displaying the L<strong>and</strong>au–Lifshitz equation because its intimidatingappearance does not add anything illuminating here.Of course, the combination of Dirac’s ad hoc renormalization procedurewith the heuristic L<strong>and</strong>au–Lifshitz approximation step does not qualify asa satisfactory “derivation” of the L<strong>and</strong>au–Lifshitz equation of motion from“first principles.” A rigorous derivation from an extended particle model hasbeen given by Spohn <strong>and</strong> collaborators, cf. [Spo2004]. This derivation establishesthe status of the L<strong>and</strong>au–Lifshitz equation as an effective equationof motion for the geometrical center of a particle with structure, not thepoint particle hoped for by Dirac. Whether it will ever have a higher (say, atleast asymptotic) status for a proper theory of charged point-particle motionremains to be seen. In any event, this equation seems to yield satisfactoryresults for many practical purposes.3. Wheeler–Feynman electrodynamicsBefore I can move on to the next stage in our quest for a proper theory ofmotion of the point charge sources of the electromagnetic fields, I need tomention the radical solution to this classical problem proposed by Fokker,Schwarzschild, <strong>and</strong> Tetrode (see [Baru1964], [Spo2004]) <strong>and</strong> its amplificationby Wheeler–Feynman [WhFe1949]. This “action-at-a-distance” theory takesover from formal Lorentz electrodynamics the Liénard–Wiechert fields associatedto each point charge, but there are no additional external Maxwell–Maxwell fields. Most importantly, a point charge is not directly affected by itsown Liénard–Wiechert field. Therefore the main problem of formal Lorentz7 This tedious calculation becomes simplified in the four-vector formulation mentioned infootnote 5. The L<strong>and</strong>au–Lifshitz reasoning for the leading order ◦◦ u yields the proper timederivative of the external Lorentz–Minkowski force, divided by m e , in which in turn the externalLorentz–Minkowski force, divided by m e , is resubstituted for the u ◦ term. Projectiononto the subspace orthogonal to u <strong>and</strong> multiplication by 2e23c 3 yields the L<strong>and</strong>au–Lifshitzapproximation to the Laue four-vector. Its space part divided by γ(t) gives the pertinentapproximation to the second line at r.h.s.(2.19) in terms of Q(t) <strong>and</strong> ˙Q(t).
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 307electrodynamics, the self-interaction of a point charge with its own Liénard–Wiechert field, simply does not exist!In the Fokker–Schwarzschild–Tetrode theory, the law of motion of anelectron is given by equations (2.17), (2.18), though with E <strong>and</strong> B now st<strong>and</strong>ingfor the arithmetic means 1 ret2(E lw + E advlw )<strong>and</strong> 1 2 (Bret lw + Badv lw )oftheretarded<strong>and</strong> advanced Liénard–Wiechert fields summed over all the other pointelectrons. Nontrivial motions can occur only in the many-particle Fokker–Schwarzschild–Tetrode theory.While the Fokker–Schwarzschild–Tetrode electrodynamics does not sufferfrom the infinities of formal Lorentz electrodynamics, it raises anotherformidable problem: its second-order equations of motion for the system ofpoint charges do not pose a Cauchy problem for the traditional classical statevariables Q(t) <strong>and</strong> ˙Q(t) of these point charges. Instead, given the classicalstate variables Q(t 0 )<strong>and</strong> ˙Q(t 0 )ofeachelectronattimet 0 , the computationof their accelerations at time t 0 requires the knowledge of the states ofmotion of all point electrons at infinitely many instances in the past <strong>and</strong> inthe future. While it would be conceivable in principle, though certainly notpossible in practice, to find some historical records of all those past events,how could we anticipate the future without computing it from knowing thepresent <strong>and</strong> the past?Interestingly enough, though, Wheeler <strong>and</strong> Feynman showed that theFokker–Schwarzschild–Tetrode equations of motion can be recast as Abraham–Lorentz–Diracequations of motion for each point charge, though withthe external Maxwell–Maxwell fields replaced by the retarded Liénard–Wiechertfields of all the other point charges, provided the Wheeler–Feynmanabsorber identity is valid. While this is still not a Cauchy problem for theclassical state variables Q(t) <strong>and</strong> ˙Q(t) of each electron, at least one does notneed to anticipate the future anymore.The Fokker–Schwarzschild–Tetrode <strong>and</strong> Wheeler–Feynman theories aremathematically fascinating in their own right, but pose very difficult problems.Rigorous studies [Bau1998], [BDD2010], [Dec2010] have only recentlybegun.4. Nonlinear electromagnetic field equationsGustav Mie [Mie1912/13] worked out the special-relativistic framework forfundamentally nonlinear electromagnetic field equations without point charges(for a modern treatment, see [Chri2000]). Twenty years later Mie’s workbecame the basis for Max Born’s assault on the infinite self-energy problemof a point charge.4.1. Nonlinear self-regularizationBorn [Bor1933] argued that the dilemma of the infinite electromagnetic selfenergyof a point charge in formal Lorentz electrodynamics is caused by
308 M. KiesslingMaxwell’s “law of the pure aether,” 8 which he suspected to be valid onlyasymptotically, viz. E m ∼D m <strong>and</strong> H m ∼B m in the weak field limit. In thevicinity of a point charge, on the other h<strong>and</strong>, where the Coulombic D mlfield diverges in magnitude as 1/r 2 , nonlinear deviations of the true law ofthe “pure aether” from Maxwell’s law would become significant <strong>and</strong> ultimatelyremove the infinite self-field-energy problems. The sought-after nonlinear“aether law” has to satisfy the requirements that the resulting electromagneticfield equations derive from a Lagrangian which(P) is covariant under the Poincaré group;(W) is covariant under the Weyl (gauge) group;(M) reduces to the Maxwell–Maxwell Lagrangian in the weak field limit;(E) yields finite field-energy solutions with point charge sources.The good news is that there are “aether laws” formally satisfying criteria (P),(W), (M), (E). The bad news is that there are too many “aether laws” whichsatisfy criteria (P), (W), (M), (E) formally, so that additional requirementsare needed.Since nonlinear field equations tend to have solutions which form singularitiesin finite time (think of shock formation in compressible fluid flows),it is reasonable to look for the putatively least troublesome nonlinearity. In1970, Boillat [Boi1970], <strong>and</strong> independently Plebanski [Ple1970], discoveredthat adding to (P), (W), (M), (E) the requirement that the electromagneticfield equations(D) are linearly completely degenerate,a unique one-parameter family of field equations emerges, indeed the one proposedby Born <strong>and</strong> Infeld [BoIn1933b, BoIn1934] 9 in “one of those amusingcases of serendipity in theoretical physics” ([BiBi1983], p.37).4.2. The Maxwell–Born–Infeld field equationsThe Maxwell–Born–Infeld field equations, here for simplicity written onlyfor a single (negative) point charge, consist of Maxwell’s general field equationswiththepointchargesourcetermsρ(t,s) ≡−eδ Q(t) (s) <strong>and</strong>j(t, s) ≡−eδ Q(t) (s) ˙Q(t),1 ∂c∂t B mbi (t, s) =−∇ ×E mbi (t, s) , (4.1)1 ∂c ∂t D mbi (t, s) =+∇ ×H mbi (t, s)+4π 1 c eδ Q(t)(s) ˙Q(t) , (4.2)∇ ·B mbi (t, s) =0, (4.3)∇ ·D mbi (t, s) =−4πeδ Q(t) (s) , (4.4)8 After its demolition by Einstein, “aether” will be recycled here as shorth<strong>and</strong> for whatphysicists call “electromagnetic vacuum.”9 While the unique characterization of the Maxwell–Born–Infeld field equations in terms of(P), (W), (M), (E), (D) was apparently not known to Born <strong>and</strong> his contemporaries, thefact that these field equations satisfy, beside (P), (W), (M), (E), also (D) is mentioned inpassing already on p. 102 in [Schr1942a] as the absence of birefringence (double refraction),meaning that the speed of light [sic!] is independent of the polarization of the wave fields.The Maxwell–Lorentz equations for a point charge satisfy (P), (W), (M), (D), but not (E).
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 309together with the electromagnetic “aether law” 10 of Born–Infeld [BoIn1934],E mbi =D mbi − 1 b 2 B mbi × (B mbi ×D mbi )√1+ 1 b 2 (|B mbi | 2 + |D mbi | 2 )+ 1 b 4 |B mbi ×D mbi | 2 , (4.5)H mbi =B mbi − 1 b 2 D mbi × (D mbi ×B mbi )√1+ 1 b 2 (|B mbi | 2 + |D mbi | 2 )+ 1 b 4 |B mbi ×D mbi | 2 , (4.6)where b ∈ (0, ∞) isBorn’s field strength, a hypothetical new “constant ofnature,” which Born determined [Bor1934, BoIn1933a, BoIn1933b, BoIn1934]as follows.4.2.1. Born’s determination of b. In the absence of any charges, these sourcefreeMaxwell–Born–Infeld field equations conserve the field energy, the fieldmomentum, <strong>and</strong> the field angular momentum, given by the following integrals,respectively (cf. [BiBi1983]),E f = b24πP f = 1 ∫4πc∫L f = 14πc∫R 3 (√1+ 1 b 2 (|B sfmbi| 2 + |D sfmbi| 2 )+ 1 b 4 |B sfmbi ×D sfmbi| 2 − 1R 3 (D sfmbi ×B sf)(t, s)d 3 s,(4.7)mbi)(t, s)d 3 s, (4.8)R 3 s × (D sfmbi ×B sfmbi)(t, s)d 3 s. (4.9)Supposing that these integrals retain their meanings also in the presenceof sources, Born computed the energy of the field pair (B Born , D Born ), withB Born = 0 <strong>and</strong>( ) ( )D Born s = DCoulomb s ≡−es /|s| 3 , (4.10)which is the unique electrostatic finite-energy solution to the Maxwell–Born–Infeld equations with a single 11 negative point charge source at the originof (otherwise) empty space; see [Pry1935b], [Eck1986]. The field energy ofBorn’s solution is( ) b 2E f 0, DBorn =4π(√)1+∫R 1 b|D 2 Born (s)| 2 − 1 d 3 s = 1 6 B( 14 , √4) 1 be3 ,3where B(p, q) is Euler’s Beta function; numerical evaluation gives(4.11)16 B( 14 , 4) 1 ≈ 1.2361. (4.12)Finally, inspired by the idea of the later 19th century that the electron’sinertia a.k.a. mass has a purely electromagnetic origin, Born [Bor1934] now10 Note that Born <strong>and</strong> Infeld viewed their nonlinear relationship between E, H on one side<strong>and</strong> B, D on the other no longer as a constitutive law in the sense of Maxwell, but as theelectromagnetic law of the classical vacuum.11 The only other explicitly known electrostatic solution with point charges was found byHoppe [Hop1994], describing an infinite crystal with finite energy per charge [Gib1998].
310 M. Kiesslingargued that E f(0, DBorn)= me c 2 , thus finding for his field strength constantb Born=36B ( 14 , 4) 1 −2m 2 c 4 e −3 . (4.13)Subsequently Born <strong>and</strong> Schrödinger [BoSchr1935] argued that this valuehas to be revised because of the electron’s magnetic moment, <strong>and</strong> they cameup with a very rough, purely magnetic estimate. Born then asked MadhavaRao [Rao1936] to improve on their estimate by computing the energy of theelectromagnetic field of a charged, stationary circular current density, 12 butRao’s computation is too approximate to be definitive.We will have to come back to the determination of b at a later time.4.2.2. The status of (P), (W), (M), (E), (D). The Maxwell–Born–Infeld fieldequations formally satisfy the postulates (P), (W), (M), (E), (D), <strong>and</strong> thereis no other set of field equations which does so. However, in order to qualifyas a proper mathematical realization of (P), (W), (M), (E), (D) they need togenerically have well-behaved solutions. In this subsubsection I briefly reviewwhat is rigorously known about generic solutions.Source-free fields. For the special case of source-free fields the mentioningof point charge sources in (E) is immaterial. The finite-energy requirementremains in effect, of course.Brenier [Bre2004] has recently given a very ingenious proof that thesource-free electromagnetic field equations (4.1)–(4.6) are hyperbolic <strong>and</strong> posea Cauchy problem; see also [Ser2004]. The generic existence <strong>and</strong> uniquenessof global classical solutions realizing (P), (W), (M), (E), (D) for initial datawhich are sufficiently small in a Sobolev norm was only recently shown, 13by Speck [Spe2010a], who also extended his result to the general-relativisticsetting [Spe2010b].The restriction to small initial data is presumably vital because theworks by Serre [Ser1988] <strong>and</strong> Brenier [Bre2004] have shown that arbitrarilyregular plane wave initial data can lead to formation of a singularity in finitetime. Now, plane wave initial data trivially have an infinite energy, but inhis Ph.D. thesis Speck also showed that the relevant plane-wave initial datacan be suitably cut off to yield finite-energy initial data which still lead to asingularity in finite time. The singularity is a divergent field-energy density,not of the shock-type. However, it is still not known whether the initial datathat lead to a singularity in finite time form an open neighborhood. If theydo, then formation of a singularity in finite time is a generic phenomenon ofsource-free Maxwell–Born–Infeld field evolutions. In this case it is importantto find out how large, in terms of b, the field strengths of the initial dataare allowed to be in order to launch a unique global evolution. Paired withempirical data this information should yield valuable bounds on b.12 This corresponds to a ring singularity in both the electric <strong>and</strong> magnetic fields.13 This result had been claimed in [ChHu2003], but their proof contained a fatal error.
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 311<strong>Field</strong>s with point charge sources. Alas, hardly anything is rigorously knownabout the Maxwell–Born–Infeld field equations with one (or more) pointcharge source term(s) for generic smooth subluminal motions t ↦→ Q(t).Hopefully in the not too distant future it will be shown that this Cauchyproblem is locally well-posed, thereby realizing (P), (W), (M), (E), (D) atleast for short times. A global well-posedness result would seem unlikely,given the cited works of Serre <strong>and</strong> Brenier.Only for the special case where all point charges remain at rest thereare generic existence <strong>and</strong> uniqueness results for electrostatic solutions. By applyinga Lorentz boost these electrostatic solutions map into unique travelingelectromagnetic solutions; 14 of course, this says nothing about solutions forgeneric subluminal motions.The first generic electrostatic results were obtained for charges with sufficientlysmall magnitudes in unbounded domains with boundary [KlMi1993,Kly1995]. Recently it has been proved [Kie2011a] that a unique electrostaticsolution realizing (P), (W), (M), (E), (D) exists for arbitrary placements,signs <strong>and</strong> magnitudes of arbitrarily (though finitely) many point charges inR 3 ; the solutions have C ∞ regularity away from the point charges. 15Incidentally, the existence of such electrostatic solutions can be perplexing.Here is Gibbons (p.19 in [Gib1998]): “[W]hy don’t the particles accelerateunder the influence of the mutual forces? The reason is that they are pinnedto their fixed position ... by external forces.” A more sober assessment of thissituation will be offered in our next section.5. Classical theory of motionWe now turn to the quest for a well-defined classical law of motion for thepoint charge sources of the electromagnetic Maxwell–Born–Infeld fields. Inthe remainder of this presentation I assume that the Cauchy problem for theMaxwell–Born–Infeld field equations with point charge sources which movesmoothly at subluminal speeds is generically locally well-posed.5.1. Orthodox approaches: a critiqueIn the beginning there was an intriguing conjecture, that because of theirnonlinearity the Maxwell–Born–Infeld field equations alone would yield a (locally)well-posed Cauchy problem for both, the fields <strong>and</strong> the point charges.In a sense this idea goes back to the works by Born <strong>and</strong> Infeld [Bor1934,BoIn1934] who had argued that the law of motion is already contained in thedifferential law of field energy-momentum conservation obtained as a consequenceof the source-free equations (or, equivalently, as a local consequence of14 Incidentally, traveling electromagnetic solutions satisfying (P), (W), (M), (E), (D) cannotbe source-free if they travel at speeds less than c [Kie2011b].15 In [Gib1998] it is suggested that such a result would follow from the results on maximalhypersurfaces described in [Bart1987]. Results by Bartnik <strong>and</strong> Simon [BaSi1982] <strong>and</strong>by Bartnik [Bart1989] are indeed important ingredients to arrive at our existence <strong>and</strong>regularity result, but in themselves not sufficient to do so.
312 M. Kiesslingthe Maxwell–Born–Infeld field equations away from point sources). A relatedsentiment can also be found in the work of Einstein, Infeld, <strong>and</strong> Hoffmann[EIH1938], who seemed to also derive further inspiration from Helmholtz’extraction of the equations of point vortex motions out of Euler’s fluiddynamicalequations.However, if this conjecture were correct, then the existence <strong>and</strong> uniquenessof well-behaved electrostatic field solutions with fixed point charges, announcedin the previous section, would allow us to immediately dismiss theMaxwell–Born–Infeld field equations as unphysical. For example, considerjust two identical point charges initially at rest <strong>and</strong> far apart, <strong>and</strong> considerfield initial data identical to the pertinent electrostatic two-charge solution. Ifthe field equations alone would uniquely determine the (local) future evolutionof both, fields <strong>and</strong> point charges, they inevitably would have to producethe unique electrostatic solution with the point charges remaining at rest,whereas a physically acceptable theory must yield that these two chargesbegin to (approximately) perform a degenerate Kepler motion, moving awayfrom each other along a straight line.The upshot is: either the Maxwell–Born–Infeld field equations alone doform a complete classical theory of electromagnetism, <strong>and</strong> then this theory isunphysical, or they are incomplete as a theory of electromagnetism, in whichcase they may very well be part of a physically acceptable classical theory ofelectromagnetism. As I’ve stated at the end of the previous section, I expectthat it will be shown that the Cauchy problem for the Maxwell–Born–Infeldfield equations is locally well-posed for generic prescribed smooth subluminalmotions of their point charge sources. Assuming this to pan out, the Maxwell–Born–Infeld field equations would have to be complemented by an additionalset of dynamical equations which characterize the classical physical motionsamongst all possible ones.While we have been using electrostatic solutions in three dimensions toargue for the incompleteness of the Maxwell–Born–Infeld field equations as aclassical theory of electromagnetism, the first person, apparently, to have realizedthe flaw in the “intriguing conjecture” was Born’s son-in-law MauricePryce who, after finding analogous electrostatic solutions in two dimensions[Pry1935a], wrote ([Pry1936], p.597): “It was clear from the start of the New<strong>Field</strong> <strong>Theory</strong> (although not fully appreciated in I <strong>and</strong> II [i.e. [Bor1934] <strong>and</strong>[BoIn1934]]) that the motion of the charges was not governed by the fieldequations alone, <strong>and</strong> that some further condition had to be added.” But thenPryce continued: “It was also clear from physical considerations of conservationof energy <strong>and</strong> momentum what this condition had to be; namely, thatthe total force (...) on each charge must vanish.” His proposal is truthfulto the revival, by Born <strong>and</strong> Infeld [BoIn1933a], [BoIn1934], of the old ideathat the inertial mass of the electron has a purely electromagnetic origin,<strong>and</strong> therefore it follows Abraham’s <strong>and</strong> Lorentz’ proposal for the equation ofmotion in a purely electromagnetic model (“the total force vanishes”). However,since, as mentioned earlier, in [ApKi2001] it was shown that the purely
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 313electromagnetic Abraham–Lorentz model is overdetermined <strong>and</strong> genericallyhas no solution, we have the benefit of hindsight <strong>and</strong> should be apprehensiveof Pryce’s proposal. Yet, until [ApKi2001] nobody had found anythingmathematically suspicious in Abraham’s <strong>and</strong> Lorentz’ manipulations up tosecond order, 16 which got repeated verbatim at least until [Jac1975], so itshould come as no surprise that Pryce’s adaptation of the Abraham–Lorentzreasoning to the Born–Infeld setting was accepted by most peers. In particular,Schrödinger [Schr1942b] picked up on Pryce’s proposal <strong>and</strong> tried to pushthe approximate evaluation by including more terms in an expansion usingspherical harmonics.Interestingly, Dirac [Dir1960] used a refinement of Born’s original approach[Bor1934] (recanted in [BoIn1933b, BoIn1934]) to arrive at the Newtonianlaw of motion (2.17), (2.18), with E mbi <strong>and</strong> B mbi for, respectively, E <strong>and</strong> Bin (2.17), then went on to define the Lorentz force with the total fields throughregularization which involved manipulations of the energy-momentum-stresstensor approach used also by Pryce. Dirac also had a “non-electromagneticmass M” form e in(2.18),butremarked:“WemayhaveM =0,whichisprobably the case for an electron.”I will now first explain why the Lorentz self-force still cannot be welldefined,<strong>and</strong> why postulating field energy-momentum conservation is not goingto help, neither in the basic version as proposed first by Born <strong>and</strong> Infeld[BoIn1934] nor in the amended manner discussed by Pryce [Pry1936]. Ouranalysis will also bear on some more recent discussions, e.g. [Gib1998].I will then describe a well-defined Hamilton–Jacobi law of motion byfollowing [Kie2004a], in fact improving over the one proposed in [Kie2004a].5.2. The inadequacy of the Lorentz self-force concept5.2.1. Ambiguity of the regularized Lorentz self-force. Even if the Maxwell–Born–Infeld field equations with point charge sources have solutions realizing(P), (W), (M), (E), (D) for generic smooth subluminal motions, asconjectured to be the case, the formal expression for the Lorentz force,“E(t, Q(t)) + 1 c˙Q(t) ×B(t, Q(t)),” is still undefined a priori when E(t, s) =E mbi (t, s) <strong>and</strong>B(t, s) =B mbi (t, s) are the total electric <strong>and</strong> magnetic fields.Also, it cannot be defined at the locations of the point charge(s) by a limits → Q(t) ofE mbi (t, s) <strong>and</strong>B mbi (t, s) attimet, because these fields cannot becontinuously extended to s = Q(t). Again one can hope to define the totalLorentz force by taking a point limit ϱ(s|Q(t)) → δ Q(t) (s) of a Lorentz forcefield E mbi (t, s)+ 1 c˙Q(t)×B mbi (t, s) averaged over some normalized regularizingdensity ϱ(s|Q(t)). While in contrast to our experience with this procedurewhen applied to the solutions of the Maxwell–Lorentz field equations onemay now obtain a finite result, because the point charge’s self-energy is nowfinite, this is still not a satisfactory definition because even such a finite resultwill still depend on the specific details of the regularization ϱ <strong>and</strong> its16 Of course, the third-order term always was a source of much discussion <strong>and</strong> confusion.And as c<strong>and</strong>idates for a fundamental model of the physical electron these classical approacheswere ab<strong>and</strong>oned long ago.
