This book provides a simple, but mathematically rigorous, introduction to perturbative renormalization. It can be read by students from the third year on and will leads them to a level where they can start reading the current research literature. The book gives a thorough introduction to the field-theoretic techniques required for renormalization. Special effort has been devoted to simplifying combinatorial proofs as much as possible by using generating function techniques. A renormalization group differential equation is used to study the flow of Wilson's effective action. This differential equation technique, developed during the last few years, is used to give simple but complete proofs of renormalizability theorems. The book contains an introduction to renormalization in models of solid-state physics.

Klappentext

This monograph is the first to present the recently discovered renormalization techniques for the Schrödinger and Dirac equations, providing a mathematically rigorous, yet simple and clear introduction to the subject. It develops field-theoretic techniques such as Feynman graph expansions and renormalization, taking pains to make all proofs as simple as possible by using generating function techniques throughout. Renormalization is performed by using an exact renormalization group differential equation, a technique that provides simple but complete proofs of the theorems.

Inhalt

1. Field Theory.- 2. Techniques.- 3. The Renormalization Group.- 4. The Fermi Surface Problem.- A. Appendix to Chapters 1-3.- A.1 A Topology on the Ring of Formal Power Series.- A.2 Fourier Transformation.- A.3 Properties of the Boson Propagator.- A.4 Wick Reordering for Bosons.- A.5 The Lower Bound for the Sunset Graph.- B. Appendix to Chapter 4.- B.1 Fermionic Fock Space.- B.2 Calculus on Grassmann Algebras.- B.3 Grassmann Gaussian Integrals.- B.4 Gram's Inequality; Bounds for Gaussian Integrals.- B.5 Grassmann Integrals for Fock Space Traces.- B.5.1 Delta Functions and Integral Kernels.- B.5.2 The Formula for the Trace.- B.5.3 The Time Continuum Limit.- B.5.4 Nambu Formalism.- B.5.5 Matsubara Frequencies.- B.6 Feynman Graph Expansions.- B.7 The Thermodynamic Limit in Perturbation Theory.- B.8 Volume Improvement Bounds.- B.8.1 The One-Loop Volume Bound.- B.8.2 The Two-Loop Volume Bound.- References.