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Local Covariance and Background Independence 21 Lie derivative on E(M) induces in a natural way an action on T E(M). Together with the adjoint action on Γ c T M we obtain an action of ΓT M on Sym(Γ c T M) ⊗ Λ(ΓT E(M)) which we denote by ρ. We can now write down how δ and 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 not difficult to see that the assignment of the graded algebra BV(M) to spacetime M can be made into a covariant functor [14]. As indicated already, the BRST method when applied to gravity has to be generalized to global objects. Otherwise the cohomology of the BRST operator s turns out to be trivial. This corresponds to the well-known fact that there are no local on-shell observables in general relativity. It was recently shown in [14] that one can introduce the BRST operator on the level of natural transformations and obtain in this way a nontrivial cohomology. Fields are now understood as natural transformations. Let D k be a functor from the category Loc to the product category Vec k , that assigns to a manifold M a k-fold product of the test section spaces D(M) × ...× D(M). Let Nat(D k , BV) denote the set of natural transformations from D k to BV. We define the extended algebra of fields as Fld = ∞⊕ Nat(D k , BV) . (4.2) k=0 It 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-hand side is the product of the algebra BV(M). Let Φ be a field; then the action of the BRST differential on it is defined by (sΦ) M (f) :=s(Φ M (f)) + (−1) |Φ| Φ M (£ (.) f) , (4.4) where |.| denotes the ghost number and the action of s on BV(M) isgiven above. The physical fields are identified with the 0-th cohomology of s on Fld. Among them we have for example scalars constructed covariantly from the metric.
22 K. Fredenhagen and K. Rejzner 5. Conclusions It was shown that a construction of quantum field theory on generic Lorentzian spacetimes is possible, in accordance with the principle of general covariance. This framework can describe a wide range of physical situations. Also a consistent incorporation of the quantized gravitational field seems to be possible. Since the theory is invariant under the action of an infinite-dimensional Lie group, the framework of infinite-dimensional differential geometry plays an important role. It provides the mathematical setting in which the BV method has a clear geometrical interpretation. The construction of a locally covariant theory of gravity in the proposed setting was already performed for the classical theory. Based on the gained insight it seems to be possible to apply this treatment also in the quantum case. One can then investigate the relations to other field-theoretical approaches to quantum gravity (Reuter [23], Bjerrum-Bohr [6], . . . ). As a conclusion we want to stress that quantum field theory should be taken serious as a third way to quantum gravity. References [1] G. Barnich, F. Brandt, 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 Mathematical and Physical Aspects of Quantum Gravity, Blaubeuren, Germany, 28 Jul - 1 Aug 2005. In: B. Fauser et al. (eds.), Quantum 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 and K. Fredenhagen, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena, June 20-25 2000 [arXiv:hepth/0101079]. [11] M. Dütsch and K. Fredenhagen, Rev. Math. Phys. 16(10) (2004) 1291-1348 [arXiv:hep-th/0403213]. [12] M. Dütsch and K. Fredenhagen, Commun. Math. Phys. 243 (2003) 275 [arXiv:hep-th/0211242]. [13] K. Fredenhagen, Locally Covariant Quantum Field Theory, Proceedings of the XIVth International Congress on Mathematical Physics, Lisbon 2003 [arXiv:hep-th/0403007]. [14] K. Fredenhagen, K. Rejzner, [arXiv:math-ph/1101.5112].
Page 4 and 5: Quantum Field Theory and Gravity Co
Page 6 and 7: Contents Preface ..................
Page 8 and 9: Preface The present volume arose fr
Page 10 and 11: Preface ix 6. (How) can we test qua
Page 12 and 13: Preface xi Felix Finster, Andreas G
Page 14 and 15: Preface xiii it is related to other
Page 16 and 17: Quantum Gravity: Whence, Whither? C
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Page 40 and 41: The “Big Wave” Theory for Dark
Page 56 and 57: Discrete and Continuum Third Quanti
Page 80 and 81: Unsharp Values, Domains and Topoi A
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Unsharp Values, Domains and Topoi 7
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Causal Boundary of Spacetimes: Revi
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Causal Boundaries 99 to the Gromov
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Causal Boundaries 101 I + (ρ) ρ P
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Causal Boundaries 103 Figure 2. The
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Causal Boundaries 105 more pairings
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Causal Boundaries 107 point of the
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Causal Boundaries 109 (0, 1) x n x
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Causal Boundaries 111 to Busemann f
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Causal Boundaries 113 respectively.
