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Quantum Gravity : Mathematical Models and Experimental Bounds

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Quantum Gravity : Mathematical Models and Experimental Bounds Quantum Gravity Mathematical Models and Experimental Bounds Bertfried Fauser Jürgen Tolksdorf Eberhard Zeidler Editors Birkhäuser Verlag Basel . Boston . Berlin Editors: Bertfried Fauser Vladimir Rabinovich Jürgen Tolksdorf Instituto Politecnico Nacional Eberhard Zeidler ESIME Zacatenco Avenida IPN Max-Planck-Institut für Mathematik in den Mexico, D. F. 07738 Naturwissenschaften Mexico Inselstrasse 22–26 e-mail: [email protected] D-04103 Leipzig Germany e-mail: [email protected] Sergei M. Grudsky [email protected] Nikolai Vasilevski [email protected] Departamento de Matematicas CINVESTAV Apartado Postal 14-740 07000 Mexico, D.F. Mexico e-mail: [email protected] [email protected] 2000 Mathematical Subject Classification: primary 81-02; 83-02; secondary: 83C45; 81T75; 83D05; 83E30; 83F05 Library of Congress Control Number: 2006937467 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 3-7643-7574-4 978-3-7643-7977-3 Birkhäuser Birkhäuser Verlag, Basel Verlag,– Boston Basel– Berlin – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. must be obtained. © 2007 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF f Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN-10: 3-7643-7977-4 e-ISBN-10: 3-7643-7978-2 ISBN-13: 978-3-7643-7977-3 e-ISBN-13: 978-3-7643-7978-0 9 8 7 6 5 4 3 2 1 www.birkhauser.ch CONTENTS Preface .......................................................................xi Bertfried Fauser, J¨urgen Tolksdorf and Eberhard Zeidler Quantum Gravity – A Short Overview.........................................1 Claus Kiefer 1. Why do we need quantum gravity? 1 2. Quantum general relativity 5 2.1. Covariant approaches 5 2.2. Canonical approaches 5 2.2.1. Quantum geometrodynamics 6 2.2.2. Connection and loop variables 7 3. String theory 8 4. Loops versus strings – a few points 9 5. Quantum cosmology 10 6. Some central questions about quantum gravity 11 References 12 The Search for Quantum Gravity.............................................15 Claus L¨ammerzahl 1. Introduction 15 2. The basic principles of standard physics 16 3. Experimental tests 17 3.1. Tests of the universality of free fall 17 3.2. Tests of the universality of the gravitational redshift 18 3.3. Tests of local Lorentz invariance 19 3.3.1. Constancy of c 20 3.3.2. Universality of c 21 3.3.3. Isotropy of c 21 3.3.4. Independence of c from the velocity of the laboratory 21 3.3.5. Time dilation 21 3.3.6. Isotropy in the matter sector 22 3.4. Implications for the equations of motion 22 3.4.1. Implication for point particles and light rays 22 1 3.4.2. Implication for spin– 2 particles 23 3.4.3. Implications for the Maxwell field 23 3.4.4. Summary 24 3.5. Implications for the gravitational field 24 3.6. Tests of predictions – determination of PPN parameters 25 3.6.1. Solar system effects 25 3.6.2. Strong gravity and gravitational waves 27 vi Contents 4. Unsolved problems: first hints for new physics? 27 5. On the magnitude of quantum gravity effects 29 6. How to search for quantum gravity effects 30 7. Outlook 31 Acknowledgements 32 References 32 Time Paradox in Quantum Gravity ...........................................41 Alfredo Mac´ıas and Hernando Quevedo 1. Introduction 41 2. Time in canonical quantization 43 3. Time in general relativity 45 4. Canonical quantization in minisuperspace 49 5. Canonical quantization in midisuperspace 51 6. The problem of time 52 7. Conclusions 56 Acknowledgements 57 References 57 Differential Geometry in Non-Commutative Worlds ..........................61 Louis H. Kauffman 1. Introduction to non-commutative worlds 61 2. Differential geometry and gauge theory in a non-commutative world 66 3. Consequences of the metric 69 Acknowledgements 74 References 74 Algebraic Approach to Quantum Gravity III: Non-Commutative Riemannian Geometry ....................................77 Shahn Majid 1. Introduction 77 2. Reprise of quantum differential calculus 79 2.1. Symplectic connections: a new field in physics 81 2.2. Differential anomalies and the orgin of time 82 3. Classical weak Riemannian geometry 85 3.1. Cotorsion and weak metric compatibility 86 3.2. Framings and coframings 87 4. Quantum bundles and Riemannian structures 89 5. Quantum gravity on finite sets 94 6. Outlook: Monoidal functors 97 References 98 Quantum Gravity as a Quantum Field Theory of Simplicial Geometry .......101 Daniele Oriti 1. Introduction: Ingredients and motivations for the group field theory 101 1.1. Why path integrals? The continuum sum-over-histories approach 102 Contents vii 1.2. Why topology change? Continuum 3rd quantization of gravity 103 1.3. Why going discrete? Matrix models and simplicial quantum gravity 105 1.4. Why groups and representations? Loop quantum gravity/spin foams 107 2. Group field theory: What is it? The basic GFT formalism 109 2.1. A discrete superspace 109 2.2. The field and its symmetries 111 2.3. The space of states or a third quantized simplicial space 112 2.4. Quantum histories or a third quantized simplicial spacetime 112 2.5. The third quantized simplicial gravity action 113 2.6. The partition function and its perturbative expansion 114 2.7. GFT definition of the canonical inner product 115 2.8. Summary: GFT as a general framework for quantum gravity 116 3. An example: 3d Riemannian quantum gravity 117 4. 