Quantum Gravity

by Fauser, Bertfried; Tolksdorf, Jurgen; Zeidler, Eberhard

• ISBN13: 9783764379773
• ISBN10: 3764379774
• Format: Hardcover
• Copyright: 2007-02-03
• Publisher: Birkhauser

Summary

The construction of a quantum theory of gravity is the most fundamental challenge confronting contemporary theoretical physics. The different physical ideas evolved in developing a theory of quantum gravity require highly advanced mathematical methods. This book provides the reader with an overview of the different mathematical attempts to quantize gravity written by leading experts in this field. Also discussed are the possible experimental bounds on quantum gravity effects. All of the contributions have been strictly refereed. The present volume emerged from the 2nd Blaubeuren Workshop "Mathematical and Physical Aspects of Quantum Gravity''. In general, these Workshops are intended to bring together experts in mathematics and physics to discuss in an open atmosphere the fundamental questions at the frontier of theoretical physics.

Table of Contents

Preface	p. xi
Quantum Gravity - A Short Overview	p. 1
Why do we need quantum gravity?	p. 1
Quantum general relativity	p. 5
Covariant approaches	p. 5
Canonical approaches	p. 5
Quantum geometrodynamics	p. 6
Connection and loop variables	p. 7
String theory	p. 8
Loops versus strings - a few points	p. 9
Quantum cosmology	p. 10
Some central questions about quantum gravity	p. 11
References	p. 12
The Search for Quantum Gravity	p. 15
Introduction	p. 15
The basic principles of standard physics	p. 16
Experimental tests	p. 17
Tests of the universality of free fall	p. 17
Tests of the universality of the gravitational redshift	p. 18
Tests of local Lorentz invariance	p. 19
Constancy of c	p. 20
Universality of c	p. 21
Isotropy of c	p. 21
Independence of c from the velocity of the laboratory	p. 21
Time dilation	p. 21
Isotropy in the matter sector	p. 22
Implications for the equations of motion	p. 22
Implication for point particles and light rays	p. 22
Implication for spin- [fraction12] particles	p. 23
Implications for the Maxwell field	p. 23
Summary	p. 24
Implications for the gravitational field	p. 24
Tests of predictions - determination of PPN parameters	p. 25
Solar system effects	p. 25
Strong gravity and gravitational waves	p. 27
Unsolved problems: first hints for new physics?	p. 27
On the magnitude of quantum gravity effects	p. 29
How to search for quantum gravity effects	p. 30
Outlook	p. 31
Acknowledgements	p. 32
References	p. 32
Time Paradox in Quantum Gravity	p. 41
Introduction	p. 41
Time in canonical quantization	p. 43
Time in general relativity	p. 45
Canonical quantization in minisuperspace	p. 49
Canonical quantization in midisuperspace	p. 51
The problem of time	p. 52
Conclusions	p. 56
Acknowledgements	p. 57
References	p. 57
Differential Geometry in Non-Commutative Worlds	p. 61
Introduction to non-commutative worlds	p. 61
Differential geometry and gauge theory in a non-commutative world	p. 66
Consequences of the metric	p. 69
Acknowledgements	p. 74
References	p. 74
Algebraic Approach to Quantum Gravity III: Non-Commutative Riemannian Geometry	p. 77
Introduction	p. 77
Reprise of quantum differential calculus	p. 79
Symplectic connections: a new field in physics	p. 81
Differential anomalies and the orgin of time	p. 82
Classical weak Riemannian geometry	p. 85
Cotorsion and weak metric compatibility	p. 86
Framings and coframings	p. 87
Quantum bundles and Riemannian structures	p. 89
Quantum gravity on finite sets	p. 94
Outlook: Monoidal functors	p. 97
References	p. 98
Quantum Gravity as a Quantum Field Theory of Simplicial Geometry	p. 101
Introduction: Ingredients and motivations for the group field theory	p. 101
Why path integrals? The continuum sum-over-histories approach	p. 102
Why topology change? Continuum 3rd quantization of gravity	p. 103
Why going discrete? Matrix models and simplicial quantum gravity	p. 105
Why groups and representations? Loop quantum gravity/spin foams	p. 107
Group field theory: What is it? The basic GFT formalism	p. 109
A discrete superspace	p. 109
The field and its symmetries	p. 111
The space of states or a third quantized simplicial space	p. 112
Quantum histories or a third quantized simplicial spacetime	p. 112
The third quantized simplicial gravity action	p. 113
The partition function and its perturbative expansion	p. 114
GFT definition of the canonical inner product	p. 115
Summary: GFT as a general framework for quantum gravity	p. 116
An example: 3d Riemannian quantum gravity	p. 117
Assorted questions for the present, but especially for the future	p. 120
Acknowledgements	p. 124
References	p. 125
