Path Integrals in Classical Theory Brownian motion: Introduction to the concept of path integration Wiener path integrals and stochastic processes Path Integrals in Quantum Mechanics Feynman path integrals Path integrals in Hamiltonian formalism Quantizations, the operator ordering problem and path integrals Path integrals and quantizations in spaces with topologiecal constraints Path integrals in curved spaces, space time transformations and the Coulomb problem Path integrals over anticommuting variables for fermions and generalizations Appendices Bibliography Index
Path Integrals in Classical Theory Brownian motion: Introduction to the concept of path integration Wiener path integrals and stochastic processes Path Integrals in Quantum Mechanics Feynman path integrals Path integrals in Hamiltonian formalism Quantizations, the operator ordering problem and path integrals Path integrals and quantizations in spaces with topologiecal constraints Path integrals in curved spaces, space time transformations and the Coulomb problem Path integrals over anticommuting variables for fermions and generalizations Appendices Bibliography Index
CONTENTS: Preface Bosonic Path Integrals Nelson's Stochastic Mechanics by Means of Feynman Path Integrals Feymman path-integrals for Brownian Classical Motions (1+1)-dimensional Burger turbulence in the framework of Hopf-Cole Transformation and Feymman Trajectories path-integrals Feynman-Wiener Path Integral Representations for the Quantum Harmonic Oscillator with Time-Dependent Stochastic Frequency: Quantum Brownian Motions and Navier-Stokes Weakly Turbulence Some studies on functional integrals representations for fluid motion with random conditions Statistics turbulence suppression by a strong magnetic field: A Feynman path integral argument A non-linear diffusion equation, Some existence and uniqueness results on weak solutions for random initial conditions and its path integral representation -- a mathematical methods approach The Exact Evalution of the Response Function for Wave Propagation in Random Media: A Functional Integral Approach On the Abstract Method of Integration for Wave Equation and some Applications for Wave Physics Feynman Path-Integral Studies for Quantum Particles Propagators Interacting with Environments The Quantum Fractional Dimensional Brownian Motion: A Path-Integral Study Ohm's Law: A Langevin Path-Integral Study at Classical and Quantum W.K.B. Regime On the Statistical of An Ideal Gas with Mass Fluctuations and the Boltzman Ergodic Theorem: The Combined Boltzman-Tsallis Statistics The Suppression of the Quantum Phenomena by an External Strong White-Noise Magnetic Field: A Path Integral Argument Index.
Concepts of Lagrangian and Hamiltonian path integrals Wiener integrals and Brownian motion representation of the statistical mechanical density matrix as a path integral propagators for local and non-local quadratic actions path integrals in polar and curvilinear coordinates and group manifolds local and global time transformation techniques approximation and numerical methods of evaluating path integrals and their applications to polaron bi polaron spectrum of disordered systems path integrals with topological constraints and their applications
Introduction Preliminary remarks The path integral formalism is one of the most important tools in modern theoretical physics. Integration in functional spaces was considered for the first time by Norbert Wiener in his studies of diffusion processes [60, 61]. The corresponding integration measure is now known as the Wiener measure, i.e. the so-called Gaussian path integrals were studied. In quantum mechanics, functional integrals were introduced by R. Feynmann [62, 63] who was inspired by important observations due to P. A. M. Dirac concerning a relation between wave function and classical action [64, 65]. In modern physics the path integral formalism is applied not only to developing a quantum description for systems with infinitely many degrees of freedom as, for example, in quantum field theories or condensed matter physics, but also to ordinary quantum mechanical problems. Moreover there is a tendency to reformulate standard quantum mechanical problems (such as a particle in a “box” or potential well [66, 67], or a particle in the Coulomb field [68]) within the framework of the path integral method. The reason for doing this is rather obvious: physicists develop a new mathematical tool, a new language and, in order to comprehend all of its subtleties, it is necessary to investigate as many concrete examples as possible. Use of the path integral formalism is widespread in quantum field theory because it is the most adequate mathematical tool for a wide range of problems which allows, not only for elegant formulation of problems, but also for a great deal of simplification of their theoretical analysis.
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.
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.
This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. It addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantum mechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann) variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature. This book is written in a concise style with careful attention to precise mathematics formulation of methods and results.Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory. Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.