314 M. Kiesslingremoval. For instance, Dirac [Dir1960] proposed that the procedure shouldbe such as to yield a vanishing contribution from the discontinuous part ofthe electromagnetic fields, but this to me seems as arbitrary as proposingthat the regularization should produce the minimal or the maximal possibleforce, or some other “distinguished” vector. Since Dirac invoked the energymomentum-stresstensor for his regularization argument, I will comment ona few more details in the next subsection. Here the upshot is: unless a compellingprinciple is found which resolves the ambiguity of the self-field contributionto the total Lorentz force on a point charge, the Lorentz force hasto be purged from the list of fundamental classical concepts.5.2.2. The postulate of local energy-momentum conservation. As I have alreadyclaimed, there exists a unique electrostatic solution of the Maxwell–Born–Infeld field equations with N point charges remaining at rest at theirinitial positions. Since the integrals of field energy <strong>and</strong> field momentum (<strong>and</strong>field angular momentum) are all conserved by any static solution of theMaxwell–Born–Infeld field equations, this shows that simply postulating theirconservation does not suffice to produce the right motions. Of course, postulatingonly the conservation of these global quantities is an infinitely muchweaker requirement than postulating detailed local balance ∂ ν Tmbi μν =0 μ everywhere,whereTmbi μν are the components of the symmetric energy(-density)-momentum(-density)-stress tensor of the electromagnetic Maxwell–Born–Infeldfield; the local law for field angular-momentum conservation follows fromthis one <strong>and</strong> needs not to be postulated separately. Note that in the absence ofpoint charges the law ∂ ν Tmbi μν =0 μ is a consequence of the field equations <strong>and</strong>not an independent postulate. Only in the presence of point charges, when∂ ν Tmbi μν =0 μ is valid a priori only away from these point charges, its continuousextension into the locations of the point charges amounts to a new postulate,indeed. Yet also this “local conservation of field energy-momentum”postulate does not deliver what it promises, <strong>and</strong> here is why.Consider the example of fields with two identical point charges (charged−e, say). The charges move smoothly at subluminal speed, <strong>and</strong> suppose thefields solve the Maxwell–Born–Infeld field equations. Introducing the electromagneticstress tensor of the Maxwell–Born–Infeld fields,(√) ]Θ mbi = 14π[E ⊗D + H⊗B−b 2 1+ 1 b(|B| 2 + |D| 2 )+ 1 2 b|B×D| 2 − 1 I ,4(5.1)with E = E mbi etc., when viewing Tmbi(t, μν s) asat-family of distributions overR 3 , the space components of ∂ ν Tmbi μν then yield the formal identity1 ∂4πc∂t (D mbi ×B mbi )(t, s) − ∇ · Θ mbi (t, s)= e[“E mbi (t, Q 1 (t))” + 1 ˙Q c 1 (t) × “B mbi (t, Q 1 (t))”]δ Q1 (t)(s)+ e[“E mbi (t, Q 2 (t))” + 1 ˙Q c 2 (t) × “B mbi (t, Q 2 (t))”]δ Q2 (t)(s),(5.2)where the quotes around the field symbols in the second line of (5.2) remindus that the electric <strong>and</strong> magnetic fields are generally ill-defined at Q 1 <strong>and</strong>
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 315Q 2 , indicating that our problem has caught up with us again. At this point,to restore complete electromagnetic energy-momentum conservation, Pryce[Pry1936] <strong>and</strong> Schrödinger [Schr1942b] rationalized that in effect one has topostulate r.h.s.(5.2)= 0. This in essence is what Pryce meant by “the totalforce on each charge must vanish.” (The law of energy conservation is dealtwith analogously.) They go on <strong>and</strong> extract from this postulate the familiarNewtonian test particle law of motion in weak applied fields, the rest massof a point charge given by its electrostatic field energy/c 2 .For Dirac, on the other h<strong>and</strong>, allowing an extra non-electromagneticmass M, r.h.s.(5.2)≠ 0 because a time-dependent extra kinetic energy associatedwith M had to be taken into account, <strong>and</strong> only total energy-momentumshould be conserved. Hence he made a different postulate to remove the ambiguityhighlighted by the quotes around the electromagnetic field at thelocations of the point charges. Eventually also Dirac obtained the familiarNewtonian test particle law of motion, but when M = 0 at the end of theday, the rest mass of the point charge became its purely electrostatic fieldenergy/c 2 ,too.To exhibit the subtle mathematical <strong>and</strong> conceptual issues in the reasoningsof Pryce, Schrödinger, <strong>and</strong> Dirac, we return (briefly) to the special casewhere the two charges are initially at rest, with electrostatic field initial data.In this particular case, by continuous <strong>and</strong> consistent extension into Q 1<strong>and</strong> Q 2 , we find that ∂ ∂t (D mbi ×B mbi )(t, s) =0 for all t <strong>and</strong> s. Nowpretendingr.h.s.(5.2) were well-defined as a vector-valued distribution, i.e. if“E mbi (0, Q 1 )” <strong>and</strong> “E mbi (0, Q 2 )” were actual vectors (note that ˙Q k = 0 now),we can use Gauss’ divergence theorem to actually compute these vectors.Thus, integrating (5.2) over any smooth, bounded, simple open domain Λcontaining, say, Q 1 but not Q 2 then yields −e“E mbi (0, Q 1 )”= ∫ ∂Λ Θ mbi · ndσ.The surface integral at the right-h<strong>and</strong> side is well-defined for any such Λ, butsince the left-h<strong>and</strong> side of this equation is independent of Λ, also the surfaceintegral must be independent of Λ. Happily it is independent of Λ (as longas Λ does not contain Q 2 ) because for the electrostatic field the distribution∇ · Θ mbi (0, s) is supported only at Q 1 <strong>and</strong> Q 2 . In the absence of anyexplicit formula for the electrostatic two-point solution one so-far relies onan approximate evaluation. Gibbons [Gib1998] suggests that for large separationsbetween the point charges the answer is Coulomb’s force formula. (In[Pry1935a] the two-dimensional electrostatic field with two point charges iscomputed exactly, <strong>and</strong> the closed line integral of the stresses around a chargeshown to yield Coulomb’s formula for distant charges.) If proven rigorouslycorrect in three dimensions, <strong>and</strong> presumably it can be shown rigorously, thiswould seem to invalidate Pryce’s (<strong>and</strong> Schrödinger’s) line of reasoning that“E mbi (0, Q 1 )” <strong>and</strong> “E mbi (0, Q 2 )” could be postulated to vanish.However, Pryce <strong>and</strong> Schrödinger were no fools. They would have pointedout that what we just explained in the previous paragraph would be an unambiguousdefinition of “E mbi (0, Q 1 )” <strong>and</strong> “E mbi (0, Q 2 )” for the electrostaticfield solution to the Maxwell–Born–Infeld field equations, because we used
316 M. Kiessling∂4πcthat for all t <strong>and</strong> s we have ∂ ∂t (D mbi ×B mbi )(t, s) =0 for the electrostaticsolution (actually, this is only needed for t ∈ (−ɛ, ɛ)). But, they would havecontinued, the electrostatic solution is unphysical <strong>and</strong> has to be ruled out,<strong>and</strong> this is precisely one of the things which postulating “E mbi (0, Q 1 )”= 0 <strong>and</strong>“E mbi (0, Q 2 )”= 0 for the physical solution accomplishes, thanks to the mathematicalresults of the previous paragraph which show that for the physicalsolution to the Maxwell–Born–Infeld field equations with electrostatic initialdata for fields <strong>and</strong> particles one cannot have ∂ ∂t (D mbi ×B mbi )(0, s) =0 for alls. This in turn implies that the point charges for the physical solution cannotremain at rest but are accelerated by the electrostatic forces so defined. Fur-1thermore, they would have insisted, since the physical∂t (D mbi ×B mbi )(t, s)has to exactly offset the distribution ∇ · Θ mbi (t, s), one obtains an equationof motion for the positions of the point charges for all times.Brilliant! But does it work? There are two issues to be addressed.First, there is the issue as to the definition of the forces on the pointcharges in general dynamical situations. Since ∫ ∂Λ kΘ mbi (t, s) · ndσ, withΛ kcontaining only Q k (t), will now generally depend on Λ k , one can at bestdefine the force on the kth charge at time t by taking the limit Λ k →{Q k (t)},provided the limit exists <strong>and</strong> is independent of the particular shapes of theshrinking Λ k . Whether this is possible is a mathematical issue, regarding thebehavior of the field solutions near the point charges that move at generic,smooth subluminal speeds. Thus, D mbi (t, s) <strong>and</strong>H mbi (t, s) mustnotdivergestronger than 1/|s−Q k (t)| 2 at each Q k (t); to get nontrivial forces, they mustin fact diverge exactly at this rate in leading order. This is an open problem,but it is not unreasonable to assume, as Pryce did, that this will be proventrue, at least for sufficiently short times.The second issue is more problematic. The distribution ∇ · Θ mbi (t, s)would, for sufficiently short times t>0 after the initial instant, be of the formf(t, s)+ ∑ k F k(t)δ Qk (t)(s), where f(t, s) is a regular force density field, whileF k (t) is the above defined force vector on the kth point charge. The field∂4πc1f(t, s) will be precisely offset by the regular part of∂t (D mbi ×B mbi )(t, s)thanks to the local conservation law of electromagnetic energy-momentumaway from the charges, as implied by the field equations alone. Thus, in orderto get an equation of motion in line with Newtonian classical physical notions,the singular (distributional) part of ∂ ∂t (D mbi ×B mbi )(t, s) atQ k (t) nowmustbe of the form δ Qk (t)(s) times a vector which depends on Q k (t) <strong>and</strong>itsfirsttwo time-derivatives — note that also the initial source terms for the fieldequations require Q k (0) <strong>and</strong> ˙Q k (0) to be prescribed, suggesting a secondorderequation of motion for the Q k (t). In particular, for the initial dataobtained from the electrostatic field solution, with charges initially at rest <strong>and</strong>very far apart, the “physical” 1 ∂4πc ∂t (D mbi ×B mbi )(0, s) has to be asymptoticto ∑ k m ¨Q e k (0)δ Qk (0)(s) as|Q 1 (0) − Q 2 (0)| →∞, if it is to reproduce thephysically correct equation of slow <strong>and</strong> gently accelerated Kepler motions oftwo physical electrons in the classical regime.
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 317I do not see how the second issue could be resolved favorably. In fact, itshould be worthwhile to try to come up with a proof that the putative equationof motion is overdetermined, in the spirit of [ApKi2001], but I haven’ttried this yet.But how could Born, Infeld, Pryce, <strong>and</strong> Schrödinger, all have convincedthemselves that this procedure will work? It is illuminating to see the answerto this question, because it will sound an alarm.Consider once again the electrostatic initial data with two identical pointcharges initially at rest <strong>and</strong> far apart. Let (R 2 =)Σ∼⊂ R 3 be the symmetryplane for this electrostatic field, <strong>and</strong> let Λ k be the open half-space containingQ k (0), k =1, 2; thus, R 3 =Λ 1 ∪Λ 2 ∪Σ. Then, by integrating their postulatedequation of motion over Λ k ,<strong>and</strong>withk + k ′ = 3, we find∫∫1 d(D mbi ×B mbi )(t, s)d 3 s4πc dt∣ = Θ mbi (0, s) · n dσΛ k t=0 ∂Λ k∼ e 2 Q k (0) − Q k ′(0)(5.3)|Q k (0) − Q k ′(0)| 3with “∼” as the distance between the charges tends →∞(the asymptoticresult is assumed to be true, <strong>and</strong> presumably rigorously provable as I’vewritten already). Pryce <strong>and</strong> his peers next argued that for gently acceleratedmotions∫we should be allowed to replace the field momentum integral14πc Λ k(D mbi ×B mbi )(t, s)d 3 s by m f (Q 1 , Q 2 )γ k (t) ˙Q k (t), where γk 2(t) =1/(1−| ˙Q∫k (t)| 2 /c 2 )<strong>and</strong>m f (Q 1 , Q 2 )c 2 (√= b24π Λ k1+b −2 |D mbi | 2 −1 ) (0, s)d 3 s,with Q( k ) st<strong>and</strong>ing for Q k (0). Lastly, we should have D mbi (0, s − Q k ) →D Born s as |Qk − Q k ′|→∞, with (the relevant) Q k at the origin, <strong>and</strong>in this sense, <strong>and</strong> with b = b Born , we would then also have m f (Q 1 , Q 2 ) → m eas |Q 1 − Q 2 |→∞. Thus, in this asymptotic regime, at the initial time, thereasoning of Pryce <strong>and</strong> his peers yields(ddt me γ k (t) ˙Q k (t) )∣ ∣t=0= e 2 Q k (0)−Q k ′ (0)|Q k (0)−Q k ′ (0)|. (5.4)3Surely this looks∫very compelling, but it is clear that the heuristic replacementof14πc Λ k(D mbi ×B mbi )(t, s)d 3 s by m f (Q 1 , Q 2 )γ k (t) ˙Q k (t) as justexplained is only a first approximation. By going one step further Schrödinger[Schr1942b] argued that the (in)famous third-order radiation reaction termwill appear, <strong>and</strong> so, if the approximation were consistent, we would now needa third initial condition, namely on ¨Q k (0). We are “back to square one” withour problems. One might argue that the third-order term is not yet the consistentapproximation <strong>and</strong> invoke the L<strong>and</strong>au–Lifshitz reasoning to get aneffective second-order equation. However, going on to higher orders wouldsuccessively bring in higher <strong>and</strong> higher derivatives of Q k (t). This looks justlike the situation in the old purely electromagnetic classical electron theoryof Abraham <strong>and</strong> Lorentz. All alarm bells should be going off by now, becausetheir equation of motion is overdetermined [ApKi2001].
318 M. KiesslingddtDirac, on the other h<strong>and</strong>, obtains as putatively exact equation of motion(Mγk (t) ˙Q k (t) ) = e k[E regmbi(t, Q k (t)) + 1 ˙Q]c k (t) ×B regmbi(t, Q k (t)) , (5.5)where the superscripts reg indicate his regularization procedure, which alsoinvolves the integration of the stress tensor over a small domain Λ k containingQ k (t), plus Gauss’ theorem, followed by the limit Λ k →{Q k (t)}. However,Dirac uses this only to split off a singular term from the ill-defined Lorentzforce, arguing that the remaining electromagnetic force field is regular; thisfield enters in (5.5). As far as I can tell, Dirac’s remainder field is generallystill singular; the critical passage is on the bottom of page 36 in [Dir1960].I end here with a comment on Dirac’s suggestion that M = 0 for anelectron. In this case, even with a regular force field, his (5.5) would beoverdetermined, because setting the coefficient of the highest derivative equalto zero amounts to a singular limit.5.2.3. On Newton’s law for the rate of change of momentum. Ihavearguedthat no matter how you cut the cake, the Lorentz force formula r.h.s.(2.17) forthe electromagnetic force on a point charge cannot be well-defined when E <strong>and</strong>B are the total fields. Since only the total fields can possibly be fundamental,while “external field” <strong>and</strong> “self-field” are only auxiliary notions, it followsthat the Lorentz force cannot play a fundamental dynamical role. This nowinevitably raises the question as to the status of Newton’s law for the rateof change of momentum, Ṗ (t) =F (t), of which (2.17) pretends to be theparticular realization in the context of classical electrodynamics.If one insists that Newton’s law Ṗ (t) = F (t) remains fundamentalthroughout classical physics, including relativistic point charge motion coupledto the electromagnetic fields, then one is obliged to continue the questfor a well-defined expression for the fundamental electromagnetic force F ona point charge.The alternative is to relegate Newton’s law Ṗ (t) =F (t) to the statusof an effective law, emerging in the regime of relativistic charged test particlemotions. In this case one needs to look elsewhere for the fundamental relativisticlaw of motion of point charges. Of course, the effective concept of theexternal Lorentz force acting on a test particle remains a beacon which anyfundamental theory must keep in sight.This is the point of view taken in [Kie2004a], <strong>and</strong> also here.5.3. Hamilton–Jacobi theory of motionAlthough we have not only ab<strong>and</strong>oned (2.17) but actually Newton’s Ṗ (t) =F (t) altogether, for now we will hold on to the second one of the two equations(2.17), (2.18) of the formal relativistic Newtonian law of motion. But then,since (2.18) expresses the rate of change of position, ˙Q(t), in terms of the“mechanical momentum” vector P (t), we need to find a new type of lawwhich gives us P (t) in a well-defined manner. Keeping in mind the moralthat “formal manipulations are not to be trusted until they can be vindicatedrigorously,” in this section we will argue that Hamilton–Jacobi theory supplies
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 319a classical law for P (t) which in fact is well-defined, provided the solutionsrealizing (P), (W), (M), (E), (D) of the Maxwell–Born–Infeld field equationswith point charge sources are as well-behaved for generic smooth subluminalmotions as they are for Born’s static solution, at least locally in time.Indeed, the electric field E Born associated to (B Born , D Born ), Born’s staticfield pair for a single (negative) point charge at the origin, is undefined at theorigin but uniformly bounded ( elsewhere; ) it exhibits a point defect at s = 0.For s ≠ 0, itisgivenbyE Born s = −∇φBorn (s), whereφ Born (s) =− √ ∫ ∞dxbe √ . (5.6)1+x4|s| √ b/eNote, φ Born (s) ∼−e|s| −1 for |s| ≫ √ e/b, <strong>and</strong> lim |s|↓0 φ Born (s) =:φ Born (0)
320 M. KiesslingThe Lorenz–Lorentz condition fixes the gauge freedom of the relativisticfour-vector potential field ( φ, A ) (t, s) to some extent, yet the equations (5.7),(5.8), <strong>and</strong> (5.9) are still invariant under the gauge transformationsφ(t, s) → φ(t, s) − 1 ∂c ∂tΥ(t, s), (5.10)A(t, s) →A(t, s)+∇Υ(t, s), (5.11)with any relativistic scalar field Υ : R 1,3 → R satisfying the wave equation1 ∂ 2c 2 ∂tΥ(t, s) =∇ 2 Υ(t, s), (5.12)2with ∇ 2 = Δ, the Laplacian on R 3 . Since a (sufficiently regular) solutionof (5.12) in R + × R 3 is uniquely determined by the initial data for Υ <strong>and</strong>its time derivative ∂ ∂tΥ, the gauge freedom that is left concerns the initialconditions of φ, A.5.3.2. Canonical momenta of point defects with intrinsic mass m. As perour (plausible but as of yet unproven) hypothesis, for generic smooth subluminalpoint charge motions the electromagnetic potential fields φ mbi <strong>and</strong>A mbi have Lipschitz continuous extensions to all of space, at any time t.In the following we shall always mean these extensions when we speak ofthe electromagnetic-field potentials. With our electromagnetic-field potentialsunambiguously defined at each location of a field point defect, we arenow able to define the so-called canonical momentum of a point defect of theelectromagnetic fields associated with a point charge moving along a smoothtrajectory t ↦→ Q(t) with subluminal speed. Namely, given a smooth trajectoryt ↦→ Q(t), consider (2.18) though now with “intrinsic inert mass m”in place of m e . Inverting (2.18) with intrinsic inert mass m we obtain the“intrinsic” momentum of the point defect,˙Q(t)P (t) =m. (5.13)√1 −|˙Q(t)| 2 /c 2Also, per our conjecture, A mbi (t, Q(t)) is well-defined at each t, <strong>and</strong>so,foranegative charge, the canonical momentumΠ(t) :=P (t) − 1 c eA mbi (t, Q(t)) (5.14)is well-defined for all t.We now turn (5.14) around <strong>and</strong>, for a negative point charge, takeP (t) =Π(t)+ 1 c eA mbi (t, Q(t)) (5.15)as the formula for P (t) that has to be coupled with (2.18). Thus, next we needto find an expression for Π(t). Precisely this is supplied by Hamilton–Jacobitheory.