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Causal Boundaries 115 6.3. Structur
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Causal Boundaries 117 Qualitative F
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Causal Boundaries 119 [26] E. Mingu
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122 D. Häfner Let us give an idea
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124 D. Häfner Figure 1. The collap
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126 D. Häfner Figure 2. The manifo
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128 D. Häfner and the angular mome
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130 D. Häfner Lemma 3.5. There exi
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132 D. Häfner Proposition 4.4. The
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134 D. Häfner 6. Strategy of the p
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136 D. Häfner [3] B. Carter, Black
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138 R. Oeckl it is less compelling
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140 R. Oeckl Depending on the theor
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142 R. Oeckl 2.3. Recovery of stand
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144 R. Oeckl where the condition A
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146 R. Oeckl Before we proceed to i
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148 R. Oeckl the complex classical
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150 R. Oeckl The corresponding anni
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152 R. Oeckl Proposition 4.1 (Coher
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154 R. Oeckl much larger than the s
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156 R. Oeckl Setting F equal to F
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158 F. Finster, A. Grotz and D. Sch
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184 C. Bär and N. Ginoux treat the
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186 C. Bär and N. Ginoux the categ
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188 C. Bär and N. Ginoux Example 2
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190 C. Bär and N. Ginoux 3.2. Diff
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192 C. Bär and N. Ginoux In other
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194 C. Bär and N. Ginoux Here (e j
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196 C. Bär and N. Ginoux Remark 3.
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198 C. Bär and N. Ginoux Proof. Th
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200 C. Bär and N. Ginoux To prove
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202 C. Bär and N. Ginoux We refer
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204 C. Bär and N. Ginoux where Glo
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206 C. Bär and N. Ginoux [24] R. M
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208 C. J. Fewster well-described by
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210 C. J. Fewster and J + M (p) ∩
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212 C. J. Fewster commute. As funct
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214 C. J. Fewster ι + ι − M +
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216 C. J. Fewster ϕ(ψ) M M ϕ(M)(
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218 C. J. Fewster theory. The resul
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220 C. J. Fewster Following [7], a
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222 C. J. Fewster Sketch of proof.
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224 C. J. Fewster The relative Cauc
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226 C. J. Fewster [2] Bär, C., Gin
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Local Covariance, Renormalization A
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Local Thermal Equilibrium and Cosmo
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Shape Dynamics. An Introduction Jul
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Shape Dynamics. An Introduction 259
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where dx bm a Shape Dynamics. An In
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300 M. Kiessling In the following,
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302 M. Kiessling where “δ( · )
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304 M. Kiessling the point charge,
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306 M. Kiessling (2.17) and (2.18)
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308 M. Kiessling Maxwell’s “law
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310 M. Kiessling argued that E f (
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312 M. Kiessling the Maxwell-Born-I
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314 M. Kiessling removal. For insta
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316 M. Kiessling ∂ 4πc that for
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318 M. Kiessling d dt Dirac, on the
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320 M. Kiessling The Lorenz-Lorentz
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322 M. Kiessling ( 1 c Lorentz and
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324 M. Kiessling potential fields
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326 M. Kiessling as I have done her
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328 M. Kiessling just the Hamilton-
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330 M. Kiessling 7. Closing remarks
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332 M. Kiessling [Bor1937] Born, M.
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334 M. Kiessling [Lié1898] Liénar
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How Unique Are Higher-dimensional B
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Equivalence Principle, Quantum Mech
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Atom Interferometry 347 The second
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Atom Interferometry 349 α CR , aim
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Atom Interferometry 351 of this mas
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Atom Interferometry 353 height B A
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Atom Interferometry 355 iff ψ = (
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Atom Interferometry 357 have be rea
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Atom Interferometry 359 where F (t
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Atom Interferometry 361 5.2. Interf
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Atom Interferometry 363 Evaluating
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Atom Interferometry 365 where the l
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Atom Interferometry 367 Then they s
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Atom Interferometry 369 [8] T.M. Fo
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Index n-point function, 51, 146, 23
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Index 373 conformal field theory, 1
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Index 375 gravitational redshift, 2
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Index 377 operator product, 138 par
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Index 379 standard model of particl
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