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Journal of Mathematical Physics

New aspects of the path integrational treatment of the Coulomb potential

Domenico P. L. Castrigiano, F. Stärk

Published Dec 1, 1989• 19 citations
Publisher

Abstract

The well-known treatment of the path integral for the Coulomb potential, by means of the Kustaanheimo–Stiefel transformation and a time transformation, is made more transparent by reducing the problem to the equality of two measures on the space of paths. For this equality two proofs are given: an elementary computational one and a short one recurring to general features of stochastic processes. It is shown that the time transformation is a special case of the well-known time change for a continuous local martingale, by which the process is changed to a Brownian motion and which is determined by its quadratic variation.

DOI
10.1063/1.528513

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The well-known treatment of the path integral for the Coulomb potential, by means of the Kustaanheimo–Stiefel transformation and a time transformation, is made more transparent by reducing the problem to the equality of two measures on the space of paths. For this equality two proofs are given: an elementary computational one and a short one recurring to general features of stochastic processes. It is shown that the time transformation is a special case of the well-known time change for a continuous local martingale, by which the process is changed to a Brownian motion and which is determined by its quadratic variation.
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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
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In this lecture a short introduction is given into the theory of the Feynman path integral in quantum mechanics. The general formulation in Riemann spaces will be given based on the Weyl- ordering prescription, respectively product ordering prescription, in the quantum Hamiltonian. Also, the theory of space-time transformations and separation of variables will be outlined. As elementary examples I discuss the usual harmonic oscillator, the radial harmonic oscillator, and the Coulomb potential. Lecture given at the graduate college ''Quantenfeldtheorie und deren Anwendung in der Elementarteilchen- und Festkörperphysik'', Universität Leipzig, 16-26 November 1992.