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 3215.3.3. Hamilton–Jacobi laws of motion. In Hamilton–Jacobi theory of singlepointmotion one introduces the single-point configuration space of genericpositions q ∈ R 3 of the point defect. A Hamilton–Jacobi law of motion consistsof two parts: (i) an ordinary differential equation for the actual positionQ(t) of the point defect, equating its actual velocity with the evaluation — atits actual position — of a velocity field on configuration space; (ii) a partialdifferential equation for this velocity field. The correct law should reduce tothe test particle theory in certain regimes, so we begin with the latter.The test charge approximation: all well again, so far! In advanced textbookson mathematical classical physics [Thi1997] one finds the equationsof relativistic Hamilton–Jacobi theory for test charge motion in the potentialsA(t, s) =A mm (t, s) <strong>and</strong>φ(t, s) =φ mm (t, s) for the actual field solutionsE mm (t, s) <strong>and</strong>B mm (t, s) of the Maxwell–Maxwell field equations, servingas “external” fields. This reproduces the test particle motions computedfrom (2.17), (2.18) with E = E mm <strong>and</strong> B = B mm . This setup is locally welldefined,for “externally generated potentials” are independent of where <strong>and</strong>how the test charge moves, <strong>and</strong> so they can be assumed to be smooth functionsof space <strong>and</strong> time. Provided the solutions E sfmbi(t, s) <strong>and</strong>B sfmbi(t, s) ofthesource-free Maxwell–Born–Infeld field equations are smooth, the Hamilton–Jacobi law of test particle motion remains locally well-defined if we setA(t, q) =A sfmbi(t, q) <strong>and</strong>φ(t, q) =φ sfmbi(t, q), the generic-q-evaluation of thesesource-free Maxwell–Born–Infeld potentials.The relativistic Hamilton–Jacobi guiding equationSuppose the actual canonical momentum of the point test charge isgiven byΠ(t) =∇ q S hj (t, Q(t)), (5.16)where q ↦→ S hj (t, q) is a time-dependent differentiable scalar field on configurationspace. By virtue of (5.16), (5.13), (5.15), with A mbi (t, Q(t)) replacedby A sfmbi(t, Q(t)), we can eliminate Π(t) in favor of ∇ q S hj (t, Q(t)) which, for anegative test charge of mass m, yields the relativistic Hamilton–Jacobi guidingequation1 dQ(t)=c dt∇ q S hj (t, Q(t)) + 1 c eAsf mbi(t, Q(t))√m 2 c 2 + ∣ ∣∇ q S hj (t, Q(t)) + 1 c eAsf mbi(t, Q(t)) ∣ ∣ 2 . (5.17)The relativistic Hamilton–Jacobi partial differential equationThe requirement that the test charge velocity is the space componentof a (future-directed) four-velocity vector divided by the relativistic γ factorquite naturally leads to the following relativistic Hamilton–Jacobi partialdifferential equation,√1 ∂c ∂t S hj(t, q) =− m 2 c 2 + ∣ ∣∇ q S hj (t, q)+ 1 c eAsf mbi(t, q) ∣ 2 + 1 c eφsf mbi(t, q).(5.18)
322 M. Kiessling( 1cLorentz <strong>and</strong> Weyl invarianceAs to Lorentz invariance, any solution to (5.18) obviously satisfies∂∂t S hj(t, q)− 1 c eφsf mbi(t, q) ) 2 ∣−∣∇q S hj (t, q)+ 1 c eAsf mbi(t, q) ∣ 2 = m 2 c 2 , (5.19)a manifestly relativistically Lorentz scalar equation.Although S hj (t, q) is a scalar configuration spacetime field, it cannot begauge-invariant, for the four-vector field ( φ, A ) (t, s) is not; recall that theLorenz–Lorentz gauge condition alone does not fix the potentials completely.Instead, if the potentials (φ, A) are transformed under the gauge transformations(5.11) with any relativistic scalar field Υ : R 1,3 → R satisfying the waveequation (5.12), then, for a negative charge, S hj needs to be transformed asS hj (t, q) → S hj (t, q) − 1 ceΥ(t, q). (5.20)This gauge transformation law also holds more generally for Υ not satisfying(5.12), meaning a change of gauge from Lorenz–Lorentz to something else.Many-body test charge theoryThe generalization to many point charges with either sign is obvious.Since test charges do not “talk back” to the “external” potentials, there is aguiding equation (5.17) coupled with a partial differential equation (5.18) forthe guiding velocity field for each test charge. Of course, they are just identicalcopies of the single-particle equations, yet it is important to keep in mind thatthe many-body theory is to be formulated on many-body configuration space.Upgrading test particle motions: self-force problems déjà vu! Since the electromagneticpotentials for the actual electromagnetic Maxwell–Born–Infeldfields with point charge sources are supposedly defined everywhere, it couldnow seem that in order to get a well-defined theory of motion of their pointcharge sources all that needs to be done is to replace A sfmbi(t, q) <strong>and</strong>φ sfmbi(t, q)by A mbi (t, q) <strong>and</strong>φ mbi (t, q) in (5.18), which yields the partial differentialequation√1 ∂c ∂t S hj(t, q) =−m 2 c 2 + ∣ ∣∇ q S hj (t, q)+ 1 c eA mbi (t, q) ∣ ∣ 2 + 1 c eφ mbi(t, q),(5.21)<strong>and</strong> to replace A sfmbi(t, Q(t)) by A mbi (t, Q(t)) in (5.17) to get the guidingequation1 dQ(t) ∇ q S hj (t, Q(t)) + 1 c=eA mbi (t, Q(t))c dt√m 2 c 2 + ∣ ∣∇ q S hj (t, Q(t)) + 1 c eA mbi (t, Q(t)) ∣ . (5.22)2So the actual electromagnetic potentials as functions of space <strong>and</strong> time areevaluated at the generic position q in (5.21) <strong>and</strong> at the actual position Q(t)in (5.22).Note that almost all flow lines of the gradient field ∇ q S hj (t, q) wouldstill correspond to test particle motions in the actual φ mbi (t, s) <strong>and</strong>A mbi (t, s)potential fields (simply because almost all generic positions q are not identicalto the actual position Q(t) of the point charge source of the actual φ mbi (t, s)
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 323<strong>and</strong> A mbi (t, s)), so one may hope that by suitably iterating the given actualmotion one can make precisely one of these test particle motions coincide withthe actual motion — which is meant by “upgrading test-particle motion.”However, this does not lead to a well-defined theory of motion of pointcharge sources! The reason is that φ mbi (t, s) <strong>and</strong>A mbi (t, s) have non-differentiable“kinks” at s = Q(t). The function S hj (t, q) picks up this nondifferentiabilityat q = Q(t) through (5.21). More precisely, (5.21) is onlywell-defined away from the actual positions of the point charges. Trying toextend the definition of ∇ q S hj (t, q) to the actual positions now leads prettymuch to the same mathematical problems as encountered when trying todefine the “Lorentz self-force” on the point charge sources of the Maxwell–Born–Infeld field equations. In particular, we could regularize the actual potentialsφ mbi (t, s) <strong>and</strong>A mbi (t, s) by averaging, thereby obtaining a regularized“upgraded test-particle Hamilton–Jacobi theory” which does yield theactual “regularized motion” amongst all “regularized test particle motions”as a nonlinear fixed point problem. Unfortunately, subsequent removal of theregularization generally does not yield a unique limit, so that any so-definedlimiting theory of point charge motion would, once again, not be well-defined.Fortunately, Hamilton–Jacobi theory offers another option. Recall thatfor the non-relativistic problem of motion of N widely separated point chargesinteracting through their Coulomb pair interactions, Hamilton–Jacobi theoryyields a gradient flow on N-particle configuration space of which each flowline represents a putative actual trajectory of the N body problem: there areno test particle trajectories! In this vein, we should focus on a formulationof Hamilton–Jacobi theory which “parallel-processes” putative actual pointcharge motions.Parallel processing of putative actual motions: success! While nontrivial motionsin a strictly non-relativistic Coulomb problem without “external” fieldscan occur only when N ≥ 2 (Kepler motions if N = 2), a system with a singlepoint charge source for the electromagnetic Maxwell–Born–Infeld fieldsgenerally should feature non-trivial motions on single-particle configurationspace because of the dynamical degrees of freedom of the electromagneticfields. So in the following we focus on the N = 1 point charge problem,although eventually we have to address the general N-body problem.Setting up a Hamilton–Jacobi law which “parallel-processes” putativeactual single point source motions in the Maxwell–Born–Infeld field equationsis only possible if there exists a generic velocity field on configuration space(here: for a negative point charge), denoted by v(t, q), which varies smoothlywith q <strong>and</strong> t, <strong>and</strong> which is related to the family of putative actual motionsby the guiding lawdQ(t)= v(t, Q(t)) , (5.23)dtyielding the actual position Q(t) for each actual initial position Q(0).Assuming such a velocity field exists, one next needs to construct configurationspace fields φ 1 (t, q) <strong>and</strong>A 1 (t, q) which are “generic-q-sourced”
324 M. Kiesslingpotential fields φ ♯ (t, s, q) <strong>and</strong>A ♯ (t, s, q) evaluated at s = q, their genericpoint source; 17 i.e. [Kie2004a]:φ 1 (t, q) ≡ φ ♯ (t, q, q) <strong>and</strong> A 1 (t, q) ≡A ♯ (t, q, q). (5.24)The “canonical” set of partial differential equations for φ ♯ (t, s, q), A ♯ (t, s, q),<strong>and</strong> their derived fields, which are compatible with the Maxwell–Born–Infeldfield equations for the actual fields of a single negative point charge, reads1 ∂c1 ∂c∂t φ♯ (t, s, q) =− 1 c v(t, q)·∇ qφ ♯ (t, s, q) − ∇·A ♯ (t, s, q), (5.25)∂t A♯ (t, s, q) =− 1 c v(t, q)·∇ qA ♯ (t, s, q) − ∇φ ♯ (t, s, q) −E ♯ (t, s, q), (5.26)1 ∂c ∂t D♯ (t, s, q) =− 1 c v(t, q)·∇ qD ♯ (t, s, q)+∇×H ♯ (t, s, q)+4πe 1 c v(t, q)δ q(s);(5.27)furthermore, D ♯ (t, s, q) obeys the constraint equation 18∇·D ♯ (t, s, q) =−4πeδ q (s). (5.28)The fields E ♯ (t, s, q) <strong>and</strong>H ♯ (t, s, q) in (5.26), (5.27) are given in terms ofD ♯ (t, s, q) <strong>and</strong>B ♯ (t, s, q) in the same way as the actual fields E mbi (t, s) <strong>and</strong>H mbi (t, s) are defined in terms of D mbi (t, s) <strong>and</strong>B mbi (t, s) through the Born–Infeld aether law (4.5), (4.6), while B ♯ (t, s, q) inturnisgivenintermsofA ♯ (t, s, q) in the same way as the actual B mbi (t, s) is given in terms of theactual A mbi (t, s) in (5.8). It is straightforward to verify that by substitutingthe actual Q(t) for the generic q in the “generic-q-sourced” ♯-fields satisfyingthe above field equations, we obtain the actual electromagnetic potentials,fields, <strong>and</strong> charge-current densities satisfying the Maxwell–Born–Infeld fieldequations (in Lorenz–Lorentz gauge). That is,φ mbi (t, s) ≡ φ ♯ (t, s, Q(t)) etc. (5.29)Next we need to stipulate a law for v(t, q).The Hamilton–Jacobi velocity fieldThe naïvely obvious thing to try is the generic velocity law∇ q S hj (t, q)+ 1 cv(t, q) =ceA 1 (t, q)√m 2 c 2 + ∣ ∣∇ q S hj (t, q)+ 1 c eA 1 (t, q) ∣ , (5.30)2corresponding to the Hamilton–Jacobi PDE√1 ∂c ∂t S hj(t, q) =− m 2 c 2 + ∣ ∣∇ q S hj (t, q)+ 1 c eA 1 (t, q) ∣ 2 + 1 c eφ 1(t, q), (5.31)17 The notation is inherited from the N-point-charge problem. In that case there are fieldsφ ♯ (t, s, q 1 , ..., q N ) etc. which, when evaluated at s = q k , give configuration space fieldsφ k (t, q 1 , ..., q N ) etc. For a system with a single point charge, k =1.18 Since the generic charge density −eδ q(s) ist-independent <strong>and</strong> ∇ qδ q(s) = −∇δ q(s),the reformulation of the continuity equation for charge conservation (in spacetime),∂∂t ρ♯ (t, s, q) =−v(t, q) ·∇ qρ ♯ (t, s, q) − ∇ · j ♯ (t, s, q), is an identity, not an independentequation.
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 325which replaces (5.18). Since A 1 (t, Q(t)) = A mbi (t, Q(t)), the guiding law(5.23) with velocity field v given by (5.30) is superficially identical to (5.22),yet note that S hj in (5.22) is not the same S hj as in (5.30), (5.23) becauseφ(t, q) <strong>and</strong>A(t, q) are now replaced by φ 1 (t, q) <strong>and</strong>A 1 (t, q). Note also thatour single-particle law of motion has a straightforward extension to the N-body problem, which I also presented in [Kie2004a].It is a reasonable conjecture that the maps q ↦→ φ 1 (t, q) <strong>and</strong>q ↦→A 1 (t, q) aregenerically differentiable, 19 in which case one obtains the firstwell-defined self-consistent law of motion of a classical point charge sourcein the Maxwell–Born–Infeld field equations [Kie2004a]. It has an immediategeneralization to N-particle systems.It is straightforward to show that this law readily h<strong>and</strong>les the simplestsituation: the trivial motion (i.e., rest) of the point charge source in Born’sstatic solution. Note that no averaging or renormalization has to be invoked!Since the nonlinearities make it extremely difficult to evaluate the modelin nontrivial situations, only asymptotic results are available so far. It isshown in [Kie2004a] <strong>and</strong> [Kie2004b] that a point charge in Maxwell–Born–Infeldfields which are “co-sourced” jointly by this charge <strong>and</strong> another one that,in a single-particle setup, is assumed to be immovable (a Born–Oppenheimerapproximation to a dynamical two-particle setup), when the charges are farapart, carries out the Kepler motion in leading order, as it should. Moreover,at least formally one can also show that in general the slow motion <strong>and</strong>gentle acceleration regime of a point charge is governed in leading order by alaw of test charge motion as introduced at the beginning of this subsection.Whether this will pan out rigorously, <strong>and</strong> if so, whether the one-body setupyields physically correct motions if we go beyond the slow motion <strong>and</strong> gentleacceleration regime has yet to be established.5.3.4. Conservation laws: re-assessing the value of b. In [Kie2004a] I explainedthat the system of Maxwell–Born–Infeld field equations with a negativepoint charge source moving according to our parallel-processing Hamilton–Jacobilaws furnishes the following conserved total energy:√E = c m 2 c 2 + ∣ ∣∇ q S hj (t, Q(t)) + 1 c eA mbi (t, Q(t)) ∣ 2+ b24π∫R 3 (√1+ 1 b 2 (|Bmbi | 2 + |D mbi | 2) + 1 b 4 |B mbi ×D mbi | 2 − 1)(t, s)d 3 s.(5.32)In [Kie2004a] I had assumed from the outset that m = m e , but that wassomewhat hidden because of the dimensionless units I chose. The assumptionm = m e caught up with me when the total rest mass of the point defect plusthe electrostatic field around it, with b = b Born , became 2m e . With hindsight,I should have allowed the “intrinsic mass of the defect” m to be a parameter,19 Normally, a Cauchy problem is locally well-posed if there exists a unique solution, locallyin time, which depends Lipschitz-continuously on the initial data. We here expect, <strong>and</strong>need, a little more regularity than what suffices for basic well-posedness.
326 M. Kiesslingas I have done here, because then this bitter pill becomes bittersweet: thereis a whole range of combinations of m <strong>and</strong> b for which E = m e c 2 ;yetitis also evident that with m>0, Born’s proposal b = b Born is untenable.More precisely, b Born is an upper bound on the admissible b values obtainedfrom adapting Born’s argument that the empirical rest mass of the physicalelectron should now be the total energy over c 2 of a single point defect in itsstatic field.What these considerations do not reveal is the relative distribution ofmass between m <strong>and</strong> b. My colleague Shadi Tahvildar-Zadeh has suggestedthat m is possibly the only surviving remnant of a general relativistic treatment,<strong>and</strong> thereby determined. I come to general relativistic issues in thenext subsection.Before I get to there, I should complete the listing of the traditionalconservation laws. Namely, with a negative point charge, the total momentum,P = [ ∫∇ q S hj + 1 c eA ]mbi (t, Q(t)) +1(D4πc mbi ×B mbi )(t, s)d 3 s, (5.33)R 3<strong>and</strong> the total angular momentum,L = Q(t) × [ ∫∇ q S hj + 1 c eA ]mbi (t, Q(t)) +14πcs × (D mbi ×B mbi )(t, s)d 3 s,R 3 (5.34)are conserved as well. In addition there are a number of less familiar conservationlaws, but this would lead us too far from our main objective.5.4. General-relativistic spacetimes with point defectsEver since the formal papers by Einstein, Infeld, <strong>and</strong> Hoffmann [EIH1938],there have been quite many attempts to prove that Einstein’s field equationsimply the equations of motion for “point singularities.” Certainly they implythe evolution equations of continuum matter when the latter is the source ofspacetime geometry, but as to true point singularities the jury is still out. Forus this means a clear imperative to investigate this question rigorously whenEinstein’s field equations are coupled with the Maxwell–Born–Infeld fieldequations of electromagnetism. Namely, if Einstein’s field equations implythe equations of motion for the point charges, as Einstein et al. would haveit, then all the developments described in the previous subsections have beenin vain. If on the other h<strong>and</strong> it turns out that Einstein’s field equations do notimply the equations of motion for the point charges, then we have the needfor supplying such — in that case the natural thing to do, for us, is to adaptthe Hamilton–Jacobi type law of motion from flat to curved spacetimes.Fortunately, the question boils down to a static problem: Does theEinstein–Maxwell–Born–Infeld PDE system with two point charge sourceshave static, axisymmetric classically regular solutions away from the twoworldlines of the point charges, no matter where they are placed? If theanswer is “Yes,” then Einstein’s equations fail to deliver the equations of motionfor the charges, for empirically we know that two physical point charges
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 327in the classical regime would not remain motionless. Shadi Tahvildar-Zadeh<strong>and</strong> myself have begun to rigorously study this question. I hope to report itsanswer in the not too distant future.Meanwhile, I list a few facts that by now are known <strong>and</strong> which makeus quite optimistic. Namely, while the Einstein–Maxwell–Maxwell equationswith point charges produce solutions with horrible naked singularities (thinkof the Reissner–Nordström spacetime with charge <strong>and</strong> mass parameter chosento match the empirical electron data), the Einstein–Maxwell–Born–Infeldequations with point charge source are much better behaved. Tahvildar-Zadeh[TaZa2011] recently showed that they not only admit a static spacetime correspondingto a single point charge whose ADM mass equals its electrostaticfield energy/c 2 , he also showed that the spacetime singularity is of themildest possible form, namely a conical singularity. Conical singularities areso mild that they lend us hope that the nuisance of “struts” between “particles,”known from multiple-black-hole solutions of Einstein’s equations, canbe avoided. Tahvildar-Zadeh’s main theorem takes more than a page to state,after many pages of preparation. Here I will have to leave it at that.6. <strong>Quantum</strong> theory of motionBesides extending the classical flat spacetime theory to curved Lorentz manifolds,I have been working on its extension to the quantum regime. In[Kie2004b] I used a method which I called least invasive quantization of theone-charge Hamilton–Jacobi law for parallel processing of putative actualmotions. Although I didn’t see it this way at the time, by now I have realizedthat this least invasive quantization can be justified elegantly in the spirit ofthe quest for unification in physics!6.1. Quest for unification: least invasive quantizationIf we accept as a reasonably well-established working hypothesis that dynamicalphysical theories derive from an action principle, we should lookfor an action principle for the Hamilton–Jacobi equation. Because of thefirst order time derivative for S hj such an action principle for the classicalS hj can be formulated only at the price of introducing a scalar companionfield R hj which complements S hj . To illustrate this explicitly it suffices toconsider a representative, nonrelativistic Hamilton–Jacobi PDE, written as∂∂t S hj(t, q)+H(q, ∇ q S hj (t, q)) = 0. Multiplying this equation by some positivefunction R 2 (t, q) <strong>and</strong> integrating over q <strong>and</strong> t (the latter over a finiteinterval I) gives the “action” integral∫ ∫ [ ∂A(R, S hj )= R 2 (t, q)I R ∂t S hj(t, q)+H(q, ∇ q S hj (t, q))]d 3 q dt =0.3Now replacing also S hj by a generic S in A <strong>and</strong> seeking the stationary pointsof A(R, S), denoted by R hj <strong>and</strong> S hj , under variations with fixed end points,we obtain the Euler-Lagrange equations ∂ ∂t S hj(t, q) +H(q, ∇S hj (t, q)) = 0<strong>and</strong> ∂ ∂t R2 hj(t, q) +∇ q · [Rhj 2 1 m ∇ qS hj ](t, q) = 0. Clearly, the S hj equation is
328 M. Kiesslingjust the Hamilton–Jacobi equation we started from, while the R hj equationis a passive evolution equation: a continuity equation.The passive evolution of R hj somehow belies the fact that R hj is neededto formulate the variational principle for S hj in the first place. This suggeststhat R hj is really a field of comparable physical significance to S hj .Sointhespirit of unification, let’s try to find a small modification of the dynamics tosymmetrize the roles of R <strong>and</strong> S at the critical points.Interestingly enough, by adding an R-dependent penalty term (a Fisherentropy, ∝ 2 ) to the action functional A(R, S), one can obtain (even inthe N-body case) a Schrödinger equation for its critical points, denotedR qm e iS qm/ = ψ, where the suffix HJ has been replaced by QM to avoidconfusion with “R hj e iS hj/ .” The important point here is that the real <strong>and</strong>imaginary parts of R qm e iS qm/ now satisfy a nicely symmetrical dynamics! Inthis sense the R qm <strong>and</strong> S qm fields have been really unified into a complex fieldψ, whereasR hj e iS hj/ , while clearly complex, is not representing a unificationof R hj <strong>and</strong> S hj . Equally important: the guiding equation, <strong>and</strong> the ontologyof points that move, is unaffected by this procedure!6.1.1. A de Broglie–Bohm–Klein–Gordon law of motion. Thesametypeofargument works for the relativistic Hamilton–Jacobi theory <strong>and</strong> yields aKlein–Gordon equation. The Klein–Gordon PDE for the complex scalar configurationspace field ψ(t, q) reads(i1 ∂2ψc ∂t + e 1 c φ 1)= m 2 c 2 ψ + ( − i∇ q + e 1 c A ) 2ψ1 (6.1)where φ 1 <strong>and</strong> A 1 are the potential fields defined as in our parallel-processingsingle-charge Hamilton–Jacobi law.To wit, least invasive quantization does not affect the underlying purposeof the theory to provide a law of motion for the point defects. For aKlein–Gordon PDE on configuration space the velocity field v for the guidingequation ˙Q(t) =v(t, Q(t)) is now given by the ratio of quantum currentvector density to density, j qu (t, q)/ρ qu (t, q), withρ qu = I ( ψ ( − ∂mc 2 ∂t + i ) ) e φmc 2 1 ψ , j qu = I ( ψ ( m ∇ q + i e mc A ) )1 ψ ,(6.2)where I means imaginary part, <strong>and</strong> ψ is the complex conjugate of ψ; thusv(t, q) ≡ c I ( ψ ( ∇ q + ie 1 c A ) )1 ψI ( ψ ( − 1 ∂c ∂t + ie 1 c φ ) )(t, q) . (6.3)1 ψThis is a familiar de Broglie–Bohm–Klein–Gordon law of motion [DüTe2009,Hol1993], except that A 1 ,φ 1 are not external fields, of course.6.1.2. A de Broglie–Bohm–Dirac law of motion. It is only a small step from aKlein–Gordon to a Dirac equation for spinor-valued ψ coupled to the genericq-sourced potential fields for a negative charge,i 1 c∂∂t ψ = mcβψ + α · ( − i∇ q + e 1 c A 1)ψ − e1c φ 1ψ; (6.4)
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 329here α <strong>and</strong> β are the familiar Dirac matrices. The guiding equation for theactual point charge motion is still (5.23), once again with v = j qu /ρ qu , butnow with the quantum density <strong>and</strong> quantum current vector density given bythe Dirac expressions, yielding the de Broglie–Bohm–Dirac guiding equation1 dQ(t)c dt= ψ† αψψ † (t, Q(t)) , (6.5)ψwhere C 4 inner product is understood in the bilinear terms at the r.h.s.This is a familiar de Broglie–Bohm–Dirac law of motion [DüTe2009, Hol1993]except, once again, that A 1 ,φ 1 are not external fields. Presumably ψ has tobe restricted to an A-dependent “positive energy subspace,” which is tricky,<strong>and</strong> we do not have space here to get into the details.6.2. Born–Infeld effects on the Hydrogen spectrumThe two-charge model with an electron <strong>and</strong> a nuclear charge in Born–Oppenheimerapproximation is formally a dynamical one-charge model with anadditional charge co-sourcing the Maxwell–Born–Infeld fields. It can be usedto investigate Born–Infeld effects on the Hydrogen spectrum.Thehardpartistofindtheelectricpotentialφ ♯ (s, q, q n ) of the electrostaticMaxwell–Born–Infeld field of an electron at q <strong>and</strong> the nucleus at q n = 0in otherwise empty space. The conceptual benefits offered by the nonlinearityof the Maxwell–Born–Infeld field equations come at a high price: in contrastto the ease with which the general solution to the Maxwell–Lorentz field equationscan be written down, there is no general formula to explicitly representthe solutions to the Maxwell–Born–Infeld field equations. So far only stationarysolutions with regular sources can be written down systematically withthe help of convergent perturbative series expansions [CaKi2010, Kie2011c].In [Kie2004b] I presented an explicit integral formula for an approximationto φ ♯ (q, q, 0) =φ 1 (q). If the point charges are slightly smeared out<strong>and</strong> b −2 is not too big, then this formula gives indeed the electric potentialfor the leading order term in the perturbative series expansion in powers ofb −2 for the displacement field D developed in [CaKi2010, Kie2011c]. Assumingthat the formula for the total electrostatic potential at the location ofthe electron is giving the leading contribution also for point charges, Born–Infeld effects on the Schrödinger spectrum of Hydrogen were computed 20 in[CaKi2006, FrGa2011]. In [FrGa2011] also the Dirac spectrum was studied.The interesting tentative conclusion from these studies is that Born’s value ofb gives spectral distortions which are too large to be acceptable. More refinedtwo-body studies are still needed to confirm this finding, but the researchclearly indicates that atomic spectral data may well be precise enough totest the viability of the Born–Infeld law for electromagnetism.20 The ground state energies as functions of Born’s b parameter agree nicely in both numericalstudies, but some of the excited states don’t, hinting at a bug in our program. Ithank Joel Franklin for pointing this out.
