Quantum Gravity : Mathematical Models and Experimental Bounds

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10 Claus Kiefer include Hawking radiation); unitarity holds only for the fully entangled quantum state of the hole plus the other fields [19]. As for the microscopic derivation of (8), the situation can be briefly summarized as follows [4], • Loop quantum gravity: The microscopic degrees of freedom are given by the spin-network states. An appropriate counting procedure for the number of the relevant horizon states leads to an entropy that is proportional to the Barbero–Immirzi parameter β. The demand for the result to be equal to (8) then fixes β. Until 2004, it was believed that the result is β = ln 2 π √ 3 ( ) ln 3 π √ , (19) 2 where the value in parentheses would refer for the relevant group to the choice of SO(3) instead of SU(2). The SO(3)-value would have exhibited an interesting connection with the quasi-normal modes for the black hole. More recently, it was found that the original estimate for the number of states was too small. A new calculation yields the following numerical estimate for β [20]: β =0.237532 ...An interpretation of this value at a more fundamental level has not yet been given. • String theory: The microscopic degrees of freedom are here the D-branes, for which one can count the quantum states in the weak-coupling limit (where no black hole is present). Increasing the string coupling, one reaches a regime where no D-branes are present, but instead one has black holes. For black holes that are extremal in the relevant string-theory charges (for extremality the total charge is equal to the mass), the number of states is preserved (‘BPS black holes’), so the result for the black-hole entropy is the same as in the D-brane regime. In fact, the result is just (8), as it must be if the theory is consistent. This remains true for non-extremal black holes, but no result has been obtained for a generic black hole (say, an ordinary Schwarzschild black hole). More recently, an interesting connection has been found between the partition function, Z BH , for a BPS black hole and a topological string amplitude, Z top ,[21], Z BH = |Z top | 2 . (20) Moreover, it has been speculated that the partition function can be identified with (the absolute square of) the Hartle–Hawking wave function of the universe [22]. This could give an interesting connection between quantum cosmology and string theory. 5. Quantum cosmology Let me turn to quantum cosmology. If one assumes the universal validity of quantum theory (as is highly suggested by the experimental situation [23]), one needs a quantum theory for the universe as a whole: All subsystems are entangled with
Quantum Gravity — A Short Overview 11 their environment, with the universe being the only strictly closed quantum system. Since gravity dominates on the largest scales, a theory of quantum gravity is needed for quantum cosmology. The main questions to be addressed are the following: [24] • How does one impose boundary conditions in quantum cosmology? • Is the classical singularity being avoided? • How does the appearance of our classical universe follow from quantum cosmology? • Can the arrow of time be understood from quantum cosmology? • How does the origin of structure proceed? • Is there a high probability for an inflationary phase? Can inflation itself be understood from quantum cosmology? • Can quantum cosmological results be justified from full quantum gravity? • Has quantum cosmology relevance for the measurement problem in quantum theory? • Can quantum cosmology be experimentally tested? There has been much progress on each of these questions, but final answers can probably only obtained after the correct quantum theory of gravity is available. For the status of each of these questions I refer to [24, 25, 1, 26]. Loop quantum gravity has recently been applied to cosmology by using spectra such as (18) for the size of the universe (‘loop quantum cosmology’ [27]). The Wheeler–DeWitt equation then assumes the form of a difference equation. It seems that one can get singularity avoidance from it. Moreover, the quantum modifications to the Friedmann equations arising from the loop approach seem to favour an inflationary scenario and could potentially be observed in the CMB anisotropy spectrum. 6. Some central questions about quantum gravity I conclude this brief review with some central questions taken from [4] concerning the development of quantum gravity: • Is unification needed to understand quantum gravity? • Into which approaches is background independence implemented? (In particular, is string theory background independent?) • In which approaches do ultraviolet divergences vanish? • Is there a continuum limit for path integrals? • Is a Hilbert-space structure needed for the full theory? (This has an important bearing on the interpretation of quantum states.) • Is Einstein gravity non-perturbatively renormalizable? (Can the cosmological constant be calculated from the infrared behaviour of renormalization-group equations?) • What is the role of non-commutative geometry? • Is there an information loss for black holes? • Are there decisive experimental tests?
Page 3 and 4: Quantum Gravity Mathematical Models
Page 5 and 6: CONTENTS Preface ..................
Page 7 and 8: Contents vii 1.2. Why topology chan
Page 9 and 10: Contents ix 7.3. Diffeomorphism inv
Page 11 and 12: Preface This Edited Volume is based
Page 13 and 14: Preface xiii In the search for quan
Page 15 and 16: Preface xv arguments from string th
Page 17 and 18: Quantum Gravity B. Fauser, J. Tolks
Page 19 and 20: Quantum Gravity — A Short Overvie
Page 21 and 22: 2. Quantum general relativity Quant
Page 25: Quantum Gravity — A Short Overvie
Page 31 and 32: 16 Claus Lämmerzahl compared with
Page 33 and 34: 18 Claus Lämmerzahl In the case th
Page 35 and 36: 20 Claus Lämmerzahl The two postul
Page 37 and 38: 22 Claus Lämmerzahl as granted. Si
Page 39 and 40: 24 Claus Lämmerzahl no other field
Page 41 and 42: 26 Claus Lämmerzahl per turn what
Page 43 and 44: 28 Claus Lämmerzahl [77, 78]. This
Page 45 and 46: 30 Claus Lämmerzahl through astrop
Page 47 and 48: 32 Claus Lämmerzahl will have same
Page 49 and 50: 34 Claus Lämmerzahl [23] K. Breche
Page 51 and 52: 36 Claus Lämmerzahl [58] J. Ehlers
Page 53 and 54: 38 Claus Lämmerzahl [92] L. Iorio.
