Ebook: Techniques and Applications of Path Integration

Title Page; Copyright Page; Dedication; Preface; Table of Contents; PART ONE -- Introduction; ONE -- Introducing and Defining the Path Integral; APPENDIX: THE TROTTER PRODUCT FORMULA; NOTES; TWO -- Probabilities and Probability Amplitudes for Paths; NOTES; THREE -- Correspondence Limit for the Path Integral (Heuristic); APPENDIX: USEFUL INTEGRALS; NOTES; FOUR -- Vector Potentials and Another Proof of the Path Integral Formula; NOTES; FIVE -- The Ito Integral and Gauge Transformations; NOTES; SIX -- Doing the Integral: Free Particle and Quadratic Lagrangians; NOTES.;A book of techniques and applications, this text defines the path integral and illustrates its uses by example. It is suitable for advanced undergraduates and graduate students in physics; its sole prerequisite is a first course in quantum mechanics. For applications requiring specialized knowledge, the author supplies background material. The first part of the book develops the techniques of path integration. Topics include probability amplitudes for paths and the correspondence limit for the path integral; vector potentials; the Ito integral and gauge transformations; free particle and quadratic Lagrangians; properties of Green's functions and the Feynman-Kac formula; functional derivatives and commutation relations; Brownian motion and the Wiener integral; and perturbation theory and Feynman diagrams. The second part, dealing with applications, covers asymptotic analysis and the calculus of variations; the WKB approximation and near caustics; the phase of the semiclassical amplitude; scattering theory; and geometrical optics. Additional topics include the polaron; path integrals for multiply connected spaces; quantum mechanics on curved spaces; relativistic propagators and black holes; applications to statistical mechanics; systems with random impurities; instantons and metastability; renormalization and scaling for critical phenomena; and the phase space path integral.