330 M. Kiessling7. Closing remarksIn the previous sections I have slowly built up a well-defined theory of motionfor point defects in the Maxwell–Born–Infeld fields, both in the classicalregime, using Hamilton–Jacobi theory, <strong>and</strong> also in the quantum regime, usingwave equations without <strong>and</strong> with spin. In either case the important notion isthe parallel processing of motions, not test particle motions or their upgradeto a fixed point problem.Unfortunately, while the nonlinearity of the Maxwell–Born–Infeld equationsmakes the introduction of such laws of motion possible in the first place,it is also an obstacle to any serious progress in computing the motions actuallyproduced by these laws. But I am sure that it is only a matter of timeuntil more powerful techniques are brought in which will clarify many of theburning open questions.So far basically everything I discussed referred to the one-charge problem.This is perfectly adequate for the purpose of studying the self-interactionproblem of a point charge which lies at the heart of the problem of its motion.But any acceptable solution to this self-interaction problem also has to be generalizedto the N-charge situation, <strong>and</strong> this is another active field of inquiry.While the jury is still out on the correct format of the many charge theory, oneaspect of it is presumably here to stay. Namely, a many-charge formulationin configuration space clearly requires synchronization of the various charges;by default one would choose to work with a particular Lorentz frame, butany other choice should be allowed as well. Actually, even the single-chargeformulation I gave here tacitly uses the synchronization of the time componentsin the four-vectors (ct, s) <strong>and</strong>(q 0 , q). In the test charge approximationsynchronization is inconsequential, but in this active charge formulation themany-charge law would seem to depend on the synchronization. Whether themotion will depend on the foliation can naturally be investigated. Even if itdoes, the law of motion would not automatically be in conflict with Lorentzcovariance. What is needed is simply a covariant foliation equation, as used ingeneral relativity [ChKl1993]. A distinguished foliation could be interpretedas restoring three-dimensionality to physical reality. This would be againstthe traditional spirit of relativity theory, i.e. Einstein’s interpretation of it asmeaning that physical reality is four-dimensional, but that’s OK.AcknowledgementI am very grateful to the organizers Felix Finster, Olaf Müller, Marc Nardmann,Jürgen Tolksdorf, <strong>and</strong> Eberhard Zeidler of the excellent “<strong>Quantum</strong><strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>” conference in Regensburg (2010), for the invitationto present my research, <strong>and</strong> for their hospitality <strong>and</strong> support. Thematerial reported here is based on my conference talk, but I have taken theopportunity to address in more detail some questions raised in response to mypresentation, there <strong>and</strong> elsewhere. I also now mentioned the definitive statusof some results that had been in the making at the time of the conference.The research has been developing over many years, funded by NSF grants
On the Motion of Point Defects in Relativistic <strong>Field</strong>s 331DMS-0406951 <strong>and</strong> DMS-0807705, which is gratefully acknowledged. I benefitedgreatly from many discussions with my colleagues, postdocs <strong>and</strong> students,in particular: Walter Appel, Holly Carley, Sagun Chanillo, DemetriosChristodoulou, Detlef Dürr, Shelly Goldstein, Markus Kunze, Tim Maudlin,Jared Speck, Herbert Spohn, Shadi Tahvildar-Zadeh, Rodi Tumulka, NinoZanghì. Lastly, I thank the anonymous referee for many helpful suggestions.References[Abr1903] Abraham, M., Prinzipien der Dynamik des Elektrons, Phys. Z. 4, 57–63(1902); Ann. Phys. 10, pp. 105–179 (1903).[Abr1905] Abraham, M., Theorie der Elektrizität, II, Teubner, Leipzig (1905).[AnTh2005] Anco, S. C., <strong>and</strong> The, D., Symmetries, conservation laws, <strong>and</strong> cohomologyof Maxwell’s equations using potentials, Acta Appl. Math. 89, 1–52(2005).[ApKi2001] Appel, W., <strong>and</strong> Kiessling, M. K.-H., Mass <strong>and</strong> spin renormalization inLorentz electrodynamics, Annals Phys. (N.Y.) 289, 24–83 (2001).[ApKi2002] Appel, W., <strong>and</strong> Kiessling, M. K.-H., Scattering <strong>and</strong> radiation dampingin gyroscopic Lorentz electrodynamics, Lett. Math. Phys. 60, 31–46 (2002).[Bart1987] Bartnik, R., Maximal surfaces <strong>and</strong> general relativity, pp. 24–49 in “Miniconferenceon Geometry/Partial Differential Equations, 2” (Canberra, June26-27, 1986), J. Hutchinson <strong>and</strong> L. Simon, Ed., Proceedings of the Center of<strong>Mathematical</strong> Analysis, Australian National Univ., 12 (1987).[Bart1989] Bartnik, R., Isolated points of Lorentzian mean-curvature hypersurfaces,Indiana Univ. Math. J. 38, 811–827 (1988).[BaSi1982] Bartnik, R., <strong>and</strong> Simon, L., Spacelike hypersurfaces with prescribedboundary values <strong>and</strong> mean curvature, Commun. Math. Phys. 87, 131–152(1982).[Baru1964] Barut, A. O., Electrodynamics <strong>and</strong> classical theory of fields <strong>and</strong> particles,Dover, New York (1964).[Bau1998] Bauer, G., Ein Existenzsatz für die Wheeler–Feynman-Elektrodynamik,Doctoral Dissertation, Ludwig Maximilians Universität, München, (1998).[BDD2010] Bauer, G., Deckert, D.-A., <strong>and</strong> Dürr, D., Wheeler–Feynman equationsfor rigid charges — Classical absorber electrodynamics. Part II.arXiv:1009.3103 (2010)[BiBi1983] Bia̷lynicki-Birula, I., Nonlinear electrodynamics: variations on a themeby Born <strong>and</strong> Infeld, pp. 31–48 in “<strong>Quantum</strong> theory of particles <strong>and</strong> fields,”special volume in honor of Jan ̷Lopuszański;eds.B.Jancewicz<strong>and</strong>J.Lukierski,World Scientific, Singapore (1983).[Boi1970] Boillat, G., Nonlinear electrodynamics: Lagrangians <strong>and</strong> equations of motion,J. Math. Phys. 11, 941–951 (1970).[Bor1933] Born, M., Modified field equations with a finite radius of the electron,Nature 132, 282 (1933).[Bor1934] Born, M., On the quantum theory of the electromagnetic field, Proc.Roy.Soc. A143, 410–437 (1934).
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On the Motion of Point Defects in Relativistic <strong>Field</strong>s 335[Spe2010b] Speck, J., The global stability of the Minkowski spacetime solution tothe Einstein-nonlinear electromagnetic system in wave coordinates,arXiv:1009.6038 (2010).[Spo2004] Spohn, H., Dynamics of charged particles <strong>and</strong> their radiation field, CambridgeUniversity Press, Cambridge (2004).[TaZa2011] Tahvildar-Zadeh, A. S., On the static spacetime of a single point charge,Rev. Math. Phys. 23, 309–346 (2011).[Thi1997] Thirring, W. E., Classical mathematical physics, (Dynamical systems <strong>and</strong>field theory), 3rd ed., Springer Verlag, New York (1997).[Tho1897] Thomson, J. J., Cathode rays, Phil. Mag. 44, 294–316 (1897).[WhFe1949] Wheeler, J. A., <strong>and</strong> Feynman, R., Classical electrodynamics in termsof direct particle interactions, Rev.Mod.Phys.21, 425–433 (1949).[Wie1896] Wiechert, E., Die Theorie der Elektrodynamik und die Röntgen’sche Entdeckung,Schriften d. Physikalisch-Ökonomischen Gesellschaft zu Königsbergin Preussen 37, 1-48 (1896).[Wie1897] Wiechert, E., Experimentelles über die Kathodenstrahlen, Schriften d.Physikalisch-Ökonomischen Gesellschaft zu Königsberg in Preussen 38, 3–16(1897).[Wie1900] Wiechert, E., Elektrodynamische Elementargesetze, Arch.Néerl. Sci. ExactesNat. 5, 549–573 (1900).[Yag1992] Yaghjian, A. D., Relativistic dynamics of a charged sphere, Lect. NotesPhys. Monographs 11, Springer, Berlin (1992).Michael K.-H. KiesslingDepartment of MathematicsRutgers University110 Frelinghuysen Rd.Piscataway, NJ 08854USAe-mail: miki@math.rutgers.edu
How Unique AreHigher-dimensional Black Holes?Stefan Holl<strong>and</strong>sAbstract. In this article, we review the classification <strong>and</strong> uniquenessof stationary black hole solutions having large abelian isometry groupsin higher-dimensional general relativity. We also point out some consequencesof our analysis concerning the possible topologies that the blackhole exteriors may have.Mathematics Subject Classification (2010). 35Q75, 53C43, 58D19.Keywords. General relativity, higher dimensions, black hole uniquenesstheorems, differential geometry, manifolds with torus action, non-linearsigma models.The idea that there might exist extra dimensions beyond the evidentthree space <strong>and</strong> one time dimension was already brought up in the early daysof General Relativity. It has since become a major ingredient in many fundamentaltheories of nature, especially those with an eye on a “unificationof forces”. To explain why the extra dimensions—if indeed they exist—haveso far gone unnoticed, it is typically supposed that they are either extremelysmall in size, or are large but somehow unable to communicate with us directly.Either way, it seems to be very difficult to probe any of these ideas experimentally,<strong>and</strong> one does not feel that such unreliable concepts as “beauty”or “naturalness” alone can be trusted as the right means to find the correcttheory. But one can at least try to underst<strong>and</strong> in more detail the status <strong>and</strong>consequences of such theories. A concrete <strong>and</strong> sensible starting point is toconsider theories of the same general type as general relativity, such as e.g.various Kaluza–Klein, or supergravity theories, on a higher-dimensional manifold,<strong>and</strong> to ask about the nature of their solutions. Of particular interestare solutions describing black holes, <strong>and</strong> among these, stationary black holesolutions are of special interest, because they might be the end point of the(classical) evolution of a dynamical black hole spacetime.So, what are for example all stationary black hole spacetimes satisfyingthe vacuum Einstein equations on a D-dimensional manifold, with say,F. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_15,© Springer Basel AG 2012337
338 S. Holl<strong>and</strong>sasymptotically flat boundary conditions at infinity? In D = 4 dimensionsthe answer to this question is provided by the famous black hole uniquenesstheorems 1 , which state that the only such solutions are provided by theKerr family of metrics, <strong>and</strong> these are completely specified by their angularmomentum <strong>and</strong> mass. Unfortunately, already in D = 5, the analogous statementis demonstrably false, as there exist different asymptotically flat blackholes—having even event horizons of different topology—with the same valuesof the angular momenta <strong>and</strong> charges 2 . Nevertheless, one might hope thata classification is still possible if a number of further invariants of the solutionsare also incorporated. This turns out to be possible [22, 23], at least ifone restricts attention to solutions which are not only stationary, but moreoverhave a comparable amount of symmetry as the Kerr family, namely thesymmetry group 3 R × U(1) D−3 . Such a spacetime cannot be asymptoticallyflat in D>5, but it can be asymptotically Kaluza-Klein, i.e. asymptotically adirect product e.g. of the form R 4,1 ×T D−5 . The purpose of this contributionis to outline the nature of this classification.Because the symmetry group has D − 2 dimensions, the metric will,in a sense, depend non-trivially only on two remaining coordinates, <strong>and</strong> theEinstein equations will consequently reduce to a coupled system of PDE’sin these variables. However, before one can study these equations, one mustunderst<strong>and</strong> more precisely the nature of the two remaining coordinates, or,mathematically speaking, the nature of the orbit space M/[R × U(1) D−3 ].The quotient by R simply gets rid of a global time coordinate, so one is leftwith the quotient of a spatial slice Σ by U(1) D−3 . To get an idea about thepossible topological properties of this quotient, we consider the following twosimple, but characteristic, examples in the case dim Σ = 4, i.e. D =5.The first example is Σ = R 4 , with one factor of U(1) × U(1) acting byrotations in the 12-plane <strong>and</strong> the other in the 34-plane. Introducing polarcoordinates (R 1 ,φ 1 )<strong>and</strong>(R 2 ,φ 2 ) in each of these planes, the group shiftsφ 1 resp. φ 2 , <strong>and</strong> the quotient is thus given simply by the first quadrant{(R 1 ,R 2 ) ∈ R 2 | R 1 ≥ 0,R 2 ≥ 0}, which is a 2-manifold whose boundaryconsists of the two semi-axes <strong>and</strong> the corner where the two axes meet. Thefirst axis corresponds to places in R 4 where the Killing field m 1 = ∂/∂φ 1vanishes, the second axis to places where m 2 = ∂/∂φ 2 vanishes. On thecorner, both Killing fields vanish <strong>and</strong> the group action has a fixed point. Thesecond example is the cartesian product of a plane with a 2-torus, Σ = R 2 ×T 2 . Letting (x 1 ,x 2 ) be cartesian coordinates on the plane, <strong>and</strong> (φ 1 ,φ 2 )angleson the torus, the group action is generated by the vector fields m 1 = ∂/∂φ 1<strong>and</strong> by m 2 = α∂/∂φ 2 +β(x 1 ∂/∂x 2 −x 2 ∂/∂x 1 ), where α, β are integers. These1 For a recent proof dealing properly with all the mathematical technicalities, see [5]. Originalreferences include [3, 36, 2, 28, 16, 17, 24].2 For a review of exact black hole solutions in higher dimensions, see e.g. [9].3 In D = 4, this is not actually a restriction, because the rigidity theorem [16, 4, 11, 35]shows that any stationary black hole solution has the additional U(1)-symmetry. In higherdimensions, there is a similar theorem [18, 19, 31], but it guarantees only one additionalU(1)-factor, <strong>and</strong> not D − 3.
How Unique Are Higher-dimensional Black Holes? 339vector fields do not vanish anywhere, but there are discrete group elementsleaving certain points invariant. The quotient is now a cone with deficit angle2π/α.The general case turns out to be locally the same as in these examples[34, 22]. In fact, one can show that the quotient Σ/[U(1) × U(1)] is a2-dimensional conifold with boundaries <strong>and</strong> corners. Each boundary segmentis characterized by a different pair (p, q) of integers such that pm 1 + qm 2 =0at corresponding points of Σ, see fig. 1.Figure 1. The numbers (p, q) may be viewed as windingnumbers associated with the generators of the 2-torus generatedby the two axial Killing fields. In such a torus, an U(1)-orbit winds around the first S 1 generator n times as it goesp times around the other S 1 -direction. Here qn ≡ 1modp.The figure shows the situation for p =3,n =7.Figure 2. A more artistic version of the previous figure,from “La Pratica della Perspectiva” (1569), by D. Barbaro.For subsequent boundary segments adjacent on a corner labeled by(p i ,q i )<strong>and</strong>(p i+1 ,q i+1 ), we have the condition∣ p ∣i p i+1∣∣∣= ±1 . (1)q i q i+1Each conical singularity is characterized by a deficit angle, i.e. another integer.In higher dimensions, there is a similar result [23]; now a boundary segmentis e.g. characterized by a (D − 3)-tuple of integers (“winding numbers”), <strong>and</strong>the compatibility condition at the corners is somewhat more complicated.
340 S. Holl<strong>and</strong>sIn the case where Σ is the spatial section of a black hole spacetime, thereare further constraints coming from Einstein’s equations <strong>and</strong> the orientabilityof the spacetime. The topological censorship theorem [12, 7] is seen to imply[23] that the 2-dimensional orbit space ˆΣ =Σ/U(1) D−3 can neither haveany conifold points, nor holes, nor h<strong>and</strong>les, <strong>and</strong> therefore has to be diffeomorphicto an upper half-plane ˆΣ ∼ = {(r, z) | r>0}. The boundary segmentscorrespond to intervals on the boundary (r = 0) of this upper half-plane<strong>and</strong> are places (“axes”) in the manifold M where a linear combination of therotational Killing fields vanishes, or to a horizon. Here the Killing fields donot vanish except where an “axis” meets a horizon. Furthermore, with eachboundary segment, one can associate its length l i ≥ 0 in an invariant way. Itcan be shown (see e.g. [6]) that the metric of a black hole spacetime with theindicated symmetries globally takes the Weyl–Papapetrou formg = − r2 dt 2det f +e−ν (dr 2 +dz 2 )+f ij (dφ i + w i dt)(dφ j + w j dt) , (2)where the metric coefficients only depend on r, z,<strong>and</strong>whereφ i are 2π-periodiccoordinates. The vacuum Einstein equations lead to two sets of differentialequations. One set can be written [27] as a sigma-model field equation for amatrix field Φ defined by( )(det f)−1−(det f)Φ=−1 χ i−(det f) −1 χ i f ij +(detf) −1 , (3)χ i χ jwhere χ i are certain potentials that are defined in terms of the metric coefficients.The other set can be viewed as determining the conformal factorν from Φ. The matrix field Φ obeys certain boundary conditions at r =0related to the winding numbers <strong>and</strong> moduli l i ∈ R + labeling the i-th boundaryinterval. Hence, it is evident 4 that any uniqueness theorem for the blackholes under consideration would have to involve these data in addition to theusual invariants, such as mass <strong>and</strong> angular momentum.In fact, what one can prove is the following theorem in D =5(modulotechnical assumptions about analyticity <strong>and</strong> certain global causal constraints)[22]:Theorem 1. Given are two asymptotically flat, vacuum black hole solutions,each with a single non-extremal horizon. If the angular momenta J 1 ,J 2 coincide,<strong>and</strong> if all the integers (“winding numbers”) 〈(p 1 ,q 1 ),...,(p N ,q N )〉 <strong>and</strong>real numbers 〈l 1 ,...,l N 〉 (“moduli”) coincide, then the solutions are isometric.The theorem has a generalization to higher dimensions D (with oursymmetry assumption) [23]; here the asymptotically flat condition must be4 This observation seems to have been made first in [15]. In this paper, the importance ofthe quantities l i , (p i ,q i ), <strong>and</strong> similarly in higher dimensions D>5, was emphasized, buttheir properties <strong>and</strong> relation to the topology <strong>and</strong> global properties of M were not yet fullyunderstood.
How Unique Are Higher-dimensional Black Holes? 341replaced by a suitable asymptotic Kaluza–Klein condition, <strong>and</strong> the windingnumbers become vectors in Z D−3 , etc. The idea behind the proof isto construct from the matrix fields Φ for the two solutions a scalar functionon the upper half-plane, which measures the pointwise distance betweenthese matrix fields. This function, u, is essentially the geodesic distance innon-compact coset-space in which one can think of Φ as taking its values.It is non-negative, satisfies a suitable Laplace-type equation with positivesource, as well as appropriate boundary conditions governed by the moduli<strong>and</strong> winding numbers. This information suffices to show that u = 0 on theupper half-plane, <strong>and</strong> this in turn implies that the metrics are isometric.The simplest case when the winding numbers <strong>and</strong> moduli are trivial (inessence the same as in D = 4) was treated in D = 5 first by [32]. The uniquenesstheorem has subsequently been generalized to extremal black holes inD =4, 5 dimensions [1, 10], using the idea to employ the “near horizon geometries”[25, 26, 20] to resolve the situation at the horizon, which shrinksto a single point in the above upper half-plane picture.The topological structure of the orbit space is easily seen to imply thatthe horizon H has to be, in D = 5, a 3-sphere S 3 ,aringS 2 × S 1 ,oralensspace L(p, q), but examples of regular black hole spacetimes are known onlyfor the first two cases. In higher dimensions, there is a similar classification.In general, it is not clear whether, for a given set (p i ,q i ) satisfying the constraint(5) at each corner, <strong>and</strong> given angular momenta J i <strong>and</strong> moduli l i ,thereactually exists a corresponding black hole solution. Thus, in this sense theabove uniqueness theorem is much weaker than that known in 4 dimensions,where there is a known black hole solution for each choice of J <strong>and</strong> l (thewinding numbers are trivial in D = 4). One can trade the parameter l for themass of the black hole if desired, or its horizon area, <strong>and</strong> a similar remark appliesto one of the parameters l i in higher dimensions. In order to attack theexistence questions for black holes in higher dimensions, one can either try tofind the solution corresponding to a given set of angular momenta, windingnumbers <strong>and</strong> moduli explicitly. Here it is of great help that the equations forν <strong>and</strong> Φ are of a very special nature <strong>and</strong> amenable to powerful methods fromthe theory of integrable systems. Another approach would be to try to findan abstract existence proof based on the boundary value problem for ν <strong>and</strong>Φ. Either way, the solution of the problem does not seem to be simple, <strong>and</strong>it is certainly not clear—<strong>and</strong> maybe even unlikely—that a solution exists forthe most general set of data J i ,l i <strong>and</strong> winding numbers.The collection of winding numbers can nicely be put in correspondencewith the topology of Σ, <strong>and</strong> hence of M ∼ = Σ × R. By exploiting the constraints(5) for adjacent winding numbers, one can prove that Σ can be decomposedinto simpler pieces. For example, in the case when D =5<strong>and</strong>thespacetime is spin 5 , one can show [34] that for spherical horizonsΣ ∼ = (R 4 \ B 4 )#(N − 2)(S 2 × S 2 ) . (4)5 This is of course physically well-motivated. In the general case, the decomposition wouldhave additional factors of ±CP 2 .
342 S. Holl<strong>and</strong>sThe black hole horizon is the boundary S 3 = ∂B 4 of a 4-dimensional ball thathas been removed. For a ringlike horizon, one has to remove S 2 × B 2 instead;the lens space case is somewhat more complicated. For all known solutions,the “h<strong>and</strong>les” S 2 × S 2 are in fact absent. Furthermore, if we introduce foreach triple of adjacent boundary segments of ˆΣ labeled by (p i ,q i ), (p i+1 ,q i+1 )<strong>and</strong> (p i+2 ,q i+2 ) the determinante i =∣ p ∣i p i+2∣∣∣∈ 2Z , (5)q i q i+2then the signature resp. Euler characteristic of the 4-manifold Σ (with blackhole glued back in) are given by 3τ(Σ) = e 1 + ··· + e N <strong>and</strong> χ(Σ) = N,where N + 2 is the number of adjacent boundary segments (“intervals”),see [29]. If it was e.g. known that Σ could carry some Einstein metric, then theHitchin–Thorpe inequality 2χ(Σ) ≥ 3|τ(Σ)| would give the further constraint2N ≥|e 1 + ···+ e N |.In dimension D ≥ 6, a similar analysis also appears to be possible, butit is more complicated. For example, when D = 7 <strong>and</strong> we consider sphericalblack holes, we have, using the results of [33]:Σ ∼ = (R 6 \ B 6 )#(N − 4)(S 2 × S 4 )#(N − 3)(S 3 × S 3 ), (6)where it has been assumed that the second Stiefel–Whitney class is w 2 (Σ) =0.In summary, in higher dimensions, there are indications [8] that thevariety of possible vacuum stationary black hole solutions could be muchgreater than in D = 4. To date it remains largely unexplored by rigorousmethods. However, if one makes more stringent symmetry assumptions thanseem to be justified on just purely dynamical grounds—i.e., than what isguaranteed by the rigidity theorem [18, 19, 31]—, then a partial classificationof stationary black holes is possible, as we have described. Also, if one assumesthat the black hole is (asymptotically flat <strong>and</strong>) even static, then a differentkind of argument [14] shows that the solution must be a higher-dimensionalanalogue of Schwarzschild. These results are not as powerful in the stationary(non-static) case as the corresponding statements in D = 4, but they goat least some way in the direction of exploring the l<strong>and</strong>scape of black holesolutions in higher dimensions.If one is not willing to make any by-h<strong>and</strong> symmetry assumptions beyondwhat is guaranteed by the rigidity theorem [18, 19, 31] in higher dimensions[which implies the existence of one Killing field ψ generating U(1)], then atpresent not much can be said about uniqueness of higher-dimensional stationary(but non-static), asymptotically flat vacuum black holes. However, certainqualitative statements of general nature can still be made. For example, bya theorem of [13], the horizon manifold must be of “positive Yamabe type”,meaning essentially that it can carry a metric of pointwise positive scalarcurvature. In D = 5, the horizon manifold H is a 3-dimensional compactRiemannian manifold, <strong>and</strong> the positive Yamabe type condition then imposessome fairly strong conditions on its topology. Actually, in that case, if the
How Unique Are Higher-dimensional Black Holes? 343positive Yamabe type is combined with the results of the rigidity theorem inD = 5, then even stronger conclusions can be drawn, namely [21]:Theorem 2. In D =5vacuum general relativity, the topology of the eventhorizon H of a compact black hole can be one of the following:1. If ψ hasazeroonH, then the topology of H must be 6H ∼ = #(l − 1) · (S 2 × S 1 )#L(p 1 ,q 1 )#···# L(p k ,q k ) . (7)Here, k is the number of exceptional orbits of the action of U(1) on Hthat is generated by ψ, <strong>and</strong>l is the number of connected components ofthe zero set of ψ.2. If ψ does not have a zero on H, thenH ∼ = S 3 /Γ, whereΓ can be certainfinite subgroups of SO(4), orH ∼ = S 2 × S 1 . This class of manifoldsincludes again the lens spaces, but also prism manifolds, the Poincaréhomology sphere, <strong>and</strong> various other quotients. All manifolds in this classare certain Seifert fibred spaces over S 2 . The precise classification of thepossibilities is given in ref. [21].References[1] Amsel, A. J., Horowitz, G. T., Marolf, D. <strong>and</strong> Roberts, M. M.: arXiv:0906.2367[gr-qc][2] Bunting, G. L.: PhD Thesis, Univ. of New Engl<strong>and</strong>, Armidale, N.S.W., 1983[3] Carter, B.: Phys. Rev. Lett. 26, 331-333 (1971)[4] Chruściel, P. T.: Commun. Math. Phys. 189, 1-7 (1997)[5] Chruściel, P. T. <strong>and</strong> Lopes Costa, J.: arXiv:0806.0016 [gr-qc][6] Chruściel, P. T.: J. Math. Phys. 50, 052501 (2009)[7] Chruściel, P. T., Galloway, G. J. <strong>and</strong> Solis, D.: Annales Henri Poincaré 10, 893(2009)[8] Emparan, R., Harmark, T., Niarchos V. <strong>and</strong> Obers, N. A.: arXiv:1106.4428[hep-th] <strong>and</strong> references therein[9] Emparan, R. <strong>and</strong> Reall, H. S.: Living Rev. Rel. 11, 6 (2008)[10] Figueras, P. <strong>and</strong> Lucietti, J.: arXiv:0906.5565 [hep-th][11] Friedrich, H., Rácz, I. <strong>and</strong> Wald, R. M.: Commun. Math. Phys. 204, 691-707(1999)[12] Galloway, G. J., Schleich, K., Witt, D. M. <strong>and</strong> Woolgar, E.: Phys. Rev. D 60,104039 (1999)[13] Galloway, G. J. <strong>and</strong> Schoen, R: Commun. Math. Phys. 266, 571576 (2006)[14] Gibbons, G. W., Ida, D. <strong>and</strong> Shiromizu, T.: Phys. Rev. Lett. 89, 041101 (2002),arXiv:hep-th/0206049[15] Harmark, T.: Phys. Rev. D 70, 124002 (2004)[16] Hawking, S. W. <strong>and</strong> Ellis, G. F. R.: Cambridge, Cambridge University Press,1973[17] Hawking, S. W.: Commun. Math. Phys. 25, 152-166 (1972)6 Here we allow that the lens space be L(0, 1) := S 3 .
