Carlip S. Quantum Gravity in 2+1 Dimensions
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This timely volume provides a broad survey of (2+l)-dimensional quantum
gravity. It emphasises the 'quantum cosmology' of closed universes and the
quantum mechanics of the (2+l)-dimensional black hole. It compares and
contrasts a variety of approaches, and examines what they imply for a realistic
theory of quantum gravity.
General relativity in three spacetime dimensions has become a popular arena
in which to explore the ramifications of quantum gravity. As a diffeomorphism-
invariant theory of spacetime structure, this model shares many of the conceptual
problems of realistic quantum gravity. But it is also simple enough that many
programs of quantization can be carried out explicitly.
After analyzing the space of classical solutions, this book introduces some
fifteen approaches to quantum gravity - from canonical quantization in York's
'extrinsic time' to Chern-Simons quantization, from the loop representation to
covariant path integration to lattice methods. Relationships among quantizations
are explored, as well as implications for such issues as topology change and the
'problem of time'.
This book is an invaluable resource for all graduate students and researchers
working in quantum gravity.
STEVEN CARLIP received an undergraduate degree in physics from Harvard in
1975.
After seven years as a printer, editor, and factory worker, he returned to
school at the University of
Texas,
where he earned his Ph.D. in 1987. Following a
stint as a postdoctoral fellow at the Institute for Advanced Study, he joined the
faculty of the University of California at Davis in 1990.
Professor Carlip's main research interest is quantum gravity. He has also
worked on string theory, quantum field theory, classical general relativity, and
the interface between physics and topology. He has received a number of honors,
including a Department of Energy Outstanding Junior Investigator award and a
National Science Foundation Young Investigator award.
When he can find the time, Professor Carlip is active in progressive politics.
His hobbies include hiking, world travel, playing the dulcimer, and scuba diving.
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Cambridge Books Online © Cambridge Univesity Press, 2009
CAMBRIDGE MONOGRAPHS ON
MATHEMATICAL PHYSICS
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W. Sciama,
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QUANTUM GRAVITY IN 2+1 DIMENSIONS
Cambridge Books Online © Cambridge Univesity Press, 2009
CAMBRIDGE MONOGRAPHS ON
MATHEMATICAL PHYSICS
J. AmbJ0rn, B. Durhuus and T. Jonsson Quantum Geometry: A Statistical Field Theory
Approach
A. M. Anile Relativistic Fluids and Magneto-Fluids
J. A. de Azcarraga and J. M. Izquierdo Lie Groups, Lie Algebras, Cohomology and Some
Applications in Physics*
J. Bernstein Kinetic Theory in the Early Universe
G. F. Bertsch and R. A. Broglia Oscillations in Finite Quantum Systems
N.
D. Birrell and P. C. W. Davies Quantum Fields in Curved Space*
D.
M. Brink Semiclassical Methods in Nucleus-Nucleus Scattering
J. C. Collins Renormalizatiori*
M. Creutz Quarks, Gluons and Lattices*
F. de Felice and C. J. S. Clarke Relativity on Curved
Manifolds*
B.
DeWitt Supermanifolds, 2nd edition*
P. G. O. Freund Introduction to
Supersymmetry*
J. Fuchs Affine Lie Algebras and Quantum
Groups*
J. Fuchs and C. Schweigert Symmetries, Lie Algebras and Representations: A Graduate
Course
for Physicists
J. A. H. Futterman, F. A. Handler and R. A. Matzner Scattering from Black Holes
R. Gambini and J. Pullin Loops, Knots, Gauge Theories and Quantum Gravity
M. Gockeler and T. Schucker Differential Geometry, Gauge Theories and Gravity*
C.
Gomez, M. Ruiz Altaba and G. Sierra Quantum Groups in Two-dimensional Physics
M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 1:
Introduction*
M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 2: Loop
Amplitudes, Anomalies and
Phenomenology*
S. W. Hawking and G. F. R. Ellis The Large-Scale Structure of
Space-Time*
F. Iachello and A. Arima The Interacting Boson Model
F. Iachello and P. van Isacker The Interacting Boson-Fermion Model
C.
Itzykson and J.-M. Drouffe Statistical Field Theory, volume 1: From Brownian Motion
to Renormalization and Lattice Gauge Theory*
C.
