This outstanding, extensive monograph deals with the phenomenon of scale invariance,

which is a property observed in the critical behaviour of systems with an infinite number

of degrees of freedom studied in quantum field theory, statistical physics, the theory of

turbulence and other fields. The method making it possible to understand such behaviour (at least on the “physical level of rigor”) is the famous renormalization group (RG) technique, developed successively over the last fifty years. From a purely mathematical point of view, there are still many hard problems in the very foundations of this theory left to be clarified and the corresponding results rigorously proven. The book is based on a series of lectures for graduate students on quantum RG techniques given by the author at St. Petersburg University. It is intended not only as a general and self-contained textbook but also as a reference text containing the final, most accurate results of deep calculations often with a long history of development. The prime goal of the book is to give a thorough and detailed explanation of all essential computational techniques.

The first chapter (“Foundations of the Theory of Critical Phenomena”) outlines the general scheme of the RG method in the theory of critical phenomena of statistical physics. It contains an introductory section with a nicely written historical review of the subject. This material is greatly expanded later in chapter 4 (“Critical Statistics”), which is the major part of the book. It deals with many aspects of RG analysis, including the ε and 1/n expansions and the problem of critical conformal invariance. Chapters 2 (“Functional and Diagrammatic technique of Quantum Field Theory”) and 3 (“Ultraviolet Renormalization”) deal with techniques of the RG in quantum field theory. Chapter 5 is devoted to problems of critical dynamics with stochastic Langevin equations, while chapter 6 deals with an important example of non Langevin type equations: the stochastic theory of turbulence. In conclusion, this is a monograph containing an impressive wealth of material, explaining in detail one of the most universal, nontrivial and powerful computational methods of contemporary theoretical physics.