344 S. Holl<strong>and</strong>s[18] Holl<strong>and</strong>s, S., Ishibashi, A. <strong>and</strong> Wald, R. M.: Commun. Math. Phys. 271, 699(2007)[19] Holl<strong>and</strong>s, S. <strong>and</strong> Ishibashi, A.: Commun. Math. Phys. 291, 403 (2009)[20] Holl<strong>and</strong>s, S. <strong>and</strong> Ishibashi, A.: arXiv:0909.3462 [gr-qc][21] Holl<strong>and</strong>s, S., Holl<strong>and</strong>, J. <strong>and</strong> Ishibashi, A.: arXiv:1002.0490 [gr-qc][22] Holl<strong>and</strong>s, S. <strong>and</strong> Yazadjiev, S.: Commun. Math. Phys. 283, 749 (2008)[23] Holl<strong>and</strong>s, S. <strong>and</strong> Yazadjiev, S.: arXiv:0812.3036 [gr-qc].[24] Israel, W.: Phys. Rev. 164, 1776-1779 (1967)[25] Kunduri, H. K., Lucietti, J. <strong>and</strong> Reall, H. S.: Class. Quant. Grav. 24, 4169(2007)[26] Kunduri, H. K. <strong>and</strong> Lucietti, J.: J. Math. Phys. 50, 082502 (2009)[27] Maison, D.: Gen. Rel. Grav. 10, 717 (1979)[28] Mazur, P. O.: J. Phys. A 15, 3173-3180 (1982)[29] Melvin, P.: Math. Proc. Camb. Phil. Soc. 91, 305-314 (1982)[30] Moncrief, V. <strong>and</strong> Isenberg, J.: Commun. Math. Phys. 89, 387-413 (1983)[31] Moncrief, V. <strong>and</strong> Isenberg, J.: Class. Quant. Grav. 25, 195015 (2008)[32] Morisawa, Y. <strong>and</strong> Ida, D.: Phys. Rev. D 69, 124005 (2004)[33] Oh, H. S.: Topology Appl. 13, 137-154 (1982)[34] Orlik, P. <strong>and</strong> Raymond, F.: Topology 13, 89-112 (1974)[35] Rácz, I.: Class. Quant. Grav. 17, 153 (2000)[36] Robinson, D. C.: Phys. Rev. Lett. 34, 905-906 (1975)Stefan Holl<strong>and</strong>sSchool of MathematicsCardiff UniversityCardiff, CF24 4AGUKe-mail: Holl<strong>and</strong>sS@Cardiff.ac.uk
Equivalence Principle, <strong>Quantum</strong> Mechanics,<strong>and</strong> Atom-Interferometric TestsDomenico GiuliniAbstract. That gravitation can be understood as a purely metric phenomenondepends crucially on the validity of a number of hypotheseswhich are summarised by the Einstein Equivalence Principle, the leastwell tested part of which being the Universality of Gravitational Redshift.A recent <strong>and</strong> currently widely debated proposal (Nature 463 (2010)926-929) to re-interpret some 10-year old experiments in atom interferometrywould imply, if tenable, substantial reductions on upper boundsfor possible violations of the Universality of Gravitational Redshift byfour orders of magnitude. This interpretation, however, is problematic<strong>and</strong> raises various compatibility issues concerning basic principles ofGeneral Relativity <strong>and</strong> <strong>Quantum</strong> Mechanics. I review some relevant aspectsof the equivalence principle <strong>and</strong> its import into quantum mechanics,<strong>and</strong> then turn to the problems raised by the mentioned proposal.I conclude that this proposal is too problematic to warrant the claimsthat were launched with it.Mathematics Subject Classification (2010). 83C99, 81Q05.Keywords. General relativity, atom interferometry, equivalence principle.1. IntroductionThat gravitation can be understood as a purely metric phenomenon dependscrucially on the validity of a number of hypotheses which are summarised bythe Einstein Equivalence Principle, henceforth abbreviated by EEP. Theseassumptions concern contingent properties of the physical world that mayI thank Felix Finster, Olaf Müller, Marc Nardmann, Jürgen Tolksdorf <strong>and</strong> Eberhard Zeidlerfor inviting me to the conference on <strong>Quantum</strong> field theory <strong>and</strong> gravity (September 27 –October 2, 2010). This paper is the written-up version of my talk: Down-to-Earth issuesin atom interferometry, given on Friday, October 1st. I thank Claus Lämmerzahl <strong>and</strong>Ernst Rasel for discussions concerning the issues <strong>and</strong> views presented here. Last not leastI sincerely thank the QUEST cluster for making this work possible.F. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3_16,© Springer Basel AG 2012345
346 D. Giuliniwell either fail to hold in the quantum domain, or simply become meaningless.If we believe that likewise <strong>Quantum</strong> <strong>Gravity</strong> is <strong>Quantum</strong> Geometry,we should be able to argue for it by some sort of extension or adaptationof EEP into the quantum domain. As a first attempt in this direction onemight ask for the status of EEP if the matter used to probe it is describedby ordinary non-relativistic <strong>Quantum</strong> Mechanics. Can the quantum natureof matter be employed to push the bounds on possible violations of EEP tohitherto unseen lower limits?In this contribution I shall discuss some aspects related to this question<strong>and</strong>, in particular, to a recent claim [15], according to which atom interferometricgravimeters have actually already tested the weakest part of EEP,the universality of gravitational redshift, <strong>and</strong> thereby improved the validityof EEP by about four orders of magnitude! I will come to the conclusionthat this claim is unwarranted. 1 But before I do this in some detail, I give ageneral discussion of the Einstein Equivalence principle, its separation intovarious sub principles <strong>and</strong> the logical connection between them, <strong>and</strong> the importof one of these sub principles, the Universality of Free Fall (UFF), into<strong>Quantum</strong> Mechanics.2. Some backgroundThe theory of General Relativity rests on a number of hypotheses, the mostfundamental of which ensure, first of all, that gravity can be described by ametric theory [27, 30]. Today these hypotheses are canonised in the EinsteinEquivalence Principle (EEP). 2 EEP consists of three parts:UFF: The Universality of Free Fall. UFF states that free fall of “test particles”(further remarks on that notion will follow) only depend on theirinitial position <strong>and</strong> direction in spacetime. Hence test particles define a pathstructure on spacetime in the sense of [6, 3] which, at this stage, need notnecessarily be that of a linear connection. In a Newtonian setting UFF statesthat the quotient of the inertial <strong>and</strong> gravitational mass is a universal constant,i.e. independent of the matter the test particle is made of. UFF is alsooftencalledtheWeak Equivalence Principle, abbreviated by WEP, but weshall stick to the label UFF which is more telling.Possible violations of UFF are parametrised by the Eötvös factor, η,which measures the difference in acceleration of two test masses made ofmaterials A <strong>and</strong> B:η(A, B) =2·|a(A) − a(B)|a(A)+a(B) ≈ ∑ α(Eα (A)η αm i (A)c − E )α(B)2 m i (B)c 2 . (2.1)1 This is the view that I expressed in my original talk for the same reasons as those laid outhere. At that time the brief critical note [32] <strong>and</strong> the reply [16] by the original proponentshad appeared in the Nature issue of September 2nd. In the meantime more critique hasbeen voiced [33, 23], though the original claim seems to be maintained by <strong>and</strong> large [11].2 Note that EEP st<strong>and</strong>s for “the Einstein Equivalence Principle” <strong>and</strong> not “Einstein’s EquivalencePrinciple” because Einstein never expressed it in the modern canonised form.
Atom Interferometry 347The second <strong>and</strong> approximate equality arises if one supposes that violationsoccur in a specific fashion, for each fundamental interaction (labelled by α)separately. More specifically, one expresses the gravitational mass of the testparticle made of material A in terms of its inertial mass <strong>and</strong> a sum of corrections,one for each interaction α, each being proportional to the fractionthat the α’s interaction makes to the total rest energy (cf. Sect. 2.4 of [30]):m g (A) =m i (A)+ ∑ E α (A)η αmα i (A)c 2 . (2.2)Here the η α are universal constants depending only on the interaction butnot on the test particle. Typical numbers from modern laboratory tests, usingrotating torsion balances, are below the 10 −12 level. Already in 1971 Braginsky<strong>and</strong> Panov claimed to have reached an accuracy η(Al, Pt) < 9×10 −13 forthe element pair Aluminium <strong>and</strong> Platinum [1]. Currently the lowest boundis reached for the elements Beryllium <strong>and</strong> Titanium [25]. 3η(Be, Ti) < 2.1 × 10 −13 . (2.3)Resolutions in terms of η α ’s of various tests are discussed in [30]. Future testslike MICROSCOPE (“MICRO-Satellite àtraînée Compensée pour l’Observationdu Principe d’Equivalence”, to be launched in 2014) aim at a lower boundof 10 −15 . It is expected that freely falling Bose-Einstein condensates will alsoallow precision tests of UFF, this time with genuine quantum matter [28].LLI: Local Lorentz Invariance. LLI states that local non-gravitational experimentsexhibit no preferred directions in spacetime, neither timelike nor spacelike.Possible violations of LLI concern, e.g., orientation-dependent variationsin the speed of light, measured by Δc/c, or the spatial orientation-dependenceof atomic energy levels. In experiments of the Michelson-Morley type, wherec is the mean for the round-trip speed, the currently lowest bound from laboratoryexperiments based on experiments with rotating optical resonatorsis [10]:Δcc < 3.2 × 10−16 . (2.4)Possible spatial orientation-dependencies of atomic energy levels have alsobeen constrained by impressively low upper bounds in so-called Hughes-Drever type experiments.LLP: Local Position Invariance. LPI is usually expressed by saying that “Theoutcome of any local non-gravitational experiment is independent of where<strong>and</strong> when in the universe it is performed” ([31], Sect. 2.1). However, in almostall discussions this is directly translated into the more concrete Universalityof Clock Rates (UCR) or the Universality of Gravitational redshift (UGR),which state that the rates of st<strong>and</strong>ard clocks agree if taken along the sameworld line (relative comparison) <strong>and</strong> that they show the st<strong>and</strong>ard redshift if3 Besides for technical experimental reasons, these two elements were chosen to maximisethe difference in baryon number per unit mass.
348 D. Giulinitaken along different worldlines <strong>and</strong> intercompared by exchange of electromagneticsignals. Suppose a field of light rays intersect the timelike worldlinesγ 1,2 of two clocks, the four-velocities of which are u 1,2 . Then the ratio of theinstantaneous frequencies measured at the intersection points of one integralcurve of k with γ 1 <strong>and</strong> γ 2 isν 2= g(u 2,k)| γ2. (2.5)ν 1 g(u 1 ,k)| γ1Note that this does not distinguish between gravitational <strong>and</strong> Doppler shifts,which would be meaningless unless a local notion of “being at rest” wereintroduced. The latter requires a distinguished timelike vector field, as e.g.in stationary spacetimes with Killing field K. Then the purely gravitationalpart of (2.5) is given in case both clocks are at rest, i.e. u 1,2 = K/‖K‖| γ1,2 ,where γ 1,2 are now two different integral lines of K <strong>and</strong> ‖K‖ := √ g(K, K):ν 2:= g( k, K/‖K‖ ) √| γ2ν 1 g ( k, K/‖K‖ ) g(K, K)| γ1=. (2.6)| γ1g(K, K)| γ2The last equality holds since g(k, K) isconstantalongtheintegralcurvesofk, sothatg(k, K)| γ1 = g(k, K)| γ2 in (2.6), as they lie on the same integralcurve of k. Writingg(K, K) =:1+2U/c 2 <strong>and</strong> assuming U/c 2 ≪ 1, we getΔνν := ν 2 − ν 1= − U 2 − U 1. (2.7)ν 1 c 2Possible deviations from this result are usually parametrised by multiplyingthe right-h<strong>and</strong> side of (2.7) with (1 + α), where α = 0 in GR. In case ofviolations of UCR/UGR, α may depend on the space-time point <strong>and</strong>/or onthe type of clock one is using. The lowest upper bound on α to date for comparing(by electromagnetic signal exchange) clocks on different worldlinesderives from an experiment made in 1976 (so-called “<strong>Gravity</strong> Probe A”) bycomparing a hydrogen-maser clock in a rocket, that during a total experimentaltime of 1 hour <strong>and</strong> 55 minutes was boosted to an altitude of about10 000 km, to a similar clock on the ground. It led to [29]α RS < 7 × 10 −5 . (2.8)The best relative test, comparing different clocks (a 199 Hg-based optical clock<strong>and</strong> one based on the st<strong>and</strong>ard hyperfine splitting of 133 Cs) along the (almost)same worldline for six years gives [8]α CR < 5.8 × 10 −6 . (2.9)Here <strong>and</strong> above “RS” <strong>and</strong> “CR” refer to “redshift” <strong>and</strong> “clock rates”, respectively,a distinction that we prefer to keep from now on in this paper, althoughit is not usually made. To say it once more: α RS parametrises possible violationsof UGR by comparing identically constructed clocks moving alongdifferent worldlines, whereas α CR parametrises possible violations of UCR bycomparing clocks of different construction <strong>and</strong>/or composition moving moreor less on the same worldline. An improvement in putting upper bounds on
Atom Interferometry 349α CR , aiming for at least 2 × 10 −6 , is expected from ESA’s ACES mission(ACES = Atomic Clock Ensemble in Space), in which a Caesium clock <strong>and</strong> aH-maser clock will be flown to the Columbus laboratory at the InternationalSpace Station (ISS), where they will be compared for about two years [2].Remark 2.1. The notion of “test particles” essentially used in the formulationof UFF is not without conceptual dangers. Its intended meaning is that ofan object free of the “obvious” violations of UFF, like higher multipole momentsin its mass distribution <strong>and</strong> intrinsic spin (both of which would coupleto the spacetime curvature) <strong>and</strong> electric charge (in order to avoid problemswith radiation reaction). Moreover, the test mass should not significantlyback-react onto the curvature of spacetime <strong>and</strong> should not have a significantmass defect due to its own gravitational binding. It is clear that the simultaneousfulfilment of these requirements will generally be context-dependent.For example, the Earth will count with reasonable accuracy as a test particleas far as its motion in the Sun’s gravitational field is concerned, but certainlynot for the Earth-Moon system. Likewise, the notion of “clock” usedin UCR/UGR intends to designate a system free of the “obvious” violations.In GR a “st<strong>and</strong>ard clock” is any system that allows to measure the length oftimelike curves. If the curve is accelerated it is clear that some systems ceaseto be good clocks (pendulum clocks) whereas others are far more robust. Animpressive example for the latter is muon decay, where the decay time isaffected by a fraction less than 10 −25 at an acceleration of 10 18 g [7]. On theother h<strong>and</strong>, if coupled to an accelerometer, eventual disturbances could inprinciple always be corrected for. At least as far as classical physics is concerned,there seems to be no serious lack of real systems that classify as testparticles <strong>and</strong> clocks in contexts of interest. But that is a contingent propertyof nature that is far from self-evident.Remark 2.2. The lower bounds for UFF, LLI, <strong>and</strong> UCR/UGR quoted aboveimpressively show how much better UFF <strong>and</strong> LLI are tested in comparisonto UCR/UGR. This makes the latter the weakest member in the chain thatconstitutes EEP. It would therefore be desirable to significantly lower theupper bounds for violations of the latter. Precisely this has recently (February2010) been claimed in [15] by remarkable four orders in magnitude - <strong>and</strong>without doing a single new experiment! This will be analysed in detail below.It can be carefully argued for (though not on the level of a mathematicaltheorem) that only metric theories can comply with EEP. In particular, theadditional requirements in EEP imply that the path structure implied byUFF alone must be that of a linear connection. Metric theories, on the otherh<strong>and</strong>, are defined by the following properties (we state them with slightlydifferent wordings as compared to [31], Sect. 2.1):M1. Spacetime is a four-dimensional differentiable manifold, which carries ametric (symmetric non-degenerate bilinear form) of Lorentzian signature,i.e. (−, +, +, +) or (+, −, −, −), depending on convention 4 .4 Our signature convention will be the “mostly minus” one, i.e. (+, −, −, −).
