Statistical approach to quantum field theory. An introduction
It is known that statistical field theory studies the behavior of classical or quantum systems consisting of a large number of degrees of freedom in and out of equilibrium. On the other hand, quantum field theory deals with a theoretical framework for constructing quantum mechanical models of systems with an infinite number of degrees of freedom. In the past decades, the powerful methods in statistical physics and Euclidean quantum field theory have common tools based on the use of path integrals. The book consists of 15 chapters. At the end of many chapters, one can find listings of computer programs either written in C or for \texttt{Matlab}. In Chapters 2 and 3, a self contained introduction to path integrals in Euclidean quantum mechanics and statistical mechanics is provided. The high-dimensional integrals can be estimated with the help of Monte Carlo simulations based on Markov processes. The method is first introduced and then applied to ordinary integrals and to quantum mechanical systems (see Chapter 4). Chapter 5 is devoted to scalar fields at low and finite temperatures. In Chapter 6, several classical spin models such as the Ising model, the Potts model, the \(\mathbb{Z}_q\)-model, the \(\mathrm{U}(1)\)-model, the \(\mathrm{O}(N)\) model, etc., are provided. Despite of their simplicity, the models are often used to illustrate the key features of statistical systems and the methods available to understand these features. Chapter 6 is devoted to mean field approximation. Basically, the above mentioned models are studied in detail. In Chapter 8, the transfer matrix method is studied. As an illustration, the Ising and Potts models are considered. Moreover, models of Euclidean quantum mechanics are studied as well. Chapter 9 is devoted to high-temperature and low-temperature expansions of the partition functions of Ising and \(\mathrm{O}(N)\) models. In Chapter 10, Peierls argument is considered. As an alternative to the lattice formulation of quantum field theories, one may use a variant of the flexible renormalization group methods. In Chapters 11 and 12, the renormalization group methods are investigated. It is known that all fundamental interactions in nature are described by gauge theories. Gauge theories can be formulated on a finite space-time lattice without spoiling the important local gauge invariance. Chapters 13, 14 and 15 are devoted to lattice gauge theories. The book is well written, and pedagogically well-organized. The book has benefited by last-year undergraduate and beginning postgraduate students in physics and applied mathematics. I think it also could be useful for researcher in the fields Note that each chapter ends with some problems which help to better understand the material given.
scientific article; zbMATH DE number 6081519