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.

Cambridge Books Online © Cambridge Univesity Press, 2009

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QUANTUM GRAVITY IN 2+1 DIMENSIONS

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CAMBRIDGE MONOGRAPHS ON

MATHEMATICAL PHYSICS

J. AmbJ0rn, B. Durhuus and T. Jonsson Quantum Geometry: A Statistical Field Theory

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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

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M. Creutz Quarks, Gluons and Lattices*

F. de Felice and C. J. S. Clarke Relativity on Curved

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DeWitt Supermanifolds, 2nd edition*

P. G. O. Freund Introduction to

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J. Fuchs Affine Lie Algebras and Quantum

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J. Fuchs and C. Schweigert Symmetries, Lie Algebras and Representations: A Graduate

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for Physicists

J. A. H. Futterman, F. A. Handler and R. A. Matzner Scattering from Black Holes

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S. W. Hawking and G. F. R. Ellis The Large-Scale Structure of

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J. I. Kapusta Finite-Temperature Field Theory*

V. E. Korepin, A. G. Izergin and N. M. Boguliubov The Quantum Inverse Scattering

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M. Le Bellac Thermal Field Theory

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H. March Liquid Metals: Concepts and Theory

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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

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* Issued as a paperback

Cambridge Books Online © Cambridge Univesity Press, 2009

QUANTUM GRAVITY

IN 2+1 DIMENSIONS

STEVEN

CARLIP

University

of

California

at

Davis

CAMBRIDGE

UNIVERSITY PRESS

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PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

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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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