An Essay on the Spectral Action and its Relation to Quantum Gravity	p. 127
Introduction	p. 127
Classical spectral triples	p. 130
On the meaning of noncommutativity	p. 134
NC description of the standard model: the physical intuition behind it	p. 136
The intuitive idea: an picture of quantum spacetime at low energies	p. 136
The postulates	p. 138
How such a noncommutative spacetime would appear to us	p. 139
Remarks and open questions	p. 140
Remarks	p. 140
Open problems, perspectives, more speculations	p. 141
Comparision: intuitive picture/other approaches to Quantum Gravity	p. 143
Towards a quantum equivalence principle	p. 145
Globally hyperbolic spectral triples	p. 145
Generally covariant quantum theories over spectral geometries	p. 147
References	p. 149
Towards a Background Independent Formulation of Perturbative Quantum Gravity	p. 151
Problems of perturbative Quantum Gravity	p. 151
Locally covariant quantum field theory	p. 152
Locally covariant fields	p. 155
Quantization of the background	p. 158
References	p. 158
Mapping-Class Groups of 3-Manifolds	p. 161
Some facts about Hamiltonian general relativity	p. 161
Introduction	p. 161
Topologically closed Cauchy surfaces	p. 163
Topologically open Cauchy surfaces	p. 166
3-Manifolds	p. 169
Mapping class groups	p. 172
A small digression on spinoriality	p. 174
General Diffeomorphisms	p. 175
A simple yet non-trivial example	p. 183
The RP[superscript 3] geon	p. 183
The connected sum RP[superscript 3 Characters not reproducible] RP[superscript 3]	p. 185
Further remarks on the general structure of G[subscript F]([Sigma])	p. 190
Summary and outlook	p. 192
Elements of residual finiteness	p. 193
References	p. 197
Kinematical Uniqueness of Loop Quantum Gravity	p. 203
Introduction	p. 203
Ashtekar variables	p. 204
Loop variables	p. 205
Parallel transports	p. 205
Fluxes	p. 205
Configuration space	p. 206
Semianalytic structures	p. 206
Cylindrical functions	p. 207
Generalized connections	p. 207
Projective limit	p. 207
Ashtekar-Lewandowski measure	p. 208
Gauge transforms and diffeomorphisms	p. 208
Poisson brackets	p. 208
Weyl operators	p. 209
Flux derivations	p. 209
Higher codimensions	p. 209
Holonomy-flux *-algebra	p. 210
Definition	p. 210
Symmetric state	p. 210
Uniqueness proof	p. 211
Weyl algebra	p. 212
Definition	p. 212
Irreducibility	p. 213
Diffeomorphism invariant representation	p. 213
Uniqueness proof	p. 213
Conclusions	p. 215
Theorem - self-adjoint case	p. 215
Theorem - unitary case	p. 215
Comparison	p. 216
Discussion	p. 216
Acknowledgements	p. 217
References	p. 218
Topological Quantum Field Theory as Topological Quantum Gravity	p. 221
Introduction	p. 221
Quantum Observables	p. 223
Link Invariants	p. 224
WRT invariants	p. 226
Chern-Simons and String Theory	p. 227
Conifold Transition	p. 228
WRT invariants and topological string amplitudes	p. 229
Strings and gravity	p. 232
Conclusion	p. 233
Acknowledgements	p. 234
References	p. 234
Strings, Higher Curvature Corrections, and Black Holes	p. 237
Introduction	p. 237
The black hole attractor mechanism	p. 240
Beyond the area law	p. 244
From black holes to topological strings	p. 247
Variational principles for black holes	p. 250
Fundamental strings and 'small' black holes	p. 253
Dyonic strings and 'large' black holes	p. 256
Discussion	p. 258
Acknowledgements	p. 259
References	p. 260
The Principle of the Fermionic Projector: An Approach for Quantum Gravity?	p. 263
A variational principle in discrete space-time	p. 264
Discussion of the underlying physical principles	p. 266
Naive correspondence to a continuum theory	p. 268
The continuum limit	p. 270
Obtained results	p. 271
Outlook: The classical gravitational field	p. 272
Outlook: The held quantization	p. 274
References	p. 280
Gravitational Waves and Energy Momentum Quanta	p. 283
Introduction	p. 283
Conserved quantities and electromagnetism	p. 285
Conserved quantities and gravitation	p. 286
The Bel-Robinson tensor	p. 287
Wave solutions	p. 289
Conclusions	p. 292
References	p. 292
Asymptotic Safety in Quantum Einstein Gravity; Nonperturbative Renormalizability and Fractal Spacetime Structure	p. 293
Introduction	p. 293
Asymptotic safety	p. 294
RG flow of the effective average action	p. 296
Scale dependent metrics and the resolution function l(k)	p. 300
Microscopic structure of the QEG spacetimes	p. 304
The spectral dimension	p. 307
Summary	p. 310
References	p. 311
Noncommutative QFT and Renormalization	p. 315
Introduction	p. 315
Noncommutative Quantum Field Theory	p. 316
Renormalization of [Phi superscript 4] -theory on the 4D Moyal plane	p. 318
Matrix-model techniques	p. 323
References	p. 324
Index	p. 327
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