Manfred Salmhofer - Böcker

This book collects lecture courses and seminars given at the Les Houches Summer School 2010 on "Quantum Theory: From Small to Large Scales". Fundamental quantum phenomena appear on all scales, from microscopic to macroscopic. Some of the pertinent questions include the onset of decoherence, the dynamics of collective modes, the influence of external randomness and the emergence of dissipative behaviour. Our understanding of such phenomena has been advanced by the study of model systems and by the derivation and analysis of effective dynamics for large systems and over long times. In this field, research in mathematical physics has regularly contributed results that were recognized as essential in the physics community. During the last few years, the key questions have been sharpened and progress on answering them has been particularly strong. This book reviews the state-of-the-art developments in this field and provides the necessary background for future studies. All chapters are written from a pedagogical perspective, making the book accessible to master and PhD students and researchers willing to enter this field.

Why another book on the renormalization of field theory? This book aims to contribute to the bridging of the gap between the treatments of renor­ malization in physics courses and the mathematically rigorous approach. It provides a simple but rigorous introduction to perturbative renormalization, and, in doing so, also equips the reader with some basic techniques which are a prerequisite for studying renormalization nonperturbatively. Beside these technical issues, it also contains a proof of renormalizability of ¢4 theory in d :5 4 dimensions and a discussion of renormalization for systems with a Fermi surface, which are realistic models for electrons in metals. Like the two courses on which it is based, the book is intended to be easily accessible to mathematics and physics students from the third year on, and after going through it, one should be able to start reading the current literature on the subject, in particular on nonperturbative renormalization. Chapter 1 provides a brief motivation for studying quantum theory by functional integrals, as well as the setup. In Chap. 2, the techniques of Gaus­ sian integration and Feynman graph expansions are introduced. I then give simple proofs of basic results, such as the theorem that the logarithm of the generating functional is a sum of values of connected Feynman graphs. In Chap. 3, the Wilson renormalization flow is defined, and perturbative renormalizability of ¢4 theory in d :5 4 dimensions is proven using a renormal­ ization group differential equation. The Feynman graph expansion of Chap.