350 D. GiuliniM2. The trajectories of freely falling test bodies are geodesics of that metric.M3. With reference to freely falling frames, the non-gravitational laws ofphysics are those known from Special Relativity.This canonisation of EEP is deceptive insofar as it suggests an essentiallogical independence of the individual hypotheses. But that is far from true.In fact, in 1960 the surprising suggestion has been made by Leonard Schiffthat UFF should imply EEP, <strong>and</strong> that hence UFF <strong>and</strong> EEP should, in fact,be equivalent; or, expressed differently, UFF should already imply LLI <strong>and</strong>LPI. This he suggested in a “note added in proof” at the end of his classicpaper [24], in the body of which he asked the important question whether thethree classical “crucial tests” of GR were actually sensitive to the precise formof the field equations (Einstein’s equations) or whether they merely testedthe more general equivalence principle. He showed that the gravitational redshift<strong>and</strong> the deflection of light could be deduced from EEP <strong>and</strong> that onlythe correct evaluation of the precession of planetary orbits needed an inputfrom Einstein’s equations. If true, it follows that any discrepancy betweentheory <strong>and</strong> experiment would have to be reconciled with the experimentallywell-established validity of the equivalence principle <strong>and</strong> special relativity.Hence Schiff concludes:“By the same token, it will be extremely difficult to design a terrestrialor satellite experiment that really tests general relativity,<strong>and</strong> does not merely supply corroborative evidence for the equivalenceprinciple [meaning UFF; D.G.] <strong>and</strong> special relativity. Toaccomplish this it will be necessary either to use particles of finiterest mass so that the geodesic equation my be confirmed beyondthe Newtonian approximation, or to verify the exceedingly smalltime or distance changes of order (GM/c 2 r) 2 . For the latter therequired accuracy of a clock is somewhat better than one part in10 18 .”Note that this essentially says that testing GR means foremost to test UFF,i.e. to perform Eötvös-type experiments.This immediately provoked a contradiction by Robert Dicke in [5],who read Schiff’s assertions as “serious indictment of the very expensivegovernment-sponsored program to put an atomic clock into an artificial satellite”.For, he reasoned, “If Schiff’s basic assumptions are as firmly establishedas he believes, then indeed this project is a waste of government funds.” Dickegoes on to point out that for several reasons UFF is not as well tested bypast Eötvös-type experiments as Schiff seems to assume <strong>and</strong> hence argues instrong favour of the said planned tests.As a reaction to Dicke, Schiff added in proof the justifying note alreadymentioned above. In it he said:“The Eötvös experiment show with considerable accuracy that thegravitational <strong>and</strong> inertial masses of normal matter are equal. Thismeans that the ground-state eigenvalue of the Hamiltonian for thismatter appears equally in the inertial mass <strong>and</strong> in the interaction
Atom Interferometry 351of this mass with a gravitational field. It would be quite remarkableif this could occur without the entire Hamiltonian being involvedin the same way, in which case a clock composed of atoms whosemotions are determined by this Hamiltonian would have its rateaffected in the expected manner by a gravitational field.”This is the origin of what is called Schiff’s conjecture in the literature. Attemptshave been made to “prove” it in special situations [14], but it is wellknown not to hold in mathematical generality. For example, consider gravity<strong>and</strong> electromagnetism coupled to point charges just as in GR, but now makethe single change that the usual Lagrangian density − 1 4 F abF ab for the freeelectromagnetic field is replaced by − 1 4 Cab cd F ab F cd , where the tensor fieldC (usually called the constitutive tensor; it has the obvious symmetries ofthe Riemann tensor) can be any function of the metric. It is clear that thischange implies that for general C the laws of (vacuum) electrodynamics ina freely falling frame will not reduce to those of Special Relativity <strong>and</strong> that,accordingly, Schiff’s conjecture cannot hold for all C. In fact, Ni proved [18]that Schiff’s conjecture holds iffC ab cd = 1 2(g ac g bd − g ad g bc) + φε abcd , (2.10)where φ is some scalar function of the metric.Another <strong>and</strong> simpler reasoning, showing that UFF cannot by itself implythat gravity is a metric theory in the semi-Riemannian sense (rather than,say, of Finslerian type) is the following: Imagine the ratio of electric charge<strong>and</strong> inertial mass were a universal constant for all existing matter <strong>and</strong> that afixed electromagnetic field existed throughout spacetime. Test particles wouldmove according to the equationẍ a +Γ a bcẋ b ẋ c =(q/m)F a b ẋ b , (2.11)where the Γ’s are the Christoffel symbols for the metric <strong>and</strong> (q/m) isthesaid universal constant. This set of four ordinary differential equations forthe four functions x a clearly defines a path structure on spacetime, but for ageneral F ab there will be no semi-Riemannian metric with respect to which(2.11) is the equation for a geodesic. Hence Schiff’s conjecture should at bestbe considered as a selection criterion.2.1. LLI <strong>and</strong> UGRWe consider a static homogeneous <strong>and</strong> downward-pointing gravitational field⃗g = −g⃗e z . We follow Section 2.4 of [30] <strong>and</strong> assume the validity of UFF <strong>and</strong>LLI but allow for violations of LPI. Then UFF guarantees the local existenceof a freely-falling frame with coordinates {x μ f}, whose acceleration is the same
352 D. Giulinias that of test particles. For a rigid acceleration we havect f =(z s + c 2 /g) sinh(gt s /c) ,x f = x s ,y f = y s ,z f =(z s + c 2 /g) coth(gt s /c) .LLI guarantees that, locally, time measured by, e.g., an atomic clock is proportionalto Minkowskian proper length in the freely falling frame. If weconsider violations of LPI, the constant of proportionality might depend onthe space-time point, e.g. via dependence on the gravitational potential φ, aswell as the type of clock:]c 2 dτ 2 = F 2 (φ) [ c 2 dt 2 f − dx 2 f − dy 2 f − dz 2 f= F 2 (φ)[ (1+ gz ) ]2s c 2 dt 2c 2 s − dx 2 s − dys 2 − dzs2(2.12). (2.13)The same time interval dt s = dt s (z s (1) )=dt s (z s(2) ) on the two static clocksat rest wrt. {x μ s }, placed at different heights z s (1) <strong>and</strong> z s(2) , correspond todifferent intervals dτ (1) ,dτ (2) of the inertial clock, giving rise to the redshift(all coordinates are {x μ s } now, so we drop the subscript s):ζ := dτ (2) − dτ (1)dτ (1) = F (z(2) )(1 + gz (2) /c 2 )F (z (1) )(1 + gz (1) /c 2 ) − 1 . (2.14)For small Δz = z (2) − z (1) this gives to first order in ΔzΔζ =(1+α)gΔz/c 2 , (2.15)whereα = c2 (⃗ez ·g⃗∇ ln(F ) ) (2.16)parametrises the deviation from the GR result. α may depend on position,gravitational potential, <strong>and</strong> the type of clock one is using.2.2. Energy conservation, UFF, <strong>and</strong> UGRIn this subsection we wish to present some well-known gedanken-experimenttypearguments [19, 9] according to which there is a link between violationsof UFF <strong>and</strong> UGR, provided energy conservation holds. Here we essentiallypresent Nordtvedt’s version; compare Figure 1.We consider two copies of a system that is capable of 3 energy statesA, B, <strong>and</strong>B ′ (white, light grey, grey), with E A
Atom Interferometry 353heightBAhν g A g BBA B ′ B A AT 1 T 2 T 3 T 4timeFigure 1. Nordtvedt’s gedanken experiment. Two systemsin three different energy states A, B, <strong>and</strong>C are consideredat four different times T 1 ,T 2 ,T 3 ,<strong>and</strong>T 4 . The initial state atT 1 <strong>and</strong> the final state at T 4 are identical, which, by meansof energy conservation, leads to an interesting quantitativerelation between possible violations of UFF <strong>and</strong> UGR.inner state A <strong>and</strong> has hit system 1 inelastically, leaving one system in stateA <strong>and</strong> at rest, <strong>and</strong> the other system in state B with an upward motion. Byenergy conservation this upward motion has a kinetic energy ofE kin = M A g A h +(E B ′ − E B ) . (2.17)This upward motion is a free fall in a gravitational field <strong>and</strong> since the systemis now in an inner state B, it is decelerated with modulus g B ,whichmaydiffer from g A .AtT 4 the system in state B has climbed to the height h,which must be the same as the height at the beginning, again by energyconservation. Hence we have E kin = M B g B h, <strong>and</strong> since moreover E B ′ −E B =(M B ′ − M B )c 2 ,wegetM A g A h + M B ′c 2 = M B c 2 + M B g B h. (2.18)Thereforeδνν := (M B ′ − M a) − (M B − M A )M B − M A= g Bhc 2 [1+]M A g B − g A,M B − M A g B(2.19)so thatM A g B − g A δg/gα ==:M B − M A g B δM/M . (2.20)This equation gives a quantitative link between violations of UFF, hererepresented by δg, <strong>and</strong> violations of UGR, here represented by α. The strengthof the link depends on the fractional difference of energies/masses δM/M,which varies according to the type of interaction that is responsible for thetransitions. Hence this equation can answer the question of how accurate atest of UGR must be in order to test the metric nature of gravity to the samelevel of accuracy than Eötvös-type experiments. Given that for the latterwe have δg/g < 10 −13 , this depends on the specific situation (interaction)through δM/M. For atomic clocks the relevant interaction is the magnetic
354 D. Giulinione, since the energies rearranged in hyperfine transitions correspond to magneticinteractions. The variation of the magnetic contribution to the overallself-energy between pairs of chemical elements (A, B) for which the Eötvösfactor has been strongly bounded above, like Aluminium <strong>and</strong> Gold (or Platinum),have been (roughly) estimated to be |δM/M| ≈5 × 10 −5 [19]. Henceone needed a precision of α RS < 5 × 10 −9 for a UGR test to match existingUFF tests.3. UFF in <strong>Quantum</strong> MechanicsIn classical mechanics the universality of free fall is usually expressed asfollows: We consider Newton’s Second Law for a point particle of inertialmass m i ,⃗F = m i¨⃗x, (3.1)<strong>and</strong> specialise it to the case in which the external force is gravitational, i.e.⃗F = F ⃗ grav ,where⃗F = m g ⃗g. (3.2)Here m g denotes the passive gravitational mass <strong>and</strong> ⃗g : R 4 → R 3 is the(generally space <strong>and</strong> time dependent) gravitational field. Inserting (3.2) into(3.1) we get)mg¨⃗x(t) =(⃗g ( t, ⃗x(t) ) . (3.3)m iHence the solution of (3.3) only depends on the initial time, spatial position,<strong>and</strong> spatial velocity iff m g /m i is a universal constant, which by appropriatechoices of units can be made unity.This reasoning is valid for point particles only. But it clearly generalisesto the centre-of-mass motion of an extended mass distribution in case ofspatially homogeneous gravitational fields, where m i <strong>and</strong> m g are then thetotal inertial <strong>and</strong> total (passive) gravitational masses. For this generalisationto hold it need not be the case that the spatial distributions of inertial <strong>and</strong>gravitational masses are proportional. If they are not proportional, the bodywill deform as it moves under the influence of the gravitational field. If theyare proportional <strong>and</strong> the initial velocities of all parts of the body are thesame, the trajectories of the parts will all be translates of one another. If theinitial velocities are not the same, the body will disperse without the actionof internal cohesive forces in the same way as it would without gravitationalfield.There is no pointlike-supported wave packet in quantum mechanics.Hence we ask for the analogy to the situation just described: How does awave packet fall in a homogeneous gravitational field? The answer is givenby the following result, the straightforward proof of which we suppress.Proposition 3.1. ψ solves the Schrödinger Equation)i∂ t ψ =(− 2Δ − F2m ⃗ (t) · ⃗x ψ (3.4)i
Atom Interferometry 355iffψ = ( exp(iα) ψ ′) ◦ Φ −1 , (3.5)where ψ ′ solves the free Schrödinger equation (i.e. without potential). HereΦ:R 4 → R 4 is the following spacetime diffeomorphism (preserving time)Φ(t, ⃗x) = ( t, ⃗x + ξ(t) ⃗ ) , (3.6)where ⃗ ξ is a solution towith ⃗ ξ(0) = ⃗0, <strong>and</strong>α : R 4 → R given byα(t, ⃗x) = m i¨⃗ξ(t) = ⃗ F (t)/m i (3.7){˙⃗ξ(t) ·(⃗x + ⃗ ξ(t))−12∫ tdt ′ ‖ ˙⃗}ξ(t ′ )‖ 2 . (3.8)To clearly state the simple meaning of (3.5) we first remark that changingatrajectoryt↦→ |ψ(t)〉 of Hilbert-space vectors to t ↦→ exp ( iα(t) ) |ψ(t)〉results in the same trajectory of states, since the state at time t is faithfullyrepresented by the ray in Hilbert space generated by the vector (unobservabilityof the global phase). As our Hilbert space is that of square-integrablefunctions on R 2 ,onlythe⃗x-dependent parts of the phase (3.8) change theinstantaneous state. Hence, in view of (3.8), the meaning of (3.5) is that thestate φ at time t is obtained from the freely evolving state at time t withthe same initial data by 1) a boost with velocity ⃗v = ˙⃗ ξ(t) <strong>and</strong> 2) a spatialdisplacement by ξ(t). ⃗ In particular, the spatial probability distributionρ(t, ⃗x) :=ψ ∗ (t, ⃗x)ψ(t, ⃗x) is of the formρ = ρ ′ ◦ Φ −1 , (3.9)where ρ ′ is the freely evolving spatial probability distribution. This impliesthat the spreading of ρ is entirely that due to the free evolution.Now specialise to a homogeneous <strong>and</strong> static gravitational field ⃗g, suchthat F ⃗ = m g ⃗g; then⃗ξ(t) =⃗vt + 1 2 ⃗at2 (3.10)with⃗a =(m g /m i )⃗g. (3.11)In this case the phase (3.8) isα(t, ⃗x) = m {i⃗v · ⃗x + ( 12 v2 + ⃗a · ⃗x ) t + ⃗v · ⃗at 2 + 1 3 a2 t 3} , (3.12)where v := ‖⃗v‖ <strong>and</strong> a := ‖⃗a‖.As the spatial displacement ξ(t) ⃗ just depends on m g /m i ,sodoesthatpart of the spatial evolution of ρ that is due to the interaction with thegravitational field. This is a quantum-mechanical version of UFF. Clearly,the inevitable spreading of the free wave packet, which depends on m i alone,is just passed on to the solution in the gravitational field. Recall also thatthe evolution of the full state involves the ⃗x-dependent parts of the phase,which correspond to the gain in momentum during free fall. That gain due to
356 D. Giuliniacceleration is just the classical δ⃗p = m i ⃗at which, in view of (3.11), dependson m g alone.Other dependencies of physical features on the pair (m g ,m i )arealsoeasily envisaged. To see this, we consider the stationary case of (3.4), wherei∂ t is replaced by the energy E, <strong>and</strong> also take the external force to correspondto a constant gravitational field in negative z-direction: F ⃗ = −m g g⃗e z .The Schrödinger equation then separates, implying free motion perpendicularto the z-direction. Along the z-direction one gets( ) d2dζ 2 − ζ ψ =0, (3.13)withζ := κz − ε, (3.14)whereκ :=[ ] 1[ ] 12mi3m g g2mi3, ε := E · . (3.15) 2 m 2 gg 2 2Ai(ζ)ζFigure 2. Airy function from ζ = −10 to ζ =5.A solution to (3.13) that falls off for ζ →∞must be proportional tothe Airy function, a plot of which is shown in Figure 2.As has been recently pointed out in [12], it is remarkable that the penetrationdepth into the classically forbidden region, which is a simple functionof the length κ −1 , depends on the product of inertial <strong>and</strong> gravitational mass.Also, suppose we put an infinite potential barrier at z = 0. Then the energyeigenstates of a particle in the region z>0 are obtained by the requirementψ(0) = 0, hence ε = −z n ,wherez n < 0isthen-th zero of the Airy function.By (3.15) this gives[ ]m2g g 2 213E n =· (−z n ) . (3.16)2m iThe energy eigenvalues of this “atom trampoline” [12] depend on the combinationm 2 g/m i . In the gravitational field of the Earth the lowest-lying energies
Atom Interferometry 357have be realised with ultracold neutrons [17]; these energies are just a few10 −12 eV.In classical physics, the return-time of a body that is projected at levelz = 0 against the gravitational field ⃗g = −g⃗e z in positive z-direction, so thatit reaches a maximal height of z = h, isgivenby[ ] 1 [ ] 1mi2 2h2T ret =2· · . (3.17)m g gNow, in <strong>Quantum</strong> Mechanics we may well expect the return time to receivecorrections from barrier-penetration effects, which one expects to delay arrivaltimes. Moreover, since the penetration depth is a function of the productrather than the quotient of m g <strong>and</strong> m i , this correction can also be expectedto introduce a more complicated dependence of the return time on the twomasses. It is therefore somewhat surprising to learn that the classical formula(3.17) can be reproduced as an exact quantum-mechanical result [4]. This isnot the case for other shapes of the potential. A simple calculation confirmsthe intuition just put forward for a step potential, which leads to a positivecorrection to the classical return which is proportional to the quantummechanicalpenetration depth <strong>and</strong> also depends on the inertial mass. Similarthings can be said of an exponential potential (see [4] for details). Clearly,these results make no more proper physical sense than the notion of timingthat is employed in these calculations. This is indeed a subtle issue whichwe will not enter. Suffice it to say that [4] uses the notion of a “Peres clock”[20] which is designed to register <strong>and</strong> store times of flight without assuminglocalised particle states. The intuitive reason why barrier penetration doesnot lead to delays in return time for the linear potential may be read offFigure 2, which clearly shows that the Airy function starts decreasing beforethe classical turning point (ζ = 0) is reached. This has been interpreted assaying that there is also a finite probability that the particle is back-scatteredbefore it reaches the classical turning point. Apparently this just cancels theopposite effect from barrier penetration in case of the linear potential, thusgiving rise to an unexpectedly close analogue of UFF in <strong>Quantum</strong> Mechanics.4. Phase-shift calculation in non-relativistic <strong>Quantum</strong>MechanicsWe consider the motion of an atom in a static homogeneous gravitationalfield ⃗g = −g⃗e z . We restrict attention to the motion of the centre of massalong the z-axis, the velocity of which we assume to be so slow that theNewtonian approximation suffices. The centre of mass then obeys a simpleSchödinger equation in a potential that depends linearly on the centre-of-masscoordinate. This suggests to obtain (exact) solutions of the time-dependentSchrödinger equation by using the path-integral method; we will largely follow[26].
358 D. GiuliniThe time evolution of a Schrödinger wave function in position representationis given by∫ψ(z b ,t b )= dz a K(z b ,t b ; z a ,t a ) ψ(z a ,t a ) , (4.1)spacewhereK(z b ,t b ; z a ,t a ):=〈z b | exp ( −iH(t b − t a )/ ) |z a 〉 . (4.2)Here z a <strong>and</strong> z b represent the initial <strong>and</strong> final position in the vertical direction.The path-integral representation of the propagator is∫K(z b ,t b ; z a ,t a )= Dz(t) exp ( iS[z(t)]/ ) , (4.3)Γ(a,b)whereΓ(a, b) := { }z :[t a ,t b ] → M | z(t a,b )=z a,b (4.4)contains all continuous paths. The point is that the path integral’s dependenceon the initial <strong>and</strong> final positions z a <strong>and</strong> z b is easy to evaluate wheneverthe Lagrangian is a potential of at most quadratic order in the positions <strong>and</strong>their velocities:L(z,ż) =a(t)ż 2 + b(t)żz + c(t)z 2 + d(t)ż + e(t)z + f(t) . (4.5)Examples are: 1) The free particle, 2) particle in a homogeneous gravitationalfield, 3) particle in a rotating frame of reference.To see why this is true, let z ∗ ∈ Γ(a, b) denote the solution to theclassical equations of motion:δSδz(t) ∣ =0. (4.6)z(t)=z∗ (t)We parametrise an arbitrary path z(t) by its difference to the classical solutionpath; that is, we writez(t) =z ∗ (t)+ξ(t) (4.7)<strong>and</strong> regard ξ(t) as path variable:∫K(z b ,t b ; z a ,t a )= Dξ(t) exp ( iS[z ∗ + ξ]/ ) . (4.8)Γ(0,0)Taylor expansion around z ∗ (t) for each value of t, taking into account (4.6),gives}iK(z b ,t b ; z a ,t a )=exp{ S ∗(z b ,t b ; z a ,t a )∫{ ∫i× Dξ(t)exp dt [ a(t)Γ(0,0)˙ξ 2 + b(t) ˙ξξ + c(t)ξ 2]} (4.9).Γ(0,0)Therefore, for polynomial Lagrangians of at most quadratic order, the propagatorhas the exact representation}iK(z b ,t b ; z a ,t a )=F (t b ,t a )exp{ S ∗(z b ,t b ; z a ,t a ) , (4.10)
Atom Interferometry 359where F (t b ,t a ) does not depend on the initial <strong>and</strong> final position <strong>and</strong> S ∗ is theaction for the extremising path (classical solution). We stress once more that(4.10) is valid only for Lagrangians of at most quadratic order. Hence we mayuse it to calculate the exact phase change for the non-relativistic Schrödingerequation in a static <strong>and</strong> homogeneous gravitational field.5. Free fall in a static homogeneous gravitational fieldWe consider an atom in a static <strong>and</strong> homogeneous gravitational field ⃗g =−g⃗e z . We restrict attention to its centre-of-mass wave function, which werepresent as that of a point particle. During the passage from the initial to thefinal location the atom is capable of assuming different internal states. Thesechanges will be induced by laser interaction <strong>and</strong> will bring about changes inthe inertial <strong>and</strong> gravitational masses, m i <strong>and</strong> m g . It turns out that for thesituation considered here (hyperfine-split ground states of Caesium) thesechanges will be negligible, as will be shown in footnote 8. Hence we can modelthe situation by a point particle of fixed inertial <strong>and</strong> gravitational mass, whichwe treat as independent parameters throughout. We will nowhere assumem i = m g .5.1. Some background from GRIn General Relativity the action for the centre-of-mass motion for the atom(here treated as point particle) is (−m 2 c) times the length functional, wherem is the mass (here m i = m g = m):S = −mc∫ λ2λ 1√ ( )ẋαdλ g αβ x(λ) (λ)ẋ β (λ) , (5.1)where, in local coordinates, x α (λ) is the worldline parametrised by λ <strong>and</strong> g αβare the metric components. Specialised to static metricswe haveg = f 2 (⃗x)c 2 dt 2 − h ab (⃗x) dx a dx b , (5.2)√∫ t2S = −mc 2 dtf ( ⃗x(t) ) 1 − ĥ(⃗v,⃗v) , (5.3)t 1c 2where ⃗v := (v 1 ,v 2 ,v 3 )withv a = dx a /dt, <strong>and</strong>whereĥ is the “optical metric”of the space sections t = const:ĥ ab := h abf 2 . (5.4)This is valid for all static metrics. Next we assume the metric to be spatiallyconformally flat, i.e., h ab = h 2 δ ab , or equivalentlyĥ ab = ĥ2 δ ab , (5.5)
360 D. Giuliniwith ĥ := h/f, so that the integr<strong>and</strong> (Lagrange function) of (5.3) takes theformL = −mc 2 f ( ⃗x(t) )√ 1 − ĥ2( ⃗x(t) ) ⃗v 2c , (5.6)2where ⃗v 2 := (v 1 ) 2 +(v 2 ) 2 +(v 3 ) 2 .We note that spherically symmetric metrics are necessarily spatially conformallyflat (in any dimension <strong>and</strong> regardless of whether Einstein’s equationsare imposed). In particular, the Schwarzschild solution is of that form, as ismanifest if written down in isotropic coordinates:[ 1 −r Sr] 2 [g =1+ r c 2 dt 2 − 1+ r ] 4S (dx 2 + dy 2 + dz 2) . (5.7)SrrHere r S is the Schwarzschild radius:r s := GM/2c 2 . (5.8)Hence, in this case,f = 1 − r S [r1+ r , h = 1+ r ][ ]2S1+r Sr 3S , ĥ =rr1 − r . (5.9)SrBack to (5.3), we now approximate it to the case of weak gravitationalfields <strong>and</strong> slow particle velocities. For weak fields, Einstein’s equations yieldto leading order:f =1+ φ c , h =1− φ 2φ, ĥ =1− 2 c2 c . (5.10)2Here φ is the Newtonian potential, i.e. satisfies Δφ =4πGT 00 /c 2 where Δ isthe Laplacian for the flat spatial metric. Inserting this in (5.6) the Lagrangiantakes the leading-order formL = −mc 2 + 1 2 mv2 − mφ . (5.11)We note that the additional constant m 2 c 2 neither influences the evolutionof the classical nor of the quantum-mechanical state. Classically this isobvious. <strong>Quantum</strong>-mechanically this constant is inherited with opposite signby the Hamiltonian:H = mc 2 + p22m+ mφ . (5.12)It is immediate that if ψ(t) is a solution to the time-dependent Schrödingerequation for this Hamiltonian, then ψ(t) := exp ( iα(t) ) ψ ′ (t) withα(t) =−t(mc 2 /), where ψ ′ solves the time-dependent Schrödinger equation withoutthe term mc 2 .Butψ <strong>and</strong> ψ ′ denote the same time sequence of states (rays).Hence we can just ignore this term. 55 In [15] this term seems to have been interpreted as if it corresponded to an inner degreeof freedom oscillating with Compton frequency, therefore making up a “Compton clock”.But as there is no periodic change of state associated to this term, it certainly does notcorrespond to anything like a clock (whose state changes periodically) in this model.
