Quantum Gravity: Mathematical Models and Experimental Bounds Book

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