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Palmer J. — Planar Ising Correlations
Palmer J. — Planar Ising Correlations

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Название: Planar Ising Correlations

Автор: Palmer J.

Аннотация:

Understanding certain exactly solvable models in statistical mechanics and quantum field theory from a mathematical physics perspective is a very important and active area of research. These models include the scaling limits of the 2-D Ising (lattice) model, and more generally, a class of 2-D quantum fields known as holonomic fields. Steady progress has been made in understanding the special mathematical features of these models. Over the years, new results have made it possible to obtain a detailed nonperturbative analysis of the many spin correlations. This book is principally devoted to an analysis showing that the scaling functions are tau functions associated with monodromy-preserving deformations of the Dirac equation. While charting a fairly direct route to this analysis via new results of Palmer and others, as well as previous research of the Kyoto School — Sato, Miwa, and Jimbo — are the primary focus of this book, other interesting mathematical insights occur all along the way. For example, the Ising model has been a source of rich mathematics from Szego limit theorems to Wick type theorems for infinite spin groups. Also, some aspects of the solution of the Ising model are elegantly expressed in terms of Pfaffian and determinant bundles over infinite dimensional Grassmannians. These construct generalize the more familar objects in finite dimensional algebraic geometry and have appeared only recently in the mathematics literature. Exploring the Ising model as a microcosm of the confluence of interesting ideas in mathematics and physics, this work will appeal to graduate students, mathematicians, and physicists interested in the mathematics of statistical mechanics andquantum field theory.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 2007

Количество страниц: 372

Добавлена в каталог: 13.05.2008

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Предметный указатель
Abraham      xii 83
Abraham and Martin Lof      xi 4 5 9 15 19 21
Addition theorem      77
Admissible frame      201
Annihilation operators      276
Anticommutation relations      278
Avogadro’s number      2
Basor and Tracy      144
Bessel function      100 163 166 175 177 229
Boltzmann weight      vi 2 69
Branched      159
Brauer — Weyl      9 15
Broken symmetry      vii
Canonical      227
Canonical basis      232 253
Canonical section      213
Cartan and Dieudonne      286
Cheng and Wu      96
Clifford algebra      277
Clifford group      278
Clifford relations      9—11
Cluster decomposition      57
Complex orthogonal      11 279
Configuration energy      1
Conformal field theory      140
Conformal regime      106
Conjugation      167 178
Contour integral representation      164 180
Convolution      79
Corner transfer matrix      xv
Correlation functions      vi
correlation length      96 105 106
Creation operators      276
Critical correlations      58
critical exponent      106
Critical temperature      vii 26
Critical two-point function      140
Curie      3
Cylinder measures      135
Cylinder sets      132
Deformation equations      223 255 260
Determinant bundle      198
Dirac equation      232
Dirac operator      161
Discriminant      65
Disorder operator      89
Disorder variables      51
DLR equations      131
Dobrushin      131
Duality      45
Elliptic integral      69
Expansion at infinity      172
Ferromagnetic      v 3
Flaschka and Newell      270
Fock representation      14 278
Fourier transform      161
Free fermion      147
Fuchs      270
Gambier      270
Garnier      270
Gaussian domination      136
Gibbs random field      131
Gibbs state      vi 4 132
GKS inequalities      145
Graph coordinates      205
Grassmann calculus      279
Grassmannian      201
Green function      151 175 178 179 236
Griffith’s inequalities      55 126 135
Haag      2
Hardy space      34
Helmholtz equation      227
Hermitian polarization      319
high-temperature expansion      47
Holomorphic differential      65
Holonomic quantum field      147
Holonomy      148
Induced rotations      11
Infinite-volume limit      8
Ising      ix
Isotropic      277
Isotropic splitting      13
Jacobian elliptic function      68
Kadanoff and Ceva      xi 49
Kadanoff and Kamoto      ix
Kato      37
Kaufmann      x 9
KdV      ix
Kovalevskaya      270
Kramers and Wannier      ix
Kramers — Wannier duality      21 29
Lanford and Ruelle      131
Laplace method      99
Lenz      ix
Line bundle      169
Local eigenfunctions      232
Local expansion      170
Local operator product      160
localization      181 188 234
Long-range order      105
low-temperature expansion      47
Luther and Peschel      xv 271
Malgrange      270
McCoy and Wu      vii
McCoy, Wu, and Perk      ix
Minkowski      155
Minlos      xii 132
Miwa, Jimbo, and Ueno      270
Modern analysis      xi
Monodromy-preserving deformation      148
Montroll, Potts, and Ward      xi 59 63 101
nearest neighbors      1
Number operator      19
One point function      42
Onsager      ix xi 3
Onsager and Kaufmann      ix
Onsager — Yang      80
Open boundary conditions      45 51
Orthogonal reflection      285
Painleve      viii 223
Painleve equation      270
Painleve property      270
Palmer and Tracy      xi
Partition function      2
Peierls      ix
Pfaffian      87 294 301 324 333
Pfaffian bundle      199
Plus boundary conditions      3 9
Poincare      xv
Polarization      13 278
Potts and Ward      xi 59 60 101
Product deformation      310
Projection      189
Pure states      57
Quantum inverse scattering      xv
Quasifree state      288
Ramification      65
Rapidity      161
Rational parametrization      162
Renormalization      293
Rotational invariance      265
Ruelle      4
Sato, Miwa, and Jimbo      viii xv 270
Scaling limit      108 124 128
Schlesinger      270
Schmidt class      36
Segal and Pressley      x
Segal — Wilson      ix xiv
Sobolev space      182
Spectral curve      64 72 76 129 161
Spectral representation      185 189
Spectral transform      78 186
Spin configuration      1
Spin correlation      2
Spin operator      6 11 33
Spin representation      279
Spontaneous magnetization      63 81
Stokes’s theorem      158
Supersymmetric      xv
susceptibility      xv 140 143
Symbol map      290
symbols      280
Symmetries      261
Szego      xii 59
Szego theorem      102
Tau functions      147 198
Thermodynamic limit      vi
Time ordering      155
Time-ordered      156 159
toeplitz      xii 43 59 60
Tracy      140 144 271
Tracy and Widom      140 271
Transfer matrix      32
Transverse      235
Trivialization      213
Two-dimensional Ising model      vii
Two-point Function      88 95 142
Two-point scaling      143
Uniformization      68 69 72 76 161 162
Vacuum vector      14 276
Wave functions      160
Wedge product      274
Whittaker and Watson      xi 163
Wick’s theorem      294
Widom and Tracy      144
Wiener — Hopf      33—35
Wu      96
Wu, McCoy, Tracy, and Barouch      viii xv 223 270
Yamada      xii 89 96
Yang      xi 43 63
Z invariant      xv
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