Progress in Mathematical Physics

By: John Palmer

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Hardcover | 29 August 2007

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384 Pages

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Steady progress in recent years has been made in understanding the special mathematical features of certain exactly solvable models in statistical mechanics and quantum field theory, including the scaling limits of the 2-D Ising (lattice) model, and more generally, a class of 2-D quantum fields known as holonomic fields. New results have made it possible to obtain a detailed nonperturbative analysis of the multi-spin correlations. In particular, the book focuses on deformation analysis of the scaling functions of the Ising model, and will appeal to graduate students, mathematicians, and physicists interested in the mathematics of statistical mechanics and quantum field theory.

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Preface	p. vii
The Thermodynamic Limit	p. 1
The Transfer Matrix in a Pure Phase	p. 1
The Semi-infinite-Volume Limit	p. 8
Induced Rotations	p. 9
Fock Representations	p. 13
The Spectrum of T_M	p. 21
The Thermodynamic Limit for T < T_c	p. 30
Duality and the Thermodynamic Limit Above T_cp45
Even Correlations at All Temperatures	p. 58
The Spontaneous Magnetization and Two-Point Spin Correlation	p. 63
The Spectral Curve for the Transfer Matrix	p. 64
The Uniformization of M	p. 68
Trivializing ¿ over M_±	p. 72
The Matrix Elements of s for the Q Polarization	p. 74
The Two-Point Function Below T_c	p. 83
The Two-Point Function Above T_c	p. 89
The Correlation Length and Two-Point Asymptotics	p. 96
The Montroll, Potts, and Ward Calculation of <¿>	p. 101
Scaling Limits	p. 105
Introduction	p. 105
Hyperbolic Formulas for the Kernels of D^-1, BD^-1, and D^-1C	p. 110
Convergence Estimates	p. 113
Pfaffian Formulas for the Scaling Functions	p. 124
A Probabilistic Interpretation for the Scaling Limits	p. 131
Expansions for the Two-Point Scaling Functions	p. 140
The One-Point Green Function	p. 147
Introduction	p. 147
Free Fermions	p. 149
Contour Integral Representations	p. 160
Local Expansions	p. 168
The Green Function for One Branch Point	p. 175
Localization and Projections	p. 181
Scaling Functions as Tau Functions	p. 197
The Subcritical Tau Function	p. 198
The Supercritical Tau Function	p. 211
Deformation Analysis of Tau Functions	p. 223
Boundary Conditions for m - D on ¿	p. 224
Existence Theory for the Subcritical Green Function	p. 236
Existence for the Basis {W_j} and the Derivative of G(x, y; ¿)	p. 240
The Derivative of log ¿	p. 247
Holonomic Systems and the Deformation Equations	p. 252
A Deformation-Theoretic Expression for d log ¿	p. 261
Rotational Invariance for ¿^[±]	p. 264
The MTWB Result for the Two-Point Scaling Functions	p. 267
Short-Distance Behavior of the Scaling Functions	p. 271
Spin Representations of the Orthogonal Group	p. 273
Grassmann Algebra	p. 273
Fock Representations of the Clifford Algebra	p. 277
The Grassmann Calculus	p. 279
Pfaffians	p. 293
Vacuum Expectations for Elements of G	p. 304
The SMJ Product Deformation Formalism	p. 310
Infinite Dimensions	p. 319
Relations Between Transforms	p. 337
Boltzmann Weights	p. 337
Representations of the Complex Orthogonal Space W	p. 338
References	p. 345
List of Notation	p. 357
Index	p. 361
Table of Contents provided by Ingram. All Rights Reserved.

Progress in Mathematical Physics

At a Glance

Hardcover

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