Page 57 and 58: Time Paradox in Quantum Gravity 43
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Differential Geometry in Non-Commut
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78 S. Majid functional path integra
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80 S. Majid In general a given alge
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82 S. Majid and its curvature (defi
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84 S. Majid using our previous nota
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86 S. Majid 3.1. Cotorsion and weak
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88 S. Majid Next, any connection ω
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90 S. Majid 1. H a Hopf algebra coa
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92 S. Majid In other words, apart f
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94 S. Majid 5. Quantum gravity on f
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96 S. Majid for Z 4 as a model of S
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98 S. Majid to a deeper conceptual
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100 S. Majid [M12] S. Majid. Noncom
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102 Daniele Oriti so on the one han
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104 Daniele Oriti included in the (
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106 Daniele Oriti than a smooth man
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108 Daniele Oriti spin foam models.
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110 Daniele Oriti obtained gluing t
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112 Daniele Oriti depends on the gr
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114 Daniele Oriti actions, and para
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116 Daniele Oriti is certainly need
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118 Daniele Oriti For the propagato
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120 Daniele Oriti irreps used) and
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122 Daniele Oriti • What is λ? I
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124 Daniele Oriti is currently in p
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126 Daniele Oriti [35] K. Krasnov,
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128 M. Paschke spontaneously break
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130 M. Paschke 2. Classical spectra
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132 M. Paschke 4. It now remains to
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134 M. Paschke Remark 2.3. Related
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136 M. Paschke We now want to descr
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138 M. Paschke On the other hand we
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140 M. Paschke 3. The spectral acti
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142 M. Paschke • We should stress
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144 M. Paschke and don’t have def
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146 M. Paschke on the common domain
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148 M. Paschke contains a full Cauc
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150 M. Paschke [24] H.-J. Matschull
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152 Romeo Brunetti and Klaus Freden
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Quantum Gravity B. Fauser, J. Tolks
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Mapping-Class Groups 163 tensors of
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Mapping-Class Groups 165 respect to
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Mapping-Class Groups 167 these surf
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Mapping-Class Groups 169 2. 3-Manif
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Mapping-Class Groups 171 S 2 σ 1 S
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Mapping-Class Groups 173 Let us foc
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Mapping-Class Groups 175 b.) G = D
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Mapping-Class Groups 177 and [21].
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Mapping-Class Groups 179 H ϕ T 1 T
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Mapping-Class Groups 181 where θ =
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Mapping-Class Groups 183 be represe
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Mapping-Class Groups 185 manifold,
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Mapping-Class Groups 187 between S
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Mapping-Class Groups 189 P S θ =co
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Mapping-Class Groups 191 Equivalent
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Mapping-Class Groups 193 clearly ju
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Mapping-Class Groups 195 Propositio
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References Mapping-Class Groups 197
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Mapping-Class Groups 199 [41] Allen
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Mapping-Class Groups 201 [76] Rafae
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204 Ch. Fleischhack such an emergen
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206 Ch. Fleischhack 4. Configuratio
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208 Ch. Fleischhack h A ∈ A. Now,
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210 Ch. Fleischhack 6. Holonomy-flu
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212 Ch. Fleischhack Let us denote t
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214 Ch. Fleischhack operators on L
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216 Ch. Fleischhack 8.3. Comparison
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218 Ch. Fleischhack References [1]
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TQFT as TQG 223 In section 2 we int
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TQFT as TQG 225 where A is the gaug
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TQFT as TQG 227 were also obtained,
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TQFT as TQG 229 to the cone z1 2 +
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TQFT as TQG 231 2. Quantum group (o
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TQFT as TQG 233 of gravitation if W
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TQFT as TQG 235 [16] I. Smith, R¿
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238 Thomas Mohaupt By the analogy t
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240 Thomas Mohaupt should be compar
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242 Thomas Mohaupt 2n+2 , then, al
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244 Thomas Mohaupt 3. Beyond the ar
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246 Thomas Mohaupt where F Υ = ∂
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248 Thomas Mohaupt Using this one e
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250 Thomas Mohaupt moduli are separ
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252 Thomas Mohaupt where F KL denot
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254 Thomas Mohaupt [53, 47] S macro
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256 Thomas Mohaupt Hence these blac
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258 Thomas Mohaupt Q =(q, p) ∈ Γ
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260 Thomas Mohaupt References [1] L
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262 Thomas Mohaupt [60] R. Dijkgraa
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264 Felix Finster For clarity, we f
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266 Felix Finster This variational
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268 Felix Finster induced on the sp
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270 Felix Finster is singular (for
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272 Felix Finster 6. Outlook: The c
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274 Felix Finster discreteness of s
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276 Felix Finster point of view to
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278 Felix Finster to our above cons
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280 Felix Finster material on the s
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Gravitational Waves and Energy Mome
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Asymptotic Safety in QEG 295 The as
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Asymptotic Safety in QEG 297 symbol
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Asymptotic Safety in QEG 299 the pr
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Asymptotic Safety in QEG 301 bounda
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Asymptotic Safety in QEG 303 This l
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Asymptotic Safety in QEG 305 with G
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Asymptotic Safety in QEG 307 Thus t
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Asymptotic Safety in QEG 309 parame
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Asymptotic Safety in QEG 311 of asy
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Asymptotic Safety in QEG 313 [39] J
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316 Harald Grosse and Raimar Wulken
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328 Index see also Bogomol’nyi-Pr
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330 Index fine structure constant,
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332 Index spinorial, 173, 190 mappi
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334 Index renormalizable nonperturb
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336 Index list of experiments, 19 v
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