Atom Interferometry 3615.2. Interferometry of freely falling atomsWe now analyse the quantum mechanical coherences of the centre-of-massmotion in a static <strong>and</strong> homogeneous gravitational field, where we generaliseto m i ≠ m g . Hence, instead of (5.11) (without the irrelevant mc 2 term) wetakeL = 1 2 m iż 2 − m g gz . (5.13)Here we restricted attention to the vertical degree of freedom, parametrisedby z, whereż := dz/dt. This is allowed since the equation separates <strong>and</strong>implies free evolution in the horizontal directions. The crucial difference to(5.11) is that we do not assume that m i = m g .z|i〉D ′ ω 1 ω 2|g 1 〉C ′D|g 2 〉 |g 2 〉|g 2 〉|g 1 〉C⃗ k1⃗ k2B ′|g 1 〉A|g 2 〉B|g 1 〉0 T 2TtFigure 3. Atomic interferometer with beam splitters at A<strong>and</strong> D <strong>and</strong> mirrors at B <strong>and</strong> C, realised by π/2– <strong>and</strong> π–pulsesof counter-propagating laser beams with ⃗ k 1 = −k 1 ⃗e z ,wherek 1 = ω 1 /c <strong>and</strong> ⃗ k 2 = k 2 ⃗e z ,wherek 2 = ω 2 /c. |g 1,2 〉 denote thehyperfine doublet of ground states <strong>and</strong> |i〉 an intermediatestate via which the Raman transitions between the groundstates occur. The solid (bent) paths show the classical trajectoriesin presence of a downward pointing gravitational field⃗g = −g⃗e z , the dashed (straight) lines represent the classicaltrajectories for g = 0. The figure is an adaptation of Fig. 9in [26].The situation described in [15], which is as in [26], is depicted in Fig. 3.A beam of Caesium atoms, initially in the lower state |g 1 〉 of the hyperfinedoublet |g 1 〉, |g 2 〉 of ground states, is coherently split by two consecutivelaser pulses which we may take to act simultaneously at time t = T :Thefirst downward-pointing pulse with ⃗ k 1 = −k 1 ⃗e z ,wherek 1 := ω 1 /c, elevatesthe atoms from the ground state |g 1 〉 to an intermediate state |i〉, thereby
362 D. Giulinitransferring momentum Δ A1 ⃗p = −k 1 ⃗e z . The second upward-pointing pulse⃗ k2 = k 2 ⃗e z ,wherek 2 := ω 2 /c, induces a transition from |i〉 to the upperlevel of the hyperfine-split ground state, |g 2 〉. The emitted photon is pointingupwards so that the atom suffers a recoil by Δ A2 ⃗p = −k 2 ⃗e z . The totalmomentum transfer at A is the sum Δ A ⃗p := Δ A1 ⃗p +Δ A2 ⃗p = −κ⃗e z withκ = k 1 + k 2 = (ω 1 + ω 2 )/c. AtA the pulses are so adjusted that eachatom has a 50% chance to make this transition |g 1 〉→|i〉 →|g 2 〉 <strong>and</strong> proceedon branch AB <strong>and</strong> a 50% chance to just stay in |g 1 〉 <strong>and</strong> proceed onbranch AC. (So-called π/2–pulse or atomic beam splitter.) At B <strong>and</strong> C thepulses are so adjusted that the atoms make transitions |g 2 〉→|i〉 →|g 1 〉 <strong>and</strong>|g 1 〉→|i〉 →|g 2 〉 with momentum changes Δ B ⃗p = κ⃗e z <strong>and</strong> Δ C ⃗p = −κ⃗e zrespectively. Here the pulses are so adjusted that these transitions occur almostwith 100% chance. (So-called π–pulses or atomic mirrors.) Finally, atD, the beam from BD, which is in state |g 1 〉, receives another π/2-pulse sothat 50% of it re-unites coherently with the transmitted 50% of the beamincoming from CD that is not affected by the π/2–pulse at D. In total, themomentum transfers on the upper <strong>and</strong> lower paths are, respectively,Δ upper ⃗p =Δ C ⃗p = −κ⃗e} {{ z , (5.14a)}at CΔ lower ⃗p =Δ A ⃗p +Δ B ⃗p +Δ D ⃗p = −κ⃗e} {{ z +κ⃗e} z −κ⃗e} {{ } z . (5.14b)} {{ }at A at B at DFollowing [26] we now wish to show how to calculate the phase differencealong the two different paths using (4.10). For this we need to know theclassical trajectories.Remark 5.1. The classical trajectories are parabolic with downward accelerationof modulus ĝ. If the trajectory is a stationary point of the classical actionwe would have ĝ = g(m g /m i ). However, the authors of [15] contemplate thepossibility that violations of UGR could result in not making this identification.6 Hence we proceed without specifying ĝ until the end, which alsohas the additional advantage that we can at each stage cleanly distinguishbetween contributions from the kinetic <strong>and</strong> contributions from the potentialpart of the action. But we do keep in mind that (4.10) only represents theright dependence on (z a ,z b )ifz(t) =z ∗ (t) holds. This will be a crucial pointin our criticism to which we return later on.Now, the unique parabolic orbit with downward acceleration ĝ throughinitial event (t a ,z a ) <strong>and</strong> final event (t b ,z b )isz(t) =z a + v a (t − t a ) − 1 2ĝ(t − t a) 2 , (5.15)wherev a := z b − z at b − t a+ 1 2ĝ(t b − t a ) . (5.16)6 In [15] the quantity that we call ĝ is called g ′ .
Atom Interferometry 363Evaluating the action along this path givesSĝ(z b ,t b ; z a ,t a )= m i2(z b − z a ) 2t b − t a− m gg2 (z b + z a )(t b − t a )+ ĝ24 (t b − t a ) 3 (m i ĝ − 2m g g) .(5.17)Remark 5.2. Note the different occurrences of g <strong>and</strong> ĝ in this equation. Werecall that g is the parameter in the Lagrangian (5.13) that parametrises thegravitational field strength, whereas ĝ denotes the modulus of the downwardacceleration of the trajectory z(t) along which the atom actually moves, <strong>and</strong>along which the action is evaluated.Remark 5.3. In (5.17) it is easy to tell apart those contributions originatingfrom the kinetic term (first term in (5.13)) from those originating from thepotential term (second term in (5.13)): The former are proportional to m i ,the latter to m g .Remark 5.4. Note also that on the upper path ACD the atom changes theinternal state at C <strong>and</strong> on the lower path ABD it changes the internal stateat all three laser-interaction points A, B, <strong>and</strong>D. According to Special Relativity,the atom also changes its inertial mass at these point by an amountΔE/mc 2 ,whereΔE ≈ 4 × 10 −5 eV, which is a fraction of 3 · 10 −16 of Caesium’srest energy mc 2 . 7 Hence this change in inertial mass, as well as achange in gravitational mass of the same order of magnitude, can be safelyneglected without making any assumptions concerning the constancy of theirquotient m i /m g .Now we come back to the calculation of phase shifts. According to (4.10),we need to calculate the classical actions along the upper <strong>and</strong> lower paths.ΔS = S(AC)+S(CD) − ( S(AB)+S(BD) )(S(AC) − S(AB))+(S(CD) − S(BD)).(5.18)Remark 5.5. Since the events B <strong>and</strong> C differ in time from A by the sameamount T , <strong>and</strong> likewise B <strong>and</strong> C differ from D bythesameamountT ,wesee that the last term on the right-h<strong>and</strong> side of (5.17) drops out upon takingthe differences in (5.18), since it only depends on the time differences but noton the space coordinates. Therefore, the dependence on ĝ also drops out of(5.17), which then only depends on g.7 Recall that the “second” is defined to be the duration of ν =9, 192, 631, 770 cycles ofthe hyperfine structure transition frequency of Caesium-133. Hence, rounding up to threedecimal places, the energy of this transition is ΔE = h × ν = 4.136 · 10 −15 eV · s ×9, 193, 631, 770 s −1 = 3.802 · 10 −5 eV. The mass of Caesium is m = 132.905 u, where1 u = 931.494 · 10 6 eV/c 2 ; hence mc 2 =1.238 · 10 11 eV <strong>and</strong> ΔE/mc 2 =3.071 · 10 −16 .
364 D. GiuliniThe calculation of (5.18) is now easy. We write the coordinates of thefour events A, B, C, D in Fig. 3 as<strong>and</strong> getA =(z A ,t A =0), B =(z B ,t B = T ) ,C =(z C ,t C = T ) , D =(z D ,t D =2T ) (5.19)ΔS = m iT (z C − z B ) [ z B + z C − z A − z D − g(m g /m i )T 2] . (5.20)Since the curved (thick) lines in Fig. 3 are the paths with downward accelerationof modulus ĝ, whereas the straight (thin) lines correspond to the pathswithout a gravitational field, the corresponding coordinates are related asfollows 8 :A ′ = ( z A ′ = z A , ¯t A ′ =0 ) ,B ′ = ( z B ′ = z B + 1 2ĝT 2 ,t B ′ = T ) ,C ′ = ( z C ′ = z C + 1 2ĝT 2 ,t C ′ = T ) (5.21),D ′ = ( z D ′ = z D +2ĝT 2 ,t D ′ =2T ) .Hencez B + z C − z A − z D = z B ′ + z C ′ − z A ′ − z D ′ +ĝT 2 =ĝT 2 , (5.22)where z B ′ + z C ′ − z A ′ − z D ′ = 0 simply follows from the fact that A ′ B ′ C ′ D ′is a parallelogram. Moreover, for the difference (z C − z B )wehavez C − z B = z C ′ − z B ′ = κ T, (5.23)m i8 If we took into account the fact that the atoms change their inner state <strong>and</strong> consequentlytheir inertial mass m i , we should also account for the possibility that the quotient m i /m gmay change. As a result, the magnitude of the downward acceleration ĝ maydependontheinner state. These quantities would then be labelled by indices 1 or 2, according to whetherthe atom is in state |g 1 〉 or |g 2 〉, respectively. The modulus of the downward accelerationalong AC <strong>and</strong> BD is then ĝ 1 , <strong>and</strong> ĝ 2 along AB <strong>and</strong> CD. Also, the momentum transfersthrough laser interactions are clearly as before, but if converted into velocity changes byusing momentum conservation one has to take into account that the inertial mass changesduring the interaction. However, the z-component of the velocity changes at A (for that partof the incoming beam at A that proceeds on AB ′ ) <strong>and</strong> C ′ will still be equal in magnitude<strong>and</strong> oppositely directed to that at B ′ , as one can easily convince oneself; its magnitudebeing Δv = ( 1 − m )i1 vA + κ/m m i2 ,wherev A is the incoming velocity in A. As a result, iti2is still true that for laser-induced Raman transitions with momentum transfer κ at t =0<strong>and</strong> t = T the two beams from C ′ <strong>and</strong> B ′ meet at time t =2T at a common point, whichis z D ′ = z A +2v A T − κ + ( m i1− 1 ) v m i2 m A T . Switching on the gravitational field has thei2effect that z C = z C ′ − 1 2 ĝ1T 2 <strong>and</strong> z B = z B ′ − 1 2 ĝ2T 2 , but that the beam from C arrivesafter time T at z (C)Dafter time T at z (B)Dby Δz D := z (C)D= z D ′ − 1 2 (ĝ 1 +ĝ 2 )T 2 − m i1m i2ĝ 1 T 2 , whereas the beam from B arrives− z(B) D= z D ′ − 1 2 (ĝ 1 +ĝ 2 )T 2 − m i2m i1ĝ 2 T 2 , which differs from the former= T 2 (mi2m i1ĝ 2 − m i1m i2ĝ 1). Assuming that ĝ 1 =(m g1 /m i1 )g <strong>and</strong>ĝ 2 =(m g2 /m i2 )g, thisisΔz D =(m i2 m g2 − m i1 m g1 )gT 2 /(m i1 m i2 ) which, interestingly,vanishes iff the product (rather than the quotient) of inertial <strong>and</strong> gravitational mass staysconstant. If the quotient stays approximately constant <strong>and</strong> so that ĝ 1 =ĝ 2 =: ĝ, we writem i2 /m i1 =1+ε, withε =ΔE/m i1 c 2 , <strong>and</strong> get to first order in ε that Δz D ≈ 2εĝT 2 .
Atom Interferometry 365where the last equality follows from the fact that along the path AB ′ theatoms have an additional momentum of −κ⃗e z as compared to the atomsalong the path AC ′ ; compare (5.14b). Using (5.22) <strong>and</strong> (5.23) in (5.20), weget:ΔS = κT 2 [ ĝ − g(m g /m i ) ] . (5.24)As advertised in Remarks 5.1, 5.2, <strong>and</strong> 5.3, we can now state individually thecontributions to the phase shifts from the kinetic <strong>and</strong> the potential parts:(Δφ) time =+κT 2 ĝ, (5.25a)(Δφ) redshift = − κT 2 g(m g /m i ) .(5.25b)Here “time” <strong>and</strong> “redshift” remind us that, as explained in Section 5.1, thekinetic <strong>and</strong> potential energy terms correspond to the leading-order specialrelativistictime dilation (Minkowski geometry) <strong>and</strong> the influence of gravitationalfields, respectively.Finally we calculate the phase shift due to the laser interactions atA, B, C <strong>and</strong> D. For the centre-of-mass wave function to which we restrictattention here, only the total momentum transfers matter which were alreadystated in (5.14). Hence we get for the phase accumulated on the upper pathACD minus that on the lower path ABD:(Δφ) light = 1 { }(ΔC ⃗p) · ⃗z C − (Δ A ⃗p) · ⃗z A − (Δ B ⃗p) · ⃗z B − (Δ D ⃗p) · ⃗z D= −κ(z B + z C − z A − z D )=− κĝT 2 , (5.25c)whereweused(5.22)inthelaststep.Taking the sum of all three contributions in (5.25) we finally getΔφ = − m g· g · κ · T 2 . (5.26)m iThis is fully consistent with the more general formula derived by other methods(no path integrals) in [13], which also takes into account possible inhomogeneitiesof the gravitational field.5.3. Atom interferometers testing UFFEquation (5.26) is the main result of the previous section. It may be usedin various ways. For given knowledge of (m g /m i ) a measurement of Δφ maybe taken as a measurement of g. Hence the atom interferometer can be usedas a gravimeter. However, in the experiments referred to in [15] there wasanother macroscopic gravimeter nearby consisting of a freely falling cornercuberetroreflector monitored by a laser interferometer. If M i <strong>and</strong> M g denotethe inertial <strong>and</strong> gravitational mass of the corner cube, its acceleration inthe gravitational field will be ˜g =(M g /M i )g. The corner-cube accelerometerallows to determine this acceleration up to Δ˜g/˜g
366 D. Giuliniin which the left-h<strong>and</strong> side <strong>and</strong> the bracketed terms on the right-h<strong>and</strong> sideare either known or measured. Using the Eötvös ratio (2.1) for the Caesiumatom (A) <strong>and</strong> the reference cube (B)η(atom, cube) = 2 · (m g/m i ) − (M g /M i )(m g /m i )+(M g /M i ) , (5.28)we havem g M i= 2+ηm i M g 2 − η =1+η + O(η2 ) . (5.29)Hence, to first order in η := η(atom, cube), we can rewrite (5.27):Δφ = − (1 + η) · (˜gκT 2) . (5.30)This formula clearly shows that measurements of phase shifts can put upperbounds on η <strong>and</strong> hence on possible violations of UFF.The experiments [21, 22] reported in [15] led to a measured redshiftper unit length (height) which, compared to the predicted values, reads asfollows:ζ meas :=−ΔφκT 2 c 2 =(1.090 322 683 ± 0.000 000 003) × 10−16 · m −1 , (5.31a)ζ pred := ˜g/c 2 =(1.090 322 675 ± 0.000 000 006) × 10 −16 · m −1 . (5.31b)This implies an upper bound ofη(atom, cube) = ζ meas− 1 < (7 ± 7) × 10 −9 , (5.32)ζ predwhich is more than four orders of magnitude worse (higher) than the lowerbounds obtained by more conventional methods (compare (2.3)). However,it should be stressed that here a comparison is made between a macroscopicbody (cube) <strong>and</strong> a genuine quantum-mechanical system (atom) in a superpositionof centre-of-mass eigenstates, whereas other tests of UFF use macroscopicbodies describable by classical (non-quantum) laws.5.4. Atom interferometers testing URS?The foregoing interpretation seems straightforward <strong>and</strong> is presumably uncontroversial;but it is not the one adopted by the authors of [15]. Rather,they claim that a measurement of Δφ can, in fact, be turned into an upperbound on the parameter α RS which, according to them, enters the formula(5.30) for the predicted value of Δφ just in the same fashion as does η. Then,since other experiments constrain η to be much below the 10 −9 level, the verysame reasoning as above now leads to the upper bound (5.32) for α RS ratherthan η, which now implies a dramatic improvement of the upper bound (2.8)by four orders of magnitude!However, the reasoning given in [15, 16] for how α RS gets into (5.30)seems theoretically inconsistent. It seems to rest on the observation that(5.25a) cancels with (5.25c) irrespectively of whether ĝ =(m g /m i )g or not,so thatΔφ =(Δφ) redshift . (5.33)
Atom Interferometry 367Then they simply assumed that if violations of UGR existed (Δφ) redshift ,<strong>and</strong>hence Δφ, simply had to be multiplied by (1 + α RS ). (Our α RS is called β in[15].)Remark 5.6. The cancellation of (5.25a) with (5.25c) for ĝ ≠(m g /m i )g isformally correct but misleading. The reason is apparent from (4.10): Theaction has to be evaluated along the solution z ∗ (t) in order to yield the dynamicalphase of the wave function. Evaluating it along any other trajectorywill not solve the Schrödinger equation. Therefore, whenever ĝ ≠(m g /m i )g,the formal manipulations performed are physically, at best, undefined. Onthe other h<strong>and</strong>, if ĝ =(m g /m i )g, then according to (5.25)(Δφ) time = −(Δφ) redshift = −(Δφ) light , (5.34)so that we may just as well say that the total phase is entirely due to theinteraction with the laser, i.e. that we have instead of (5.33)Δφ =(Δφ) light (5.35)<strong>and</strong> no α RS will enter the formula for the phase.Remark 5.7. The discussion in Section 2.2 suggests that violations of UGRare quantitatively constrained by violations of UFF if energy conservationholds. If the upper bound for violations of UFF are assumed to be on the10 −13 level (compare (2.3)), this means that violations of UGR cannot exceedthe 10 −9 level for magnetic interactions. Since the latter is just the new levelallegedly reached by the argument in [15], we must conclude by Nordtvedt’sgedanken experiment that the violations of UGR that are effectively excludedby the argument of [15] are those also violating energy conservation.Remark 5.8. Finally we comment on the point repeatedly stressed in [15]that (4.10) together with the relativistic form of the action (5.1) shows thatthe phase change due to the free dynamics (Δφ) free := Δφ − (Δ) light ,whichin the leading order approximation is just the dt-integral over (5.11), can bewritten as the integral of the eigentime times the constant Compton frequencyω C = m i c 2 /:∫(Δφ) free = ω C dτ . (5.36)The authors of [15] interpret this as timing the length of a worldline by a“Compton clock” ([15], p. 927). 9 For Caesium atoms this frequency is about2 × 10 26 · s −1 , an enormous value. However, there is no periodic change ofany physical state associated to this frequency, unlike in atomic clocks, wherethe beat frequency of two stationary states gives the frequency by which thesuperposition state (ray in Hilbert space) periodically recurs. Moreover, thefrequency ω C apparently plays no rôle in any of the calculations performedin [15], nor is it necessary to express Δφ in terms of known quantities. Iconclude that for the present setting this reference to a “Compton clock” ismisleading.9 “The essential realisation of this Letter is that the non-relativistic formalism hides thetrue quantum oscillation frequency ω C .” ([15], pp. 928 <strong>and</strong> 930)
368 D. Giulini6. ConclusionI conclude from the discussion of the previous section that the arguments presentedin [15] are inconclusive <strong>and</strong> do not provide sufficient reason to claiman improvement on upper bounds on possible violations of UGR—<strong>and</strong> henceon all of EEP—by four orders of magnitude. This would indeed have been amajor achievement, as UCR/UGR is by far the least well-tested part of EEP,which, to stress it once more, is the connector between gravity <strong>and</strong> geometry.Genuine quantum tests of EEP are most welcome <strong>and</strong> the experiment describedin [15] is certainly a test of UFF, but not of UGR. As a test of UFFit is still more than four orders of magnitude away from the best non-quantumtorsion-balance experiments. However, one should stress immediately that itputs bounds on the Eötvös factor relating a classically describable piece ofmatter to an atom in a superposition of spatially localised states, <strong>and</strong> assuch it remains certainly useful. On the other h<strong>and</strong>, as indicated in Fig. 3,the atoms are in energy eigenstates |g 1,2 〉 between each two interaction pointson the upper <strong>and</strong> on the lower path. Hence we do not have a genuine quantumtest of UGR where a quantum clock (being in a superposition of energyeigenstates) is coherently moving on two different worldlines. This seems tohave been the idea of [15] when calling each massive system a “Comptonclock”. It remains to be seen whether <strong>and</strong> how this idea can eventually berealised with real physical quantum clocks, i.e. quantum systems whose state(ray!) is periodically changing in time. 10References[1] Vladimir B. Braginsky <strong>and</strong> Vladimir I. Panov. The equivalence of inertial <strong>and</strong>passive gravitational mass. General Relativity <strong>and</strong> Gravitation, 3(4):403–404,1972.[2] Luigi Cacciapuoti <strong>and</strong> Christophe Salomon. Space clocks <strong>and</strong> fundamentaltests: The aces experiment. The European Physical Journal - Special Topics,172(1):57–68, 2009.[3] Robert Alan Coleman <strong>and</strong> Herbert Korte. Jet bundles <strong>and</strong> path structures.Journal of <strong>Mathematical</strong> Physics, 21(6):1340–1351, 1980.[4] Paul C.W. Davies. <strong>Quantum</strong> mechanics <strong>and</strong> the equivalence principle. Classical<strong>and</strong> <strong>Quantum</strong> <strong>Gravity</strong>, 21(11):2761–2771, 2004.[5] Robert H. Dicke. Eötvös experiment <strong>and</strong> the gravitational red shift. AmericanJournal of Physics, 28(4):344–347, 1960.[6] Jürgen Ehlers <strong>and</strong> Egon Köhler. Path structures on manifolds. Journal of <strong>Mathematical</strong>Physics, 18(10):2014–2018, 1977.[7] Anton M. Eisele. On the behaviour of an accelerated clock. Helvetica PhysicaActa, 60:1024–1037, 1987.10 After this contribution had been finished I learned that the third version of [23] makessuggestions in precisely this direction (they call it “clock interferometry”).