Itzykson and J.-M. Drouffe Statistical Field Theory, volume 2: Strong Coupling, Monte
Carlo Methods, Conformal Field Theory, and Random Systems*
J. I. Kapusta Finite-Temperature Field Theory*
V. E. Korepin, A. G. Izergin and N. M. Boguliubov The Quantum Inverse Scattering
Method and Correlation Functions
M. Le Bellac Thermal Field Theory
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H. March Liquid Metals: Concepts and Theory
I. M. Montvay and G. Munster Quantum Fields on a Lattice
A. Ozorio de Almeida Hamiltonian Systems: Chaos and Quantization*
R. Penrose and W. Rindler Spinors and Space-time, volume 1: Two-Spinor Calculus and
Relativistic
Fields*
R. Penrose and W. Rindler Spinors and Space-time, volume 2: Spinor and Twistor
Methods in Space-Time Geometry^
S. Pokorski Gauge Field Theories*
V. N. Popov Functional Integrals and Collective Excitations*
R. Rivers Path Integral Methods in Quantum Field Theory*
R. G. Roberts The Structure of the Proton*
J. M. Stewart Advanced General Relativity
A. Vilenkin and E. P. S. Shellard Cosmic Strings and Other Topological Defects
R. S. Ward and R. O. Wells Jr Twistor Geometry and Field Theories*
* Issued as a paperback
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QUANTUM GRAVITY
IN 2+1 DIMENSIONS
STEVEN
CARLIP
University
of
California
at
Davis
CAMBRIDGE
UNIVERSITY PRESS
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©Cambridge University Press 1998
This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1998
First paperback edition 2003
Typeset in llpt Times [TAG]
A catalogue record for this book is available from the British Library
Library of Congress Cataloguing
in
Publication Data
Carlip, Steven (Steven Jonathan),
1953—
Quantum gravity in
2+1
dimensions / Steven Carlip.
p.
cm.
Includes bibliographical references and index.
ISBN 0 521 56408 5 hardback
1.
Quantum gravity. 2. General relativity (Physics). 3. Space and
time.
I. Title
QC178.C185 1998
530.14'3-dc21 97-42893 CIP
ISBN 0 521 56408 5 hardback
ISBN 0 521 54588 9 paperback
Cambridge Books Online © Cambridge Univesity Press, 2009
Contents
Preface xi
1
1.1
1.2
1.3
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3
3.1
3.2
3.3
3.4
4
4.1
4.2
4.3
4.4
4.5
4.6
Why (2+l)-dimensional gravity?
General relativity in 2+1 dimensions
Generalizations
A note on units
Classical general relativity in 2+1 dimensions
The topological setting
The ADM decomposition
Reduced phase space and moduli space
Diffeomorphisms and conserved charges
The first-order formalism
Boundary terms and the WZW action
Comparing generators of invariances
A field guide to the (2+l)-dimensional spacetimes
Point sources
The (2+l)-dimensional black hole
The torus universe
Other topologies
Geometric structures and Chern-Simons theory
A static solution
Geometric structures
The space of Lorentzian structures
Adding a cosmological constant
Closed universes as quotient spaces
Fiber bundles and flat connections
1
3
6
7
9
9
12
15
20
25
29
34
38
38
45
50
57
60
60
64
67
69
71
77
Vll
Cambridge Books Online © Cambridge University Press, 2009
viii
Contents
4.7 The Poisson algebra of the holonomies 80
5 Canonical quantization in reduced phase space 87
5.1 Conceptual issues in quantum gravity 87
5.2 Quantization of the reduced phase space 89
5.3 Automorphic forms and Maass operators 93
5.4 A general ADM quantization 96
5.5 Pros and cons 97
6 The connection representation 100
6.1 Covariant phase space 100
6.2 Quantizing geometric structures 104
6.3 Relating quantizations 106
6.4 Ashtekar variables 112
6.5 More pros and cons 114
7 Operator algebras and loops 117
7.1 The operator algebra of Nelson and Regge 118
7.2 The connection representation revisited 122
7.3 The loop representation 124
8 The Wheeler-DeWitt equation 131
8.1 The first-order formalism 132
8.2 A quantum Legendre transformation 134
8.3 The second-order formalism 135
8.4 Perturbation theory 140
9 Lorentzian path integrals 143
9.1 Path integrals and ADM quantization 143
9.2 Covariant metric path integrals 149
9.3 Path integrals and first-order quantization 152
9.4 Topological field theory 158
10 Euclidean path integrals and quantum cosmology 163
10.1 Real tunneling geometries 164
10.2 The Hartle-Hawking wave function 165
10.3 The sum over topologies 168
11 Lattice methods 171
11.1 Regge calculus 172
11.2 The Turaev-Viro model 176
11.3 A Hamiltonian lattice formulation 183
11.4 't Hooft's polygon model 186
11.5 Dynamical triangulation 191
Cambridge Books Online © Cambridge University Press, 2009
Contents ix
12 The (2+l)-dimensional black hole 194
12.1 A brief introduction to black hole thermodynamics 195
12.2 The Lorentzian black hole 196
12.3 The Euclidean black hole 202
12.4 Black hole statistical mechanics 209
13 Next steps 212
Appendix A: The topology of manifolds 217
A.I Homeomorphisms and diffeomorphisms 217
A.2 The mapping class group 218
A.3 Connected sums 219
A.4 The fundamental group 220
A.5 Covering spaces 222
A.6 Quotient spaces 224
A.7 Geometrization 226
A.8 Simplices and Euler numbers 227
A.9 An application: topology of surfaces 229
Appendix B: Lorentzian metrics and causal structure 236
B.I Lorentzian metrics 236
B.2 Lorentz cobordism 239
B.3 Closed timelike curves and causal structure 240
Appendix C: Differential geometry and fiber bundles 243
C.I Parallel transport and connections 243
C.2 Holonomy and curvature 245
C.3 Frames and spin connections 246
C.4 The tangent bundle 246
C.5 Fiber bundles and connections 247
References 250
Index 267
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