Quantum gravity at a Lifshitz point

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PETR HOŘAVA PHYSICAL REVIEW D 79, 084008 (2009)ture—that of a codimension-one foliation [13]. This foliationstructure F is to be viewed as a part of the topologicalstructure of M, before any notion of a Riemannian metricis introduced. The leaves of this foliation are the hypersurfacesof constant time. Coordinate transformationsadapted to the foliation are of the form~x i ¼ ~x i ðx j ;tÞ; ~t ¼ ~tðtÞ: (13)Thus, the transition functions are foliation-preserving diffeomorphisms.We will denote the group of foliationpreservingdiffeomorphisms of M by Diff F ðMÞ. In thelocal adapted coordinate system, the infinitesimal generatorsof Diff F ðMÞ are given byx i ¼ i ðt; xÞ; t ¼ fðtÞ: (14)We will simplify our presentation by further assuming thatthe spacetime foliation is topologically given byM ¼ R ; (15)with all leaves of the foliation topologically equivalent to afixed D-dimensional manifold .Differential geometry of foliations is a well-developedbranch of mathematics, and represents the proper mathematicalsetting for the class of gravity theories studiedhere. We will not review the geometric theory of foliationsin any detail here, instead referring the reader to [14–16].For example, there are two natural classes of functions thatcan be defined on a foliation: In addition to functions thatare allowed to depend on all coordinates, there is a specialclass of functions which take constant values on each leafof the foliation. We will call such functions ‘‘projectable.’’Foliations can be equipped with a Riemannian structure.A Riemannian structure compatible with our codimensiononefoliation of M consists of three objects: g ij , N i , and N,with N a projectable function; both N and N i transform asvectors under the reparametrizations of time. As pointedout above, these fields can be viewed as a decomposition ofa Riemannian metric on M into the metric g ij inducedalong the leaves, the shift variable N i , and the lapse field N.The generators of Diff F ðMÞ act on the fields viag ij ¼ @ i k g jk þ @ j k g ik þ k @ k g ij þ f _g ij ;N i ¼ @ i j N j þ j @ j N i þ _j g ij þ fN _ i þ f N_i ;N ¼ j @ j N þ fN _ þ f N: _(16)In [3], these transformation rules were derived by startingwith the action of spacetime diffeomorphisms on the relativisticmetric in the ADM decomposition, and taking thec !1 limit. We also saw in [3] that N i and N can benaturally interpreted as gauge fields associated with thetime-dependent spatial diffeomophisms and the time reparametrizations,respectively. In particular, since N is thegauge field associated with the time reparametrization fðtÞ,it appears natural to restrict it to be a projectable functionon the spacetime foliation F .If we wish instead to treat N as an arbitrary function ofspacetime, we have essentially two options. First, we canallow an arbitrary spacetime-dependent N as a backgroundfield, but integrate only over space-independent fluctuationsof N in the path integral. As the second option, wewill encounter situations in which N must be allowed to bea general function of spacetime, because it participates inan additional gauge symmetry. When that happens, we willintegrate over the fluctuations of N in the path integral. Anexample of such an extra symmetry is the invariance underanisotropic Weyl transformations discussed in Sec. II C 3below, and in Sec. 5.2 of [3].B. LagrangiansWe formally define our quantum field theory of gravityby a path integral,ZDgij DN i DN expfiSg: (17)Here Dg ij DN i DN denotes the path-integral measurewhose proper treatment involves the Faddeev-Popov gaugefixing of the gauge symmetry Diff F ðMÞ, and S is the mostgeneral action compatible with the requirements of gaugesymmetry (and further restricted by unitarity). As is oftenthe case, this path integral is interpreted as the analyticcontinuation of the theory which has been Wick rotated toimaginary time ¼ it.Our next step is to construct the action S compatible withour symmetry requirements. For simplicity, we will assumethat all global topological effects can be ignored, freelydropping all total derivative terms and not discussing possibleboundary terms in the action. This is equivalent toassuming that our space is compact and its tangentbundle topologically trivial. The refinement of our constructionwhich takes into account global topology andboundary terms is outside of the scope of the present work.1. The kinetic termThe kinetic term in the action will be given by the mostgeneral expression which is (i) quadratic in first timederivatives _g ij of the spatial metric, and (ii) invariant underthe gauge symmetries of foliation-preserving diffeomorphismsDiff F ðMÞ. The object that transforms covariantlyunder Diff F ðMÞ is not _g ij , but instead the second fundamentalformK ij ¼ 12N ð _g ij r i N j r j N i Þ: (18)This tensor measures the extrinsic curvature of the leavesof constant time in the spacetime foliation F . In terms ofK ij and its trace K g ij K ij , the kinetic term is given by084008-4
