Quantum Gravity in 2+1 Dimensions

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Cambridge University Press, 2003 M12 4 - 276 páginas
This timely volume provides a broad survey of (2+1)-dimensional quantum gravity. It emphasises the 'quantum cosmology' of closed universes and the quantum mechanics of the (2+1)-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 analysing 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.
 

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Contenido

Why 2+1dimensional gravity?
1
11 General relativity in 2+1 dimensions
3
12 Generalizations
6
13 A note on units
7
Classical general relativity in 2+1 dimensions
9
22 The ADM decomposition
12
23 Reduced phase space and moduli space
15
24 Diffeomorphisms and conserved charges
20
84 Perturbation theory
140
Lorentzian path integrals
143
92 Covariant metric path integrals
149
93 Path integrals and firstorder quantization
152
94 Topological field theory
158
Euclidean path integrals and quantum cosmology
163
101 Real tunneling geometries
164
102 The HartleHawking wave function
165

25 The firstorder formalism
25
26 Boundary terms and the WZW action
29
27 Comparing generators of invariances
34
A field guide to the 2+1dimensional spacetimes
38
32 The 2+1dimensional black hole
45
33 The torus universe
50
34 Other topologies
57
Geometric structures and ChernSimons theory
60
42 Geometric structures
64
43 The space of Lorentzian structures
67
44 Adding a cosmological constant
69
45 Closed universes as quotient spaces
71
46 Fiber bundles and flat connections
77
47 The Poisson algebra of the holonomies
80
Canonical quantization in reduced phase space
87
52 Quantization of the reduced phase space
89
53 Automorphic forms and Maass operators
93
54 A general ADM quantization
96
55 Pros and cons
97
The connection representation
100
62 Quantizing geometric structures
104
63 Relating quantizations
106
64 Ashtekar variables
112
65 More pros and cons
114
Operator algebras and loops
117
71 The operator algebra of Nelson and Regge
118
72 The connection representation revisited
122
73 The loop representation
124
The WheelerDeWitt equation
131
81 The firstorder formalism
132
82 A quantum Legendre transformation
134
83 The secondorder formalism
135
103 The sum over topologies
168
Lattice methods
171
111 Regge calculus
172
112 The TuraevViro model
176
113 A Hamiltonian lattice formulation
183
114 t Hoofts polygon model
186
115 Dynamical triangulation
191
The 2+1dimensional black hole
194
121 A brief introduction to black hole thermodynamics
195
122 The Lorentzian black hole
196
123 The Euclidean black hole
202
124 Black hole statistical mechanics
209
Next steps
212
The topology of manifolds
217
A2 The mapping class group
218
A3 Connected sums
219
A4 The fundamental group
220
A5 Covering spaces
222
A6 Quotient spaces
224
A7 Geometrization
226
A8 Simplices and Euler numbers
227
topology of surfaces
229
Lorentzian metrics and causal structure
236
B2 Lorentz cobordism
239
B3 Closed timelike curves and causal structure
240
Differential geometry and fiber bundles
243
C2 Holonomy and curvature
245
C3 Frames and spin connections
246
C5 Fiber bundles and connections
247
References
250
Index
267
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