This book covers the fundamentals of path integrals, both the Wiener and Feynman types, and their many applications in physics. It deals with systems that have an infinite number of degrees of freedom. The book discusses the general physical background and concepts of the path integral approach used, followed by the most typical and important applications as well as problems with either their solutions or hints how to solve them. Each chapter is self-contained and can be considered as an independent textbook. It provides a comprehensive, detailed, and systematic account of the subject suitable for both students and experienced researchers.
In the second volume of this book (chapters 3 and 4) we proceed to discuss path-integral applications for the study of systems with an infinite number of degrees of freedom. An appropriate description of such systems requires the use of second quantization, and hence, field theoretical methods. The starting point will be the quantum-mechanical phase-space path integrals studied in volume I, which we suitably generalize for the quantization of field theories.
One of the central topics of chapter 3 is the formulation of the celebrated Feynman diagram technique for the perturbation expansion in the case of field theories with constraints (gauge-field theories), which describe all the fundamental interactions in elementary particle physics. However, the important applications of path integrals in quantum field theory go far beyond just a convenient derivation of the perturbation theory rules. We shall consider, in this volume, various modern non-perturbative methods for calculations in field theory, such as variational methods, the description of topologically non-trivial field configurations, the quantization of extended objects (solitons and instantons), the 1/N-expansion and the calculation of quantum anomalies. In addition, the last section of chapter 3 contains elements of some advanced and currently developing applications of the path-integral technique in the theory of quantum gravity, cosmology, black holes and in string theory.
For a successful reading of the main part of chapter 3, it is helpful to have some acquaintance with a standard course of quantum field theory, at least at a very elementary level. However, some parts (e.g., quantization of extended objects, applications in gravitation and string theories) are necessarily more fragmentary and presented without much detail. Therefore, their complete understanding can be achieved only by rather experienced readers or by further consultation of the literature to which we refer. At the same time, we have tried to present the material in such a form that even those readers not fully prepared for this part could get an idea about these modern and fascinating applications of path integration.
As we stressed in volume I, one of the most attractive features of the path-integral approach is its universality. This means it can be applied without crucial modifications to statistical (both classical and quantum) systems. We discuss how to incorporate the statistical properties into the path-integral formalism for the study of many-particle systems in chapter 4. Besides the basic principles of pathintegral calculations for systems of indistinguishable particles, chapter 4 contains a discussion of various problems in modern statistical physics (such as the analysis of critical phenomena, calculations in field theory at non-zero temperature or at fixed energy, as well as the study of non-equilibrium systems and the phenomena of superfluidity and superconductivity). Therefore, to be tractable in a single book, these examples contain some simplifications and the material is presented in a more fragmentary style in comparison with chapters 1 and 2 (volume I). Nevertheless, we have again tried to make the text as self-contained as possible, so that all the crucial points are covered. The reader will find references to the appropriate literature for further details.
Masud Chaichian, Andrei Demichev