QUANTUM GRAVITY AT A LIFSHITZ POINT PHYSICAL REVIEW D 79, 084008 (2009)S K ¼ 2 Zdtd D p 2 x ffiffiffi g NðKij K ij K 2 Þ: (19)S V ¼ Z dtd D px ffiffiffi g NV½gij Š; (23)This kinetic term contains two coupling constants: and .The dimension of depends on the spatial dimension D:Since the dimension of the volume element is½dtd D xŠ¼ D z; (20)and each time derivative contributes ½@ t Š¼z, the scalingdimension of is½Š ¼ z D : (21)2As intended, this coupling will be dimensionless in 3 þ 1spacetime dimensions if z ¼ 3.The presence of an additional, dimensionless coupling reflects the fact that each of the two terms in (19) isseparately invariant under Diff F ðMÞ. In other words, therequirement of Diff F ðMÞ symmetry allows the generalizedDe Witt ‘‘metric on the space of metrics’’G ijk‘ ¼ 1 2 ðgik g j‘ þ g i‘ g jk Þ g ij g k‘ (22)to contain a free parameter . It is this generalized De Wittmetric that defines the form quadratic in K ij which appearsin the kinetic term (see [3]).In general relativity, the requirement of invariance underall spacetime diffeomophisms forces ¼ 1. In our theorywith Diff F ðMÞ gauge invariance, represents a dynamicalcoupling constant, susceptible to quantum corrections.It is interesting to note that the kinetic term S K isuniversal, and independent of both the desired value of zand the dimension of spacetime. The only place where thevalue of z shows up in S K is in the scaling dimension of theintegration measure (20), which in turn determines thedimension (21)of. The main difference between theorieswith different z will be in the pieces of the action which areindependent of time derivatives.2. The potentialThe logic of effective field theory suggests that thecomplete action should contain all terms compatible withthe imposed symmetries, which are of dimension equal toor less than the dimension of the kinetic term, ½K ij K ij Š¼2z. In addition to S K , which contains the two independentterms of second order in the time derivatives of the metric,the general action will also contain terms that are independentof time derivatives. Since our framework is fundamentallynonrelativistic, we will refer to all terms in theaction which are independent of the time derivatives (butdo depend on spatial derivatives) simply as the ‘‘potential.’’There is a simple way to construct potential terms invariantunder our gauge symmetry Diff F ðMÞ: Startingwith any scalar function V½g ij Š which depends only onthe metric and its spatial derivatives, the following potentialterm,will be invariant under Diff F ðMÞ.Throughout this paper, our strategy is to focus first onthe potential terms of the same dimension as ½K ij K ij Š,atfirst ignoring all possible relevant terms of lower dimensionsin V. This is equivalent to focusing first on the highenergylimit, where such highest-dimension terms dominate.Once the high-energy behavior of the theory is understood,one can restore the relevant terms, and study theflows of the theory away from the UV fixed point that suchrelevant operators induce in the infrared.With our choice of D ¼ 3 and z ¼ 3, there are manyexamples of terms in V of the same dimension as thekinetic term in (19). Some such terms are quadratic incurvature,r k R ij r k R ij ; r k R ij r i R jk ; RR; R ij R ij ;(24)they will not only add interactions but also modify thepropagator. Other terms, such asR 3 ; R i j Rj k Rk i ; RR ijR ij ; (25)are cubic in curvature, and therefore represent pure interactingterms. Some of the terms of the correct dimensionare related by the Bianchi identity and other symmetries ofthe Riemann tensor, or differ only up to a total derivative.Additional constraints on the possible values of the couplingswill likely follow from the requirements of stabilityand unitarity of the quantum theory. However, the list ofindependent operators appears to be prohibitively large,implying a proliferation of couplings which makes explicitcalculations rather impractical.C. UV theory with detailed balanceIn order to reduce the number of independent couplingconstants, we will impose an additional symmetry on thetheory. The reason for this restriction is purely pragmatic,to limit the proliferation of independent couplings mentionedin the previous paragraph. The way in which thisrestriction will be implemented, however, is very reminiscentof methods used in nonequilibrium critical phenomenaand quantum critical systems. As a result, it isnatural to suspect that there might also be conceptualreasons behind restricting the general class of classicaltheories to conform to this framework in systems withgravity as well.We will require the potential term to be of a special form,S V ¼ 28Zdtd D px ffiffiffi g NE ij G ijk‘ E k‘ ; (26)and will further demand that E ij itself follow from a variationalprinciple,084008-5
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Quantum gravity at a Lifshitz point

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