Path Integrals and Stochastic Processes in Theoretical Physics

Масуд Чайчиан, Andrey P. Demichev

Horacio S. Wio

Масуд Чайчиан, Andrey P. Demichev

Path Integrals in Physics: Volume I, Stochastic Processes and Quantum Mechanics presents the fundamentals of path integrals, both the Wiener and Feynman type, and their many applications in physics. Accessible to a broad community of theoretical physicists, the book deals with systems possessing a infinite number of degrees in freedom. It discusses the general physical background and concepts of the path integral approach used, followed by a detailed presentation of the most typical and important applications as well as problems with either their solutions or hints how to solve them. It describes in detail various applications, including systems with Grassmann variables. Each chapter is self-contained and can be considered as an independent textbook. The book provides a comprehensive, detailed, and systematic account of the subject suitable for both students and experienced researchers.

Sumita Datta

Abstract Feynman developed the path integral approach to quantum mechanics as an alternative to wave or matrix mechanics that has found fruitful implementation in quantum mechanics, quantum electrodynamics, and statistical physics. Despite its many successes, the usual Feynman path integral formalism suffers from a lack of mathematical rigor. Questions of mathematical rigor still exist, but Mark Kac provided a rigorous path integral approach to quantum mechanics. The Kac formulation for quantum systems is given in terms of Wiener measure for a path integral and employs the Brownian motion distribution. In this paper, we review the Feynman–Kac approach and other stochastic processes related to Brownian motion. We also give an account of the successes and difficulties we faced during our journey through numerical path integral problems involved in physics.

Walter T. Strunz

The general theory of path integral propagators for the solution of linear quantum state diffusion (LQSD) stochastic Schr\"odinger equations describing open quantum systems is developed. Both Hamiltonian and, where possible, Lagrangian path integrals are derived and their connection established. The Hamiltonian version turns out to be more suitable. The results also show how the stochastic terms in the LQSD equation introduce a weight functional under the path integral, thus restricting the set of contributing paths. The center of this weight functional is determined by the stochastic processes governing the LQSD equation. In general, this picture holds in a semiclassical limit only. Some peculiarities of stochastic path integrals are pointed out. We evaluate the stochastic path integral in closed form for soluble models, gaining further insight into the behavior of the solutions of the LQSD equation. \textcopyright{} 1996 The American Physical Society.