Non-perturbative renormalization

Preface	p. v
Introduction to Renormalization	p. 1
1 Basic Notions	p. 3
1.1 Relativistic quantum field theory	p. 3
1.1.1 Quantum fields	p. 3
1.1.2 Functional integrals	p. 5
1.1.3 Perturbative renormalization	p. 8
1.2 Classical statistical mechanics	p. 12
1.2.1 Phase transitions	p. 12
1.2.2 Universality and non-universality	p. 14
1.3 Condensed Matter	p. 16
1.3.1 Electrons in a crystal	p. 16
1.3.2 The free Fermi gas	p. 19
1.3.3 Fermi liquids	p. 21
1.3.4 Luttinger liquids and BCS superconductors	p. 23
2 Fermionic Functional Integrals	p. 27
2.1 Grassmann variables	p. 27
2.2 Grassmann measures	p. 29
2.3 Truncated expectations	p. 31
2.4 Properties of Grassmann integrals	p. 32
2.5 Gallavotti-Nicolo tree expansion	p. 33
2.6 Feynman graphs	p. 39
2.7 Determinant bounds for simple expectations	p. 42
2.8 The Brydges-Battle-Federbush representation	p. 45
2.9 The Gawedzki-Kupiainen-Lesniewski formula	p. 51
Quantum Field Theory	p. 57
3 The Ultraviolet Problem in Massive QED2	p. 59
3.1 Regularization and cut-offs	p. 59
3.2 Integration of the bosons	p. 61
3.3 Propagator decomposition	p. 63
3.4 Renormalized expansion	p. 66
3.5 Feynman graph expansion	p. 68
3.6 Convergence of the renormalized expansion	p. 69
3.7 Determinant bounds	p. 72
3.8 The short memory property	p. 75
3.9 Extraction of loop lines	p. 75
3.10 The 2-point Schwinger function	p. 79
3.11 The Yukawa model	p. 79
4 Infrared Problem and Anomalous Behavior	p. 81
4.1 Anomalous dimension	p. 81
4.2 Renormalization	p. 82
4.3 Modification of the fermionic interaction	p. 83
4.4 Bounds for the renormalized expansion	p. 89
4.5 The beta function at lowest orders	p. 96
4.6 Boundedness of the flow	p. 99
4.7 The 2-point Schwinger function	p. 101
5 Ward Identities and Vanishing of the Beta Function	p. 105
5.1 Schwinger functions and running couplings	p. 105
5.2 Ward identities in presence of cut-offs	p. 107
5.3 The correction identity	p. 109
5.4 The Schwinger-Dyson equation	p. 115
5.5 Analysis of the cut-off corrections	p. 118
5.6 Vanishing of Beta function	p. 120
5.7 Non-perturbative Adler-Bardeen theorem	p. 122
5.8 Further remarks	p. 123
6 Thirring and Gross-Neveu Models	p. 125
6.1 The Thirring model	p. 125
6.2 Removing the fermionic ultraviolet cut-off before the bosonic one	p. 126
6.3 Removing the bosonic ultraviolet cut-off before the fermionic one	p. 128
6.4 The Gross-Neveu model	p. 132
7 Axioms Verification and Wilson Fermions	p. 133
7.1 Osterwalder-Schrader axioms	p. 133
7.2 Lattice regularization and fermion doubling	p. 135
7.3 Integration of the doubled fermions	p. 137
7.4 Lattice fermions	p. 138
8 Infraed QED4 with Large Photon Mass	p. 143
8.1 Regularization	p. 143
8.2 Tree expansion	p. 145
Lattice Statistical Mechanics	p. 147
9 Universality in Generalized Ising Models	p. 149
9.1 The nearest neighbor Ising model	p. 149
9.2 Heavy and light Majorana fermions	p. 153
9.3 Generalized Ising models	p. 156
9.4 Fermionic representation of the generalized Ising model	p. 157
9.5 Integration of the x-variables	p. 159
9.6 Integration of the light fermions	p. 160
9.7 Correlation functions and the specific heat	p. 164
10 Nonuniversality in Vertex or Isotropic Ashkin-Teller Models	p. 165
10.1 Ashkin-Teller or Vertex models	p. 165
10.2 Fermionic representation	p. 167
10.3 Anomalous behaviour	p. 170
10.4 Simmetry properties	p. 171
10.5 Integration of the light fermions	p. 175
10.6 The specific heat	p. 177
11 Universality-Nonuniversality Crossover in the Ashkin-Teller Model	p. 181
11.1 The anisotropic AT model	p. 181
11.2 Anomalous universality	p. 183
11.3 Integration of the x variables	p. 185
11.4 Integration of the [psi] variables: first regime	p. 188
11.5 Integration of the [psi] variables: second regime	p. 192
11.6 Critical behaviour	p. 195
Quantum Liquids	p. 199
12 Spinless Luttinger Liquids	p. 201
12.1 Fermions on a chain	p. 201
12.2 Grassman representation	p. 203
12.3 Luttinger liquid behavior	p. 204
12.4 The ultraviolet integration	p. 207
12.5 Quasi-particle fields	p. 209
12.6 The flow of the running coupling constants	p. 213
12.7 Density correlations	p. 215
12.8 Quantum spin chains	p. 219
12.9 Crystals and quasi-crystals	p. 222
13 The 1d Hubbard Model	p. 225
13.1 Spinning fermions	p. 225
13.2 The effective potential	p. 226
13.3 The flow of the running coupling constants	p. 228
13.4 The auxiliary model	p. 231
13.5 The effective renormalizations	p. 236
13.6 Attractive interactions	p. 237
14 Fermi Liquids in Two Dimensions	p. 239
14.1 Interacting Fermions in d = 2	p. 239
14.2 Multiscale integration	p. 242
14.3 Bounds for the Feynman graphs	p. 245
14.4 The sector decomposition	p. 246
14.5 The sector lemma	p. 250
14.6 Bounds for the tree expansion	p. 253
14.7 Flow of runing coupling constants	p. 257
14.8 Other results in d = 2	p. 260
15 BCS Model with Long Range Interaction	p. 263
15.1 BCS model	p. 263
15.2 Partial Hubbard-Stratonovich transformation	p. 266
15.3 Corrections to the mean field	p. 268
Appendix A The Ising Model Fermionic Representation	p. 273
A.1 The Grassmann representation of the 2d Ising model with open boundary conditions	p. 273
A.2 The Grassmann representation of the 2d Ising model with periodic boundary conditions	p. 283
Bibliography	p. 287

Available:*

On Order

Table of Contents