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370 D. Giulini[26] Pippa Storey <strong>and</strong> Claude Cohen-Tannoudji. The Feynman path integral approachto atomic interferometry. A tutorial. Journal de Physique, 4(11):1999–2027, 1994.[27] Kip S. Thorne, David L. Lee, <strong>and</strong> Alan P. Lightman. Foundations for a theoryof gravitation theories. Physical Review D, 7(12):3563–3578, 1973.[28] Tim van Zoest. Bose-Einstein condensation in microgravity. Science, 328(5985):1540–1543, 2010.[29] Robert Vessot et al. Test of relativistic gravitation with a space-borne hydrogenmaser. Physical Review Letters, 45(26):2081–2084, 1980.[30] Clifford M. Will. <strong>Theory</strong> <strong>and</strong> Experiment in Gravitational Physics. CambridgeUniversity Press, Cambridge, 2nd revised edition, 1993. First edition 1981.[31] Clifford M. Will. The confrontation between general relativity <strong>and</strong> experiment.Living Reviews in Relativity, 9(3), 2006.[32] Peter Wolf, Luc Blanchet, Serge Bordé, Christian J. Reynaud, Christophe Salomon,<strong>and</strong> Claude Cohen-Tannoudji. Atom gravimeters <strong>and</strong> gravitational redshift.Nature, 467:E1, 2010.[33] Peter Wolf, Luc Blanchet, Serge Bordé, Christian J. Reynaud, Christophe Salomon,<strong>and</strong> Claude Cohen-Tannoudji. Does an atom interferometer test thegravitational redshift at the Compton frequency? arXiv:1012.1194.Domenico GiuliniInstitute for Theoretical PhysicsAppelstraße 2D-30167 HannoverGermanyCenter of Applied Space Technology <strong>and</strong> MicrogravityAm FallturmD-28359 BremenGermanye-mail: giulini@itp.uni-hannover.de
Indexn-point function, 51, 146, 233Abraham–Lorentz–Dirac equation, 305acceleration, 3, 346acceleration parameter, 27accelerator, 2ACES = Atomic Clock Ensemble in Space,349action, 145Baierlein–Sharp–Wheeler, 285diffeomorphism-invariant, 20Einstein–Hilbert, 4, 8, 19, 285action principle, 157admissible embedding, 16AdS, see anti-de SitterAdS/CFT correspondence, 10, 98Alex<strong>and</strong>rov topology, 80, 105algebrabosonic, see observable(s)fermionic, see observable(s)field algebra, 203Weyl, 198algebra of observables, see observablealgebraic quantum field theory, 16, 142,207amplitude, 60, 141amplitude map (GBF), 140angular momentum, 123, 338annihilation operator, 150anomalous acceleration, see galaxiesanomaly, 235conformal, 235curvature-induced, 235divergence, 235gluing, 140trace, 8, 235anthropic principle, 10anti-de Sitter spacetime, 10, 16, 100antinormal-ordered product, see productapproximation, 73Born–Oppenheimer, 7, 325Foldy–Wouthuysen, 2hydrodynamic, 60mean-field, 60Newtonian, 357semi-classical, 11, 61AQFT, see algebraic quantum field theoryAraki–Haag–Kastler axioms, see Haag–Kastlerareaof black hole horizon, 341quantized, 9surface, 9Arnowitt–Deser–Misner, 276arrow of time, see time, arrow ofAshtekar’s new variables, 8asymptotic safety, 5asymptotically flat, 338atom interferometer, 365atoms (in gravitational field), 2axiomatic quantum field theory, see AQFTbackground (-free, independence), 2–5,7, 10, 16, 19, 59, 139, 207, 231, 257Baierlein–Sharp–Wheeler action, see actionBarbero–Immirzi parameter, 8, 11barrier penetration, 357Batalin–Vilkovisky formalism, 19Bekenstein–Hawking entropy, see entropyBerezin–Toeplitz quantization, see quantizationbest matching, 267, 283BFV = Brunetti–Fredenhagen–Verch, seelocally covariant quantum field theorybig bang, 2, 26“big wave” theory, 25Birkhoff’s theorem, 124Bisognano–Wichmann theorem, 250black hole, 10evaporation, 11extremal, 11, 341generated in accelerator, 3in dimension > 4, 337interior, 2Kerr, 338Kerr–Newman, 121particle creation, 121primordial, 3radiation, 3Reissner–Nordström, 124, 327Schwarzschild, 3, 121, 342, 360singularity, 2stationary, 337surface, 3temperature, 3thermodynamics, 3Bogoliubov’s formula, 231Boltzmann, 3Born–Oppenheimer approximation, 7, 325F. Finster et al. (eds.), <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>and</strong> <strong>Gravity</strong>: <strong>Conceptual</strong> <strong>and</strong> <strong>Mathematical</strong>Advances in the Search for a Unified Framework, DOI 10.1007/978-3-0348-0043- 3,© Springer Basel AG 2012371
372 IndexBose–Einstein condensate, 347boson, 20massive vector, 192boundary, 10, 18, 239accessible, 106asymptotic Busemann, 111bundle, 100Busemann, 111Cauchy–Gromov, 109causal, 97conformal, 97future/past, 102geodesic, 100holographic, 98proper Gromov, 109boundary condition, 10MIT, 122Boyer–Lindquist coordinates, see coordinate(s)brane, see D-braneBRST method, 19BSW action, see Baierlein–Sharp–WheelerBuchdahl operator, see operatorBusemann completion, see completionBusemann function, 110BV, see Batalin–VilkoviskyC ∗ -algebra, 16, 186, 210c-boundary, see causal boundaryc-completion, see completioncanonical anticommutation relations =CAR, 183canonical commutation relations = CCR,9, 183canonical quantization, see quantizationcanonical quantum gravity, see quantumgravitycanonical variables, see variables, canonicalCAR, see canonical anticommutation relationsCAR representation, see representationCarter constant, 128Casimir energy density, 209Cauchy completion, see completionCauchy problem, 183, 302Cauchy surface, 16, 189, 213Cauchy–Gromov boundary, see boundarycausal boundary, see boundarycausal fermion system, 157discrete spacetime, 168particle representation, 161spacetime representation, 164causal sets, 4, 173causal variational principle, 157, 168causality axiom, 17causally compatible subset, 189causally convex, 16, 210CCR, see canonical commutation relationsCEAT, see Alex<strong>and</strong>rov topologycenter of expansion, 30CFT, see conformal field theorycharacteristicCauchy problem, 134hypersurface, 134charge, 121, 338point, see point chargechemical potential, 122chronology, 102“classical electron theory”, 303classical gravity, 20classical theory, 137classificationof black holes, 338CMB, see cosmic microwave backgroundcobordism, 18coherent factorization property, 152cohomology, 223, see also BRSTColella–Overhauser–Werner, see experimentcollapse of star, 121collapse of wave function, 179commutation relation(s), 150, see alsoCAR, CCRequal-time, 138compactificationEberlein–O’Neill, 110Gromov, 98, 108completionBusemann, 111c-, 102Cauchy, 98future/past, 102Gromov, 109complex structure, 148composition correspondence, 146Compton clock, 360, 367Compton frequency, see frequencyCompton wavelength, 4condensate, see spacetimecondensed-matter system, 60configuration space, 20, 52cartesian, 260relative, 260configuration variable, see variable, configurationconformal boundary, see boundary
Index 373conformal field theory, 10conformal superspace, see superspaceconformal time, see timeconifold, 339connection, 6, 8metric (causal fermion systems), 173spin, 8, 169, 172, 176connection variables, 43conservation law, 26constants, fundamental, 10constraint, 6, 8, 271closure, 48diffeomorphism = momentum, 6, 8, 9,19, 44Gauss, 9, 44, 286Hamiltonian, 6, 7, 9, 43quantum, 9, 53constraint equation(s), see constraintcontext category, 68continuum, 66continuum limit, see limitcoordinate transformation, 7coordinate(s)Boyer–Lindquist, 121isotropic, 360Regge–Wheeler, 122Weyl–Papapetrou, 340correspondence principle, 137, 146cosmic microwave backgroundanisotropy spectrum, 8cosmological constant, 25, 45, 208, 285cosmological principle, 25cosmological time, see timecosmology, 10, 11, 209, 229, 246, see alsost<strong>and</strong>ard modelquantum, 7, 42semi-classical, 236Coulomb force, 315Coulomb potential, see potentialcovariance, 17covariantperturbation theory, 4quantumgravity,4,8creation operator, 150curvature, 26, 54, 234, 245, 351discrete, 51, 173extrinsic, 6, 8scalar, 235, 342cylindrical function, 43, 55d’Alembert operator, see operatorD-brane, 11Dappiaggi–Fredenhagen–Pinamonti, 250dark energy, 25, 251daseinisation, 69Davies–Unruh temperature, see temperaturedcpo, 72de Broglie–Bohm–Dirac law of motion,300, 328de Broglie–Bohm–Klein–Gordon law ofmotion, 328de Sitter spacetime, 8, 16, 208decoherence, 11definite type, see operatordegrees of freedomfundamental (of QG), 11degrees of freedom, fundamental (of QG),11DeWitt metric, see supermetricdiamond, 220diffeomorphism constraint, see constraintdifference equation, 11Dirac equation, 2, 122, 158, 328Dirac Hamiltonian, see HamiltonianDirac operator, see Dirac equationDirac–Maxwell equations, 180discrete models, 4, 5discrete-continuum transition, 61discretization, 51, 93dissipation, 26distance, see also symmetrizeddivergences, 4, 5, 178domain theory, 66Doppler effect, 123, 348double-scaling limit, see limitdynamical locality, see localitydynamical triangulation, 5Eötvös factor, 346Eötvös-type experiment, see experimentedge length, 5EEP, see Einstein equivalence principleeffective (theory etc.), 5, 19, 58, 305, 318Einstein equation, 19, 25, 337, 360semi-classical, 233Einstein equivalence principle, 345Einstein–Euler equations, 29Einstein–Hilbert action, see actionEinstein–Infeld–Hoffmann theory, 312, 326Einstein–Maxwell–Born–Infeld theory, 300Einstein–Maxwell–Maxwell equations, 327electric field, colored, see fieldelectromagnetic field, see fieldelectron, extended, 305emergence of time, 265energy-momentum tensor, see stress-energytensorentropy, 26Bekenstein–Hawking, 3, 11
374 Indexentropy flux density, 242Epstein–Glaser renormalization, see renormalizationequilateral simplices, 5equilibrium, see stateEuclidean path integral, see path integralEuler characteristic, 46Euler operator, see operatorevent horizon, 338exp<strong>and</strong>ing wave solutions, see “big wave”theoryexpectation value, 143, 230vacuum, 147experiment, 1–3, 6Colella–Overhauser–Werner (COW), 3Eötvös-type (torsion balance), 347Hughes–Drever type, 347Michelson–Morley type, 347repeating of, 16, 209spaceshuttlehelium,5extended locality, see localityextra dimensions, 337fermion, 20, 121, see also causal fermionsystemfermionic operator, 162fermionic projector, 157, 166Feynman diagram, 45, 180Feynman rules, 4fieldP (φ), 230colored electric, 6Dirac, 230, 245electromagnetic, 230, 299free scalar, 223minimally coupled scalar, 232non-gravitational, 5, 7, 8Proca, 230quantized scalar, 230quantum, 230scalar, 11, 44, 208, 212, 230, 286Yang–Mills, 230, 286field algebra, see algebrafield quantization, see quantizationfield theory, see also quantumfree, 147scalar, 147Finsler geometry, 98, 112first quantization, see quantizationfixed point, ultraviolet, 5flux,6,43Fock representation, see representationFock space, 57, 149, 185Focker–Schwarzschild–Tetrode electrodynamics,306Foldy–Wouthuysen approximation, see approximationfoliation, 132, 300, 330constant mean curvature, 258Fourier transform, 47free fall, 346frequency, 44, 123, 348Compton, 367Friedmann–Robertson–Walker spacetime,11, 25, 250FRW, see Friedmann–Robertson–Walkerfuture setindecomposable (IF), 102proper (PIF), 102terminal (TIF), 102future-compact subset, 189galaxies, 25anomalous acceleration, 25galaxy cluster, 4, 30gauge group, 54gauge invariance, 9, 163, 223gauge theories, 180, 259gauge transformation, local, 163gauge, harmonic, 20Gauss constraint, see constraintGaussian measure, see measureGBF, see general boundary formulationgeneral boundary formulation, 139general relativity, 2, 31, 41, 138, 257, 3462-dimensional Riemannian, 453-dimensional Riemannian, 47geodesic principle, 264geometric quantization, see quantizationgeometrodynamics, 6, 42, 275quantum, 6GFT, see group field theoryghost field, 20ghost number, 20global hyperbolicity, 16Global Positioning System, see GPSgluing-anomaly factor, 140GNS representation, 222GNS vector, 221GPS, 2GR, see general relativitygraph (combinatorial), 43gravimeter, 365gravitationeffective/not fundamental, 4gravitational constant, 1gravitational field, 355external, 2
Index 375gravitational redshift, 2gravitational wave, 4graviton, 4, 9, 300gravity, 1semi-classical, 236<strong>Gravity</strong> Probe A, 348Green function/operator, 45, 190, 212Green-hyperbolic operator, see operatorGromov compactification, see compactificationgroup field theory, 4, 41Haag–Kastler axioms, 16, 208Hadamard condition, 233Hadamard manifold, 110Hadamard parametrix subtraction, seesymmetricHadamard state, see stateHamilton–Jacobi equation(s), 6, 7Hamilton–Jacobi law of motion, 300, 321“parallel-processing”, 323Hamiltonian, 7Dirac, 124formulation of GR, 6Hamiltonian constraint, see constraintHawking, 5, 11, see also entropyeffect, 3, 235radiation, 121temperature, 3, see temperaturehigher dimensions, 9higher tensor models, 47Hilbert space, 184, 200, 355causal fermion systems, 158formalism, 66in LQG, GFT, 9, 43in the GBF, 140Hodge–d’Alembert operator, see operatorholographic principle, 10holography, see boundaryholomorphic quantization, see quantizationholonomy, 6, 8, 43holonomy-flux representation, 9homology sphere, 343horizon, see event horizonouter, 122hot bang state, see stateHubble parameter, 8Hughes–Drever type experiments, see experimentHuygens’ principle, 123“hydrodynamic” theory of gravity, 4hydrodynamic approximation, 60hydrogen spectrum, 329hyperbolic embedding, 210, 230hypersurface, 139IF, see future setindecomposable future set, see future setindecomposable past set, see past setinflation, 8, 11, 251interaction(s), 347all, 1, 2, 4, 9weak, strong, 299interferometry, see atom interferometerinterval domain, 75intrinsic time, see time, intrinsicIP, see past setirreversibility, 26Jacobi’s principle, 266Kaluza–Klein theories, 337Kerr metric, see black holeKerr–Newman metric, see black holekinematics, 9Klein–Gordon equation, 44, 147, 212, 223,328KMS state, see stateKochen–Specker theorem, 69Krein space, 162Lagrangian, 98Lagrangian subspace, 148Lagrangian theories, 208L<strong>and</strong>au–Lifshitz equation, 306l<strong>and</strong>scape, 10lapse, 278large-N limit, see limitlaser, 3Lax–Glimm theory, 26lens space, 341Liénard–Wiechert fields, 302limitcontinuum, 42, 179double-scaling, 47large-N, 47thermodynamic, 60LLI, see local Lorentz invariancelocal correlation matrix, 168local correlation operator, 165local Lorentz invariance, 347local position invariance, 347local thermal equilibrium, see statelocality, 17dynamical, 220, 222extended, 221locally covariant quantum field theory,16, 183, 208, 229loop dynamics, 6
376 Indexloop order (of diagram), 4, 5, 8, 10loop quantum cosmology, see cosmologyloop quantum gravity, 8, 11, 15, 41Lorentz electrodynamics, 300Lorentz self-force, 303Lorenz–Lorentz gauge, 319LPI, see local position invarianceLQG, see loop quantum gravityLTB model, 11LTE, see local thermal equilibriumluminosity, 25M-theory, 4Mach’s principle, 257Maldacena, 10many-fingered time, 7many-particle physics, 60Mashhoon effect, 3mass, 37, 127, 224, 246, 286, 304, 338bare, 304gravitational, 347inertial, 347matrix model, 45Maxwell’s equations, 300Maxwell–Born–Infeld equations, 308Maxwell–Lorentz field equations, 301Maxwell–Maxwell field equations, 301mean-field approximation, 60measure, 54, 145discrete minimizing, 178Gaussian, 148path integral, 5universal, 161measurement, 66, 141, 179measurement(s), 138composition, 138consecutive, 138metric3-dimensional, 6Michelson–Morley type experiments, seeexperimentmicrolocal analysis, 16microlocal spectrum condition, see spectrumconditionMICROSCOPE, 347minisuperspace, see superspaceMinkowski spacetime, 3, 16, 44, 142, 158,208, 232, 291, 300MIT boundary condition, see boundaryconditionmomentum constraint, see constraintmomentum space, 147momentum transfer, 5momentum variable, see variable, momentumMonte-Carlo simulations, 5multi-diamond, 220Nakanishi–Lautrup field, 20naked singularity, see singularityneutron, 2, 357interferometry experiment, 3Newman–Penrose formalism, 126Newton potential, see potentialNewton’s first law, 258Newton’s second law, 265, 318Newtonian approximation, 357Newtonian gravity, 2, 259, 360Newtonian time, see timeno-space state, see statenon-analytic, 5non-commutative algebra, 138non-commutative geometry, 4non-gravitational field, 8non-perturbative, 5string theory, 10non-relativistic theory, 137non-renormalizable, 4, 5, see renormalizationNordtvedt’s gedanken experiment, 352normal-ordered product, see productnormal-ordered quantization, see quantizationnormally hyperbolic, see operatorobservable(s)algebra, 16, 132, 137, 183, 210algebra, bosonic, 183algebra, fermionic, 183composition, 143GFT, 51gluing of, 143thermal, 236thermometer, 237Occam’s razor, 37one-field-two-field model, 216operational meaningoperator product, 138operatorBuchdahl, 194d’Alembert, 192Dirac-type, 183, 193Euler, 194Hodge–d’Alembert, 192of definite type, 200, 203Proca, 183, 192Rarita–Schwinger, 183, 195twisted Dirac, 194wave = normally hyperbolic, 191operator ordering, 53
Index 377operator product, 138particle concept, 16, 44particle representation (causal fermion system),see causal fermion systemparticle space, 161particle, point, see point particlepartition function, 45past setindecomposable (IP), 102proper (PIP), 102terminal (TIP), 102past-compact subset, 189path integral, 5, 8, 145, 358Euclidean, 5quantum gravity, 45Penrose diagram, 100Peres, 6Peresclock,357perturbation theory, 9, 19, 44, 180, 209,230covariant, 4phase space, see spacephase transition, 5, 60photon, 300, 352PIF, see future setPIP, see past setPlanck scale, see scalePoincaré covariance, 208Poincaré principle, 264point charge, 299point defect, 300point particle, 301Poisson bracket, 43polarization, 155Ponzano–Regge model, 51poset = partially ordered set, 67potentialCoulomb, 6Newton, 6pp-wave, 98, 115prediction, 3, 4, 6preparation, 141prism manifold, 343probability, 141probability interpretation, 16problem of time, 2Proca equation, see Proca operatorProca operator, see operatorproductantinormal-ordered, 151normal-ordered, 154propagator, 44, 122properly timelike separated points, 171QFT, see quantum field theory, see quantumfield theoryquantity-value object, 66quantization, 258Berezin–Toeplitz, 150bosonic, 196canonical, 42fermionic, 200field, 178first, 44geometric, 148, 155holomorphic, 148least invasive, 327normal-ordered, 154of gravity, 207Schrödinger–Feynman, 145second, 41third, 41quantization rules, 4, 6quantization scheme, 137quantum causality, 199, 202quantum constraint, see constraint,quantumquantum cosmology, 5, 11, see cosmology,66quantum energy inequalities, 246quantum field theory, 3, 7, 16, 43, 66,138, 178, 229, 299, see also locallycovariantperturbative, see perturbation theoryquantum geometrodynamics, 6, 9, 11quantum geometrycausal fermion systems, 169quantum gravity, 1, 4, 8, 10, 11, 15, 41,258, 299, 3462-dimensional Riemannian, 453-dimensional Riemannian, 474-dimensional, 52canonical, 4, 6, 8, 11, 42covariant,4,8perturbative, 4simplicial, 41quantum matter, 347quantum mechanicsnon-relativistic, 2, 346quantum of area, see area, quantizedquantum spacetime, see spacetimequantum theory, 1, 2, 66, 137quantum topology, 4question (observable), 141radiation phase, 26Rarita–Schwinger operator, see operatorreal numbers, 66redshift, 25, 347
378 Indexgravitational, 2Reeh–Schlieder property, 221Reeh–Schlieder theorem, 209Regge calculus, 5Regge–Wheeler coordinate, see coordinate(s)regularization, 5, 53, 303Reissner–Nordström metric, see black holerelational quantities, 258relative Cauchy evolution, 231renormalization, 16, 59, 154, 304ambiguity, 235, 252Epstein–Glaser, 19normal ordering, 232renormalization group, 5representationCAR, 132, 184Fermi–Dirac–Fock, 132Fock, 148Schrödinger, 145self-dual CAR, 186RG, see renormalization grouprigidity theorem, 338S-matrix, 16, 138, 145safety, see asymptotic safetySagnac effect, 3scalar curvature, see curvaturescalar field, see field, scalarscalegalactic, 2galaxy-cluster, 4macroscopic, 4, 5Planck, 1, 10small, 8string, 10scale factor, 246scale factor of the universe, 7, 11scale invariance, 273scattering amplitude, 5scattering theory, 121Schiff’s conjecture, 351Schrödinger equation, 7, 354Schrödinger representation, see representationSchrödinger–Feynman quantization, seequantizationSchwarzschild coordinates, 27Schwarzschild metric, see black holeScott topology, 73second quantization, see quantizationSeifert fiber space, 343self-similarity, 27semi-classicalapproximation, 11, 61expansion, 5limit, 7, 9, 11shape dynamics, 257shape space, 260sharp temperature, see temperatureshift, 278shock wave, 26SHP, see symmetric Hadamard parametrixsubtractionsigma model, 5, 340signature, 82simplex, 5simplicial geometry, 41simplicial quantum gravity, see quantumgravitysingularity, 11, 127, 246naked, 101theorems, 2spaceconfiguration, see configuration spacephase space, 43space of connections, 42space of geometries, 41spacelike compact subset, 189spacetime, 2, see backgroundas a condensate, 60discrete, 4, 157, 168four-dimensionality, 5microscopic two-dimensionality, 5quantum, 157strongly causal, 100spacetime representation (causal fermionsystem), see causal fermion systemSPASs (same physics in all spacetimes),214, 222special relativity, 138, 300specific heat exponent, 5spectral presheaf, 68spectrum condition, 16, 208microlocal, 233spectrum, discrete, 9spin 1 2 ,2,121spin 2, 4spin connection, see connection, spinspin dimension, 164spin foam, 4, 51spin network, 9, 11, 50spin structure, 132spin-statistics theorem, 184, 209splice map, 176spontaneous structure formation (causalfermion systems), 169SSC, see Schwarzschild coordinatesst<strong>and</strong>ard model of cosmology, 25
Index 379st<strong>and</strong>ard model of particle physics, 180,299statecoherent, 61, 147GBF, 139Hadamard, 235hot bang, 239initial/final, 141invariant, 230KMS, 122, 236local thermal equilibrium, 229locally covariant QFT, 230natural, 221no-space, 51object, 67quantum, 139sharp temperature, 244thermal, 121vacuum, 16, 208, 222, 232static, 98, 359stationary, 98statistical physics, 60Stone–von Neumann theorem, 9storage ring, see acceleratorstress-energy tensor, 19, 208, 214, 314quantized, 231string field theory, 10string length, 10string theory, 4, 9, 11, 15, 98non-perturbative, 10strong interaction, see interactionstrongly causal, see spacetimesubtheory, 215sum over histories, 44superfluid helium, 5supergravity, 4, 337supermetric, 7, 43, 265, 288supernova, 25superselection sectors, 209superspace, 42conformal, 277mini-, 45supersymmetry, 4, 9surface, 432-dimensional, 6of black hole, 3, 122surface area, 9symmetric Hadamard parametrix subtraction,244symmetrized distance, 113symmetry group of black hole, 338symplectic structure, 148synchronization map, 171temperature, 3, 122, 236Davies–Unruh, 3Hawking, 3, 250sharp, see stateUnruh, 3tensor functor, 17test particle, 302, 347theory of everything, 9, see also interactions,allthermal equilibrium, see statethermal observable, see observablethermal state, see statethermodynamic limit, see limitthermometer observable, see observable(s)third quantization, see quantizationTIF, see future settimearrow of, 11conformal, 246cosmological, 246external, 10intrinsic, 7many-fingered, 7Newtonian, 2problem of, 2time-evolution operator, 141time-ordered product, 138time-slice axiom, 16, 17, 199, 202, 213,231TIP, see past settopological field theory, 18topological quantum field theory, 139topology change, 41topos approach to quantum theory, 66trace anomaly, see anomaly, tracetransition amplitude, see amplitudetriad field, 43trivial theory, 221two-point function, see n-point functionUCR, see universality of clock ratesUFF, see universality of free fallUGR, see universality of gravitational redshiftultraviolet divergences, see divergencesultraviolet fixed point, see fixed pointunder-density of galaxies, 30universal measure, see measureuniversality of clock rates, 347universality of free fall, 346universality of gravitational redshift, 346universe, 25, 66, 209, 242expansion, 258Unruh effect, 235Unruh temperature, see temperatureunsharp values, 66
380 IndexUV, see ultravioletvacuum, 10, 16, 122, see also statevacuum Einstein equation, see equationvariable(s)canonical, 6configuration, 8momentum, 8, 9von Neumann algebra, 68wave operator, see operatorwave-particle duality, 179weak equivalence principle, see UFFweak gravitational field, 360weak interaction, see interactionWEP, see weak equivalence principleWeyl algebra, see algebraWeyl–Papapetrou coordinates, see coordinatesWheeler–DeWitt equation, 7, 8, 10, 11,42Wheeler–Feynman electrodynamics, 306Wick square, 236Wightman–Gårding axioms, 208winding number, 339WKBmethod,61Yamabe type, 342Yang–Mills field, see also fieldYang–Mills theory, 4, 230