Non-Perturbative Renormalization

by Mastropietro, Vieri

• ISBN13: 9789812792396
• ISBN10: 9812792392
• Format: Hardcover
• Copyright: 2008-03-04
• Publisher: World Scientific Pub Co Inc

Summary

The notion of renormalization is at the core of several spectacular achievements of contemporary physics, and in the last years powerful tehcniques have been developed allowing to put renormalization on a firm mathematical basis. This book provides a self-consistent and accessible introduction to the sophisticated tools used in the modern theory of non-perturbative renormalization, allowing an unified and rigorous treatment of Quantum Field Theory, Statistical Physics and Condensed Matter models. In particular the first part of this book is devoted to Constructive Quantum Field Theory, providing a mathematical construction of models at low dimensions and discussing the removal of the ultraviolet and infrared cut-off, the verification of the axioms and the validity of Ward Identities with the relative anomalies. The second part is devoted to lattice 2D Statistical Physics, analyzing in particular the theory of universality in perturbed lsing models and the computation of the non-universal critical indices in Vertex or Ashkin-Teller models. Finally the third part is devoted to the analysis of complex quantum fluids showing Luttinger of Fermi liquid behavior, like the 1D or 2D Hubbard model. Book jacket.

Table of Contents

Preface	p. v
Introduction to Renormalization	p. 1
Basic Notions	p. 3
Relativistic quantum field theory	p. 3
Quantum fields	p. 3
Functional integrals	p. 5
Perturbative renormalization	p. 8
Classical statistical mechanics	p. 12
Phase transitions	p. 12
Universality and non-universality	p. 14
Condensed Matter	p. 16
Electrons in a crystal	p. 16
The free Fermi gas	p. 19
Fermi liquids	p. 21
Luttinger liquids and BCS superconductors	p. 23
Fermionic Functional Integrals	p. 27
Grassmann variables	p. 27
Grassmann measures	p. 29
Truncated expectations	p. 31
Properties of Grassmann integrals	p. 32
Gallavotti-Nicolo tree expansion	p. 33
Feynman graphs	p. 39
Determinant bounds for simple expectations	p. 42
The Brydges-Battle-Federbush representation	p. 45
The Gawedzki-Kupiainen-Lesniewski formula	p. 51
Quantum Field Theory	p. 57
The Ultraviolet Problem in Massive QED2	p. 59
Regularization and cut-offs	p. 59
Integration of the bosons	p. 61
Propagator decomposition	p. 63
Renormalized expansion	p. 66
Feynman graph expansion	p. 68
Convergence of the renormalized expansion	p. 69
Determinant bounds	p. 72
The short memory property	p. 75
Extraction of loop lines	p. 75
The 2-point Schwinger function	p. 79
The Yukawa model	p. 79
Infrared Problem and Anomalous Behavior	p. 81
Anomalous dimension	p. 81
Renormalization	p. 82
Modification of the fermionic interaction	p. 83
Bounds for the renormalized expansion	p. 89
The beta function at lowest orders	p. 96
Boundedness of the flow	p. 99
The 2-point Schwinger function	p. 101
Ward Identities and Vanishing of the Beta Function	p. 105
Schwinger functions and running couplings	p. 105
Ward identities in presence of cut-offs	p. 107
The correction identity	p. 109
The Schwinger-Dyson equation	p. 115
Analysis of the cut-off corrections	p. 118
Vanishing of Beta function	p. 120
Non-perturbative Adler-Bardeen theorem	p. 122
Further remarks	p. 123
Thirring and Gross-Neveu Models	p. 125
The Thirring model	p. 125
Removing the fermionic ultraviolet cut-off before the bosonic one	p. 126
Removing the bosonic ultraviolet cut-off before the fermionic one	p. 128
The Gross-Neveu model	p. 132
Axioms Verification and Wilson Fermions	p. 133
Osterwalder-Schrader axioms	p. 133
Lattice regularization and fermion doubling	p. 135
Integration of the doubled fermions	p. 137
Lattice fermions	p. 138
Infraed QED4 with Large Photon Mass	p. 143
Regularization	p. 143
Tree expansion	p. 145
Lattice Statistical Mechanics	p. 147
Universality in Generalized Ising Models	p. 149
The nearest neighbor Ising model	p. 149
Heavy and light Majorana fermions	p. 153
Generalized Ising models	p. 156
Fermionic representation of the generalized Ising model	p. 157
Integration of the x-variables	p. 159
Integration of the light fermions	p. 160
Correlation functions and the specific heat	p. 164
Nonuniversality in Vertex or Isotropic Ashkin-Teller Models	p. 165
Ashkin-Teller or Vertex models	p. 165
Fermionic representation	p. 167
Anomalous behaviour	p. 170
Simmetry properties	p. 171
Integration of the light fermions	p. 175
The specific heat	p. 177
Universality-Nonuniversality Crossover in the Ashkin-Teller Model	p. 181
The anisotropic AT model	p. 181
Anomalous universality	p. 183
Integration of the x variables	p. 185
Integration of the [psi] variables: first regime	p. 188
Integration of the [psi] variables: second regime	p. 192
Critical behaviour	p. 195
Quantum Liquids	p. 199
Spinless Luttinger Liquids	p. 201
Fermions on a chain	p. 201
Grassman representation	p. 203
Luttinger liquid behavior	p. 204
The ultraviolet integration	p. 207
Quasi-particle fields	p. 209
The flow of the running coupling constants	p. 213
Density correlations	p. 215
Quantum spin chains	p. 219
Crystals and quasi-crystals	p. 222
The 1d Hubbard Model	p. 225
Spinning fermions	p. 225
The effective potential	p. 226
The flow of the running coupling constants	p. 228
The auxiliary model	p. 231
The effective renormalizations	p. 236
Attractive interactions	p. 237
Fermi Liquids in Two Dimensions	p. 239
Interacting Fermions in d = 2	p. 239
Multiscale integration	p. 242
Bounds for the Feynman graphs	p. 245
The sector decomposition	p. 246
The sector lemma	p. 250
Bounds for the tree expansion	p. 253
Flow of runing coupling constants	p. 257
Other results in d = 2	p. 260
BCS Model with Long Range Interaction	p. 263
BCS model	p. 263
Partial Hubbard-Stratonovich transformation	p. 266
Corrections to the mean field	p. 268
The Ising Model Fermionic Representation	p. 273
The Grassmann representation of the 2d Ising model with open boundary conditions	p. 273
The Grassmann representation of the 2d Ising model with periodic boundary conditions	p. 283
Bibliography	p. 287
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