General Aspects of the Space and Time Transformations for the Coulomb Propagator

Abstract

IntroