Contents -- Preface -- Introduction to Renormalization -- 1. Basic Notions -- 1.1 Relativistic quantum eld theory -- 1.1.1 Quantum fields -- 1.1.2 Functional integrals -- 1.1.3 Perturbative renormalization -- 1.2 Classical statistical mechanics -- 1.2.1 Phase transitions -- 1.2.2 Universality and non-universality -- 1.3 Condensed Matter -- 1.3.1 Electrons in a crystal -- 1.3.2 The free Fermi gas -- 1.3.3 Fermi liquids -- 1.3.4 Luttinger liquids and BCS superconductors -- 2. Fermionic Functional Integrals -- 2.1 Grassmann variables -- 2.2 Grassmann measures -- 2.3 Truncated expectations -- 2.4 Properties of Grassmann integrals -- 2.5 Gallavotti-Nicolo tree expansion -- 2.6 Feynman graphs -- 2.7 Determinant bounds for simple expectations -- 2.8 The Brydges-Battle-Federbush representation -- 2.9 The Gawedzki-Kupiainen-Lesniewski formula -- Quantum Field Theory -- 3. The Ultraviolet Problem in Massive QED2 -- 3.1 Regularization and cut-offs -- 3.2 Integration of the bosons -- 3.3 Propagator decomposition -- 3.4 Renormalized expansion -- 3.5 Feynman graph expansion -- 3.6 Convergence of the renormalized expansion -- 3.7 Determinant bounds -- 3.8 The short memory property -- 3.9 Extraction of loop lines -- 3.10 The 2-point Schwinger function -- 3.11 The Yukawa model -- 4. Infrared Problem and Anomalous Behavior -- 4.1 Anomalous dimension -- 4.2 Renormalization -- 4.3 Modification of the fermionic interaction -- 4.4 Bounds for the renormalized expansion -- 4.5 The beta function at lowest orders -- 4.6 Boundedness of the flow -- 4.7 The 2-point Schwinger function -- 5. Ward Identities and Vanishing of the Beta Function -- 5.1 Schwinger functions and running couplings -- 5.2 Ward identities in presence of cut-offs -- 5.3 The correction identity -- 5.4 The Schwinger-Dyson equation -- 5.5 Analysis of the cut-off corrections.

5.6 Vanishing of Beta function -- 5.7 Non-perturbative Adler-Bardeen theorem -- 5.8 Further remarks -- 6. Thirring and Gross-Neveu Models -- 6.1 The Thirring model -- 6.2 Removing the fermionic ultraviolet cut-off before the bosonic one -- 6.3 Removing the bosonic ultraviolet cut-off before the fermionic one -- 6.4 The Gross-Neveu model -- 7. Axioms Verification and Wilson Fermions -- 7.1 Osterwalder-Schrader axioms -- 7.2 Lattice regularization and fermion doubling -- 7.3 Integration of the doubled fermions -- 7.4 Lattice fermions -- 8. Infraed QED4 with Large Photon Mass -- 8.1 Regularization -- 8.2 Tree expansion -- Lattice Statistical Mechanics -- 9. Universality in Generalized Ising Models -- 9.1 The nearest neighbor Ising model -- 9.2 Heavy and light Majorana fermions -- 9.3 Generalized Ising models -- 9.4 Fermionic representation of the generalized Ising model -- 9.5 Integration of the -variables -- 9.6 Integration of the light fermions -- 9.7 Correlation functions and the specific heat -- 10. Nonuniversality in Vertex or Isotropic Ashkin-Teller Models -- 10.1 Ashkin-Teller or Vertex models -- 10.2 Fermionic representation -- 10.3 Anomalous behaviour -- 10.4 Simmetry properties -- 10.5 Integration of the light fermions -- 10.6 The specific heat -- 11. Universality-Nonuniversality Crossover in the Ashkin- Teller Model -- 11.1 The anisotropic AT model -- 11.2 Anomalous universality -- 11.3 Integration of the variables -- 11.4 Integration of the variables: rst regime -- 11.5 Integration of the variables: second regime -- 11.6 Critical behaviour -- Quantum Liquids -- 12. Spinless Luttinger Liquids -- 12.1 Fermions on a chain -- 12.2 Grassman representation -- 12.3 Luttinger liquid behavior -- 12.4 The ultraviolet integration -- 12.5 Quasi-particle fields -- 12.6 The flow of the running coupling constants -- 12.7 Density correlations.

12.8 Quantum spin chains -- 12.9 Crystals and quasi-crystals -- 13. The 1d Hubbard Model -- 13.1 Spinning fermions -- 13.2 The effective potential -- 13.3 The flow of the running coupling constants -- 13.4 The auxiliary model -- 13.5 The effective renormalizations -- 13.6 Attractive interactions -- 14. Fermi Liquids in Two Dimensions -- 14.1 Interacting Fermions in d = 2 -- 14.2 Multiscale integration -- 14.3 Bounds for the Feynman graphs -- 14.4 The sector decomposition -- 14.5 The sector lemma -- 14.6 Bounds for the tree expansion -- 14.7 Flow of runing coupling constants -- 14.8 Other results in d = 2 -- 15. BCS Model with Long Range Interaction -- 15.1 BCS model -- 15.2 Partial Hubbard-Stratonovich transformation -- 15.3 Corrections to the mean field -- Appendix A The Ising Model Fermionic Representation -- A.1 The Grassmann representation of the 2d Ising model with open boundary conditions -- A.2 The Grassmann representation of the 2d Ising model with periodic boundary conditions -- Bibliography.