Non-Perturbative Renormalization

Vieri Mastropietro9812792392, 9789812792396, 9789812792402

The notion of renormalization is at the core of several spectacular achievements of contemporary physics, and in the last years powerful techniques 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 Ising 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.
Contents: Introduction to Renormalization: Basic Notions; Fermionic Functional Integrals; Quantum Field Theory: The Ultraviolet Problem in Massive QED2; Infrared Problem and Anomalous Behavior; Ward Identities and Vanishing of the Beta Function; Thirring and Gross-Neveu Models; Axioms Verification and Wilson Fermions; Infrared QED4 with Large Photon Mass; Lattice Statistical Mechanics: Universality in Generalized Ising Models; Nonuniversality in Vertex or Isotropic Ashkin-Teller Models; Universality-Nonuniversality Crossover in the Ashkin-Teller Model; Quantum Liquids: Spinless Luttinger Liquids; The 1d Hubbard Model; Fermi Liquids in Two Dimensions; BCS Model with Long Range Interaction.

---

Table of contents :
Contents……Page 8
Preface……Page 6
Introduction to Renormalization……Page 14
1.1.1 Quantum fields……Page 16
1.1.2 Functional integrals……Page 18
1.1.3 Perturbative renormalization……Page 21
1.2.1 Phase transitions……Page 25
1.2.2 Universality and non-universality……Page 27
1.3.1 Electrons in a crystal……Page 29
1.3.2 The free Fermi gas……Page 32
1.3.3 Fermi liquids……Page 34
1.3.4 Luttinger liquids and BCS superconductors……Page 36
2.1 Grassmann variables……Page 40
2.2 Grassmann measures……Page 42
2.3 Truncated expectations……Page 44
2.4 Properties of Grassmann integrals……Page 45
2.5 Gallavotti-Nicolo tree expansion……Page 46
2.6 Feynman graphs……Page 52
2.7 Determinant bounds for simple expectations……Page 55
2.8 The Brydges-Battle-Federbush representation……Page 58
2.9 The Gawedzki-Kupiainen-Lesniewski formula……Page 64
Quantum Field Theory……Page 70
3.1 Regularization and cut-offs……Page 72
3.2 Integration of the bosons……Page 74
3.3 Propagator decomposition……Page 76
3.4 Renormalized expansion……Page 79
3.5 Feynman graph expansion……Page 81
3.6 Convergence of the renormalized expansion……Page 82
3.7 Determinant bounds……Page 85
3.9 Extraction of loop lines……Page 88
3.11 The Yukawa model……Page 92
4.1 Anomalous dimension……Page 94
4.2 Renormalization……Page 95
4.3 Modification of the fermionic interaction……Page 96
4.4 Bounds for the renormalized expansion……Page 102
4.5 The beta function at lowest orders……Page 109
4.6 Boundedness of the flow……Page 112
4.7 The 2-point Schwinger function……Page 114
5.1 Schwinger functions and running couplings……Page 118
5.2 Ward identities in presence of cut-offs……Page 120
5.3 The correction identity……Page 122
5.4 The Schwinger-Dyson equation……Page 128
5.5 Analysis of the cut-off corrections……Page 131
5.6 Vanishing of Beta function……Page 133
5.7 Non-perturbative Adler-Bardeen theorem……Page 135
5.8 Further remarks……Page 136
6.1 The Thirring model……Page 138
6.2 Removing the fermionic ultraviolet cut-off before the bosonic one……Page 139
6.3 Removing the bosonic ultraviolet cut-off before the fermionic one……Page 141
6.4 The Gross-Neveu model……Page 145
7.1 Osterwalder-Schrader axioms……Page 146
7.2 Lattice regularization and fermion doubling……Page 148
7.3 Integration of the doubled fermions……Page 150
7.4 Lattice fermions……Page 151
8.1 Regularization……Page 156
8.2 Tree expansion……Page 158
Lattice Statistical Mechanics……Page 160
9.1 The nearest neighbor Ising model……Page 162
9.2 Heavy and light Majorana fermions……Page 166
9.3 Generalized Ising models……Page 169
9.4 Fermionic representation of the generalized Ising model……Page 170
9.5 Integration of the -variables……Page 172
9.6 Integration of the light fermions……Page 173
9.7 Correlation functions and the specific heat……Page 177
10.1 Ashkin-Teller or Vertex models……Page 178
10.2 Fermionic representation……Page 180
10.3 Anomalous behaviour……Page 183
10.4 Simmetry properties……Page 184
10.5 Integration of the light fermions……Page 188
10.6 The specific heat……Page 190
11.1 The anisotropic AT model……Page 194
11.2 Anomalous universality……Page 196
11.3 Integration of the variables……Page 198
11.4 Integration of the variables: rst regime……Page 201
11.5 Integration of the variables: second regime……Page 205
11.6 Critical behaviour……Page 208
Quantum Liquids……Page 212
12.1 Fermions on a chain……Page 214
12.2 Grassman representation……Page 216
12.3 Luttinger liquid behavior……Page 217
12.4 The ultraviolet integration……Page 220
12.5 Quasi-particle fields……Page 222
12.6 The flow of the running coupling constants……Page 226
12.7 Density correlations……Page 228
12.8 Quantum spin chains……Page 232
12.9 Crystals and quasi-crystals……Page 235
13.1 Spinning fermions……Page 238
13.2 The effective potential……Page 239
13.3 The flow of the running coupling constants……Page 241
13.4 The auxiliary model……Page 244
13.5 The effective renormalizations……Page 249
13.6 Attractive interactions……Page 250
14.1 Interacting Fermions in d = 2……Page 252
14.2 Multiscale integration……Page 255
14.3 Bounds for the Feynman graphs……Page 258
14.4 The sector decomposition……Page 259
14.5 The sector lemma……Page 263
14.6 Bounds for the tree expansion……Page 266
14.7 Flow of runing coupling constants……Page 270
14.8 Other results in d = 2……Page 273
15.1 BCS model……Page 276
15.2 Partial Hubbard-Stratonovich transformation……Page 279
15.3 Corrections to the mean field……Page 281
A.1 The Grassmann representation of the 2d Ising model with open boundary conditions……Page 286
A.2 The Grassmann representation of the 2d Ising model with periodic boundary conditions……Page 296
Bibliography……Page 300