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 ......127 Mario Paschke 1. Introduction 127 2. Classical spectral triples 130 3. On the meaning of noncommutativity 134 4. NC description of the standard model: the physical intuition behind it 136 4.1. The intuitive idea: an picture of quantum spacetime at low energies 136 4.2. The postulates 138 4.3. How such a noncommutative spacetime would appear to us 139 5. Remarks and open questions 140 5.1. Remarks 140 5.2. Open problems, perspectives, more speculations 141 5.3. Comparision: intuitive picture/other approaches to Quantum Gravity143 6. Towards a quantum equivalence principle 145 6.1. Globally hyperbolic spectral triples 145 6.2. Generally covariant quantum theories over spectral geometries 147 References 149 Towards a Background Independent Formulation of Perturbative Quantum Grav- ity ..........................................................................151 Romeo Brunetti and Klaus Fredenhagen 1. Problems of perturbative Quantum Gravity 151 2. Locally covariant quantum field theory 152 3. Locally covariant fields 155 4. Quantization of the background 158 viii Contents References 158 Mapping-Class Groups of 3-Manifolds .......................................161 Domenico Giulini 1. Some facts about Hamiltonian general relativity 161 1.1. Introduction 161 1.2. Topologically closed Cauchy surfaces 163 1.3. Topologically open Cauchy surfaces 166 2. 3-Manifolds 169 3. Mapping class groups 172 3.1. A small digression on spinoriality 174 3.2. General Diffeomorphisms 175 4. A simple yet non-trivial example 183 4.1. The RP3 geon 183 4.2. The connected sum RP3 RP3 185 5. Further remarks on the general structure of GF(Σ) 190 6. Summary and outlook 192 Appendix: Elements of residual finiteness 193 References 197 Kinematical Uniqueness of Loop Quantum Gravity ..........................203 Christian Fleischhack 1. Introduction 203 2. Ashtekar variables 204 3. Loop variables 205 3.1. Parallel transports 205 3.2. Fluxes 205 4. Configuration space 206 4.1. Semianalytic structures 206 4.2. Cylindrical functions 207 4.3. Generalized connections 207 4.4. Projective limit 207 4.5. Ashtekar-Lewandowski measure 208 4.6. Gauge transforms and diffeomorphisms 208 5. Poisson brackets 208 5.1. Weyl operators 209 5.2. Flux derivations 209 5.3. Higher codimensions 209 6. Holonomy-flux ∗ -algebra 210 6.1. Definition 210 6.2. Symmetric state 210 6.3. Uniqueness proof 211 7. Weyl algebra 212 7.1. Definition 212 7.2. Irreducibility 213 Contents ix 7.3. Diffeomorphism invariant representation 213 7.4. Uniqueness proof 213 8. Conclusions 215 8.1. Theorem – self-adjoint case 215 8.2. Theorem – unitary case 215 8.3. Comparison 216 8.4. Discussion 216 Acknowledgements 217 References 218 Topological Quantum Field Theory as Topological Quantum Gravity.........221 Kishore Marathe 1. Introduction 221 2. Quantum Observables 223 3. Link Invariants 224 4. WRT invariants 226 5. Chern-Simons and String Theory 227 6. Conifold Transition 228 7. WRT invariants and topological string amplitudes 229 8. Strings and gravity 232 9. Conclusion 233 Acknowledgements 234 References 234 Strings, Higher Curvature Corrections, and Black Holes.....................237 Thomas Mohaupt 1. Introduction 237 2. The black hole attractor mechanism 240 3. Beyond the area law 244 4. From black holes to topological strings 247 5. Variational principles for black holes 250 6. Fundamental strings and ‘small’ black holes 253 7. Dyonic strings and ‘large’ black holes 256 8. Discussion 258 Acknowledgements 259 References 260 The Principle of the Fermionic Projector: An Approach for Quantum Gravity? ........................................263 Felix Finster 1. A variational principle in discrete space-time 264 2. Discussion of the underlying physical principles 266 3. Naive correspondence to a continuum theory 268 4. The continuum limit 270 5. Obtained results 271 xContents 6. Outlook: The classical gravitational field 272 7. Outlook: The field quantization 274 References 280 Gravitational Waves and Energy Momentum Quanta ........................283 Tekin Dereli and Robin W.
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