The main subject of this book can be viewed in various ways. From the standpoint of functional analysis, it studies spectral properties of a certain class of linear operators; from the viewpoint of probability theory, it is concerned with the analysis of singular Markov processes; and, from the vantage point of mathematical physics, it analyzes the dynamics of equilibrium systems in quantum statistical physics and quantum field theory. Malyshev and Minlos describe two new approaches to the subject which have not been previously treated in monograph form. They also present background material which makes the book accessible and useful to researchers and graduate students working in functional analysis, probability theory, and mathematical physics.

Suitable for advanced undergraduates and graduate students, this compact treatment examines linear space, functionals, and operators; diagonalizing operators; operator algebras; and equations of motion. 1969 edition.

Mathematical physics has made enormous strides over the past few decades, with the emergence of many new disciplines and with revolutionary advances in old disciplines. One of the especially interesting features is the link between developments in mathematical physics and in pure mathematics. Many of the exciting advances in mathematics owe their origin to mathematical physics — superstring theory, for example, has led to remarkable progress in geometry — while very pure mathematics, such as number theory, has found unexpected applications.The beginning of a new millennium is an appropriate time to survey the present state of the field and look forward to likely advances in the future. In this book, leading experts give personal views on their subjects and on the wider field of mathematical physics. The topics covered range widely over the whole field, from quantum field theory to turbulence, from the classical three-body problem to non-equilibrium statistical mechanics.

This book is an introduction to the remarkable work of Vaughan Jones and Victor Vassiliev on knot and link invariants and its recent modifications and generalizations, including a mathematical treatment of Jones-Witten invariants. The mathematical prerequisites are minimal compared to other monographs in this area. Numerous figures and problems make this book suitable as a graduate level course text or for self-study.

This book is devoted to a systematic analysis of asymptotic behavior of distributions of various typical functionals of Gaussian random variables and fields. The text begins with an extended introduction, which explains fundamental ideas and sketches the basic methods fully presented later in the book. Good approximate formulas and sharp estimates of the remainders are obtained for a large class of Gaussian and similar processes. The author devotes special attention to the development of asymptotic analysis methods, emphasizing the method of comparison, the double-sum method and the method of moments. The author has added an extended introduction and has significantly revised the text for this translation, particularly the material on the double-sum method.

This monograph offers a state-of-the-art mathematical account of functional integration methods in the context of self-adjoint operators and semigroups using the concepts and tools of modern stochastic analysis. These ideas are then applied principally to a rigorous treatment of some fundamental models of quantum field theory. In this self-contained presentation of the material both beginners and experts are addressed, while putting emphasis on the interdisciplinary character of the subject.

This book reviews some of the classic aspects in the theory of phase transitions and critical phenomena, which has a long history. Recently, these aspects are attracting much attention due to essential new contributions. The topics presented in this book include : mathematical theory of the Ising model; equilibrium and non-equilibrium criticality of one-dimensional quantum spin chains; influence of structural disorder on the critical behaviour of the Potts model; criticality, fractality and multifractality of linked polymers; field-theoretical approaches in the super conducting phase transitions. The book is based on the review lectures that were given in Lviv (Ukraine) in March 2002 at the "Ising lectures" - a traditional annual workshop on phase transitions and critical phenomena which aims to bring together scientists working in the field of phase transitions with university students and those who are interested in the subject.