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A concise summary of one of the 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...

To achieve a meaningful generalization of the classical theory of Riemann surfaces to the case of infinite genus, one must impose restrictions on the asymptotic behavior of the Riemann surface. This book geometrically constructs Riemann surfaces of infinite genus by pasting together plane domains and handles.

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 Nicolo 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...