The importance of path-integral methods in theoretical physics can hardly be disputed. Their applications in most branches of modern physics have proved to be extremely fruitful not only for solving already existing problems but also as a guide for the formulation and development of essentially new ideas and approaches in the description of physical phenomena.
This book expounds the fundamentals of path integrals, of both the Wiener and Feynman type, and their numerous applications in different fields of physics. The book has emerged as a result of many courses given by the authors for students in physics and mathematics, as well as for researchers, over more than 25 years and is based on the experience obtained from their lectures.
The mathematical foundations of path integrals are summarized in a number of books. But many results, especially those concerning physical applications, are scattered in a variety of original papers and reviews, often rather difficult for a first reading. In writing this book, the authors’ aim was twofold: first, to outline the basic ideas underlying the concept, construction and methods for calculating the Wiener, Feynman and phase-space quantum-mechanical path integrals; and second, to acquaint the reader with different aspects concerning the technique and applications of path integrals.
It is necessary to note that, despite having almost an 80-year history, the theory and applications of path integrals are still a vigorously developing area. In this book we have selected for presentation the more or less traditional and commonly accepted material. At the same time, we have tried to include some major achievements in this area of recent years. However, we are well aware of the fact that many important topics have been either left out or are only briefly mentioned. We hope that this is partially compensated by references in our book to the original papers and appropriate reviews.
The book is intended for those who are familiar with basic facts from classical and quantum mechanics as well as from statistical physics. We would like to stress that the book is not just a linearly ordered set of facts about path integrals and their applications, but the reader may find more effective ways to learn a desired topic. Each chapter is self-contained and can be considered as an independent textbook: it contains general physical background, the concepts of the path-integral approach used, followed by most of the typical and important applications presented in detail. In writing this book, we have endeavored to make it as comprehensive as possible and to avoid statements such as ‘it can be shown’ or ‘it is left as an exercise for the reader’, as much as it could be done.
A beginner can start with any of the first two chapters in volume I (which contain the basic concepts of path integrals in the theory of stochastic processes and quantum mechanics together with essential examples considered in full detail) and then switch to his/her field of interest. A more educated user, however, can start directly with his/her preferred field in more advanced areas of quantum field theory and statistical physics (volume II), and eventually return to the early chapters if necessary.