Joel S. Feldman

The authors give a detailed and pedagogically well written proof of the renormalizability of quantum electrodynamics in four dimensions. The proof is based on the free expansion of Gallavotti and Nicolò and is mathematically rigorous as well as impressively general. It applies to rather general models of quantum field theory including models with infrared or ultraviolet singularities, as shown in this monograph for the first time. Also discussed are the loop regularization for renormalized graphs and the Ward identities. The authors also establish that in QED in four dimensions only gauge invariant counterterms are required. This seems to be the first proof which will be accessible not only to the expert but also to the student.

This book, written by well-known experts in the field, offers a concise summary of one of the latest and most significant developments in the theoretical analysis of quantum field theory. The renormalization group is the name given to a technique for analyzing the qualitative behavior of a class of physical systems by iterating a map on the vector space of interactions for the class. In a typical nonrigorous application of this technique, one assumes, based on one's physical intuition, that only a certain finite dimensional subspace (usually of dimension three or less) is important. The material in this book concerns a technique for justifying this approximation in a broad class of fermionic models used in condensed matter and high energy physics. This volume is based on the Aisenstadt Lectures given by Joel Feldman at the Centre de Recherches Mathematiques (Montreal, Canada). It is suitable for graduate students and research mathematicians interested in mathematical physics. Included are many problems and solutions.