Renormalization

This book addresses the gap between the treatments of renormalization in physics courses and a mathematically rigorous approach. It offers a clear introduction to perturbative renormalization while equipping readers with essential techniques for studying nonperturbative renormalization. The text includes a proof of the renormalizability of φ^4 theory in d = 4 dimensions and discusses renormalization in systems with a Fermi surface, relevant for modeling electrons in metals. Targeted at third-year mathematics and physics students, the book aims to make the subject accessible, preparing readers to engage with current literature, particularly on nonperturbative renormalization. Chapter 1 motivates the study of quantum theory through functional integrals and outlines the setup. Chapter 2 introduces Gaussian integration techniques and Feynman graph expansions, providing straightforward proofs of fundamental results, including the theorem that the logarithm of the generating functional equals the sum of connected Feynman graph values. In Chapter 3, the Wilson renormalization flow is defined, and the perturbative renormalizability of φ^4 theory in d = 4 dimensions is established using a renormalization group differential equation.