Quantum Geometry : A Statistical Field Theory Approach (Cambridge Monographs on Mathematical Physics)

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Quantum Geometry : A Statistical Field Theory Approach (Cambridge Monographs on Mathematical Physics)

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  • 製本 Paperback:紙装版/ペーパーバック版/ページ数 363 p.
  • 言語 ENG
  • 商品コード 9780521017367
  • DDC分類 530

基本説明

New in paperback. Hardcover was published in 1997. This graduate level text describes in a unified fashion the statistical mechanics of random walks, random surfaces and random higher dimensional manifolds with an emphasis on the geometrical aspects of the theory and applications to the quantization of strings, gravity and topological field theory.

Full Description


This graduate/research level text describes in a unified fashion the statistical mechanics of random walks, random surfaces and random higher dimensional manifolds with an emphasis on the geometrical aspects of the theory and applications to the quantisation of strings, gravity and topological field theory. With chapters on random walks, random surfaces, two- and higher dimensional quantum gravity, topological quantum field theories and Monte Carlo simulations of random geometries, the text provides a self-contained account of quantum geometry from a statistical field theory point of view. The approach uses discrete approximations and develops analytical and numerical tools. Continuum physics is recovered through scaling limits at phase transition points and the relation to conformal quantum field theories coupled to quantum gravity is described. The most important numerical work is covered, but the main aim is to develop mathematically precise results that have wide applications. Many diagrams and references are included.

Table of Contents

Preface                                            xi
Notation xiii
Introduction 1 (10)
Random walks 11 (55)
Parametrized random walks 12 (7)
The Wiener measure 12 (3)
Universality of the Wiener measure 15 (4)
Geometric random walks 19 (13)
Embedded random walks 19 (5)
Riemannian random walks 24 (8)
Rigid random walks 32 (14)
Curvature-dependent action 32 (2)
The two-point function 34 (2)
The scaling limits 36 (5)
The tangent-tangent correlation function 41 (5)
Fermionic random walks 46 (7)
Branched polymers 53 (11)
Extrinsic properties 53 (8)
Intrinsic properties 61 (2)
Generalizations 63 (1)
Notes 64 (2)
Random surfaces 66 (83)
Introduction 66 (2)
The dynamically triangulated random 68 (5)
surface model
Triangulations and Regge calculus 73 (5)
Basic properties of the loop functions 78 (26)
Convergence of the loop functions 79 (6)
The susceptibility exponent y 85 (5)
Branched polymer surfaces 90 (5)
Mass and string tension 95 (5)
The Hausdorff dimension 100 (2)
Scaling and the continuum limit in the 102 (2)
DTRS-model
Random surfaces on a lattice 104 (19)
Definition of the lattice surface model 105 (5)
Mass, susceptibility and string tension 110 (5)
Critical behaviour and continuum limit 115 (8)
Rigid surfaces 123 (12)
Motivation 123 (2)
Curvature-dependent action 125 (5)
The crumpling transition 130 (5)
Crystalline surfaces 135 (10)
The kinematics of crumpling 138 (5)
A lower bound on the size of 143 (2)
crystalline surfaces
Notes 145 (4)
Two-dimensional gravity 149 (102)
The continuum formalism 149 (5)
The combinatorial solution 154 (19)
Regularization 155 (5)
Counting planar graphs 160 (5)
Generalization 165 (2)
An easy example 167 (4)
The general model 171 (2)
Counting higher-genus surfaces 173 (9)
The loop equation for genus h > 0 173 (2)
Solution of the loop equation for h 175 (3)
> 0
The generating function lh for closed 178 (3)
triangulations
The number of triangulations of genus h 181 (1)
The continuum limit 182 (7)
Renormalization of the cosmological 182 (1)
constant
Continuum results for genus zero 183 (3)
Continuum results for higher-genus 186 (3)
surfaces
Multi-critical models 189 (10)
General considerations 189 (2)
The dimer model 191 (4)
Connection with conformal field theory 195 (4)
The continuum loop equation 199 (13)
The two-point function 212 (14)
General considerations 212 (3)
A differential equation for the 215 (5)
geodesic two-loop function
Solution of the differential equation 220 (3)
A transfer matrix approach 223 (3)
Matrix models 226 (11)
Counting triangulations using matrix 226 (4)
models
The loop equations 230 (3)
Non-perturbative quantum gravity? 233 (1)
The Kontsevich model 234 (3)
More on matter and gravity 237 (10)
Coupling matter fields to gravity 237 (1)
The Ising model 238 (3)
Multiple-spin systems 241 (6)
Notes 247 (4)
Monte Carlo simulations of random geometry 251 (20)
Basic principles 251 (3)
Updating geometry 254 (6)
Finite-size scaling 260 (3)
Two-dimensional geometry 263 (7)
Notes 270 (1)
Gravity in higher dimensions 271 (26)
Basic problems in quantum gravity 271 (4)
Simplicial quantum gravity in dimensions 275 (9)
d > 2
Simplicial complexes and triangulations 275 (3)
The metric structure 278 (4)
Generalized matrix models 282 (2)
Algorithmic recognizability and numerical 284 (6)
methods
Numerical results 290 (5)
Notes 295 (2)
Topological quantum field theories 297 (42)
Introduction 297 (1)
Generalities 298 (5)
The axioms 298 (3)
Some properties of TQFTs 301 (2)
Two-dimensional TQFT 303 (7)
TQFT on triangulations 303 (4)
The unitary case 307 (3)
Three-dimensional unitary TQFT 310 (27)
TQFT and three-dimensional gravity 311 (6)
The discrete framework 317 (6)
Construction in terms of 6j-symbols 323 (14)
Notes 337 (2)
References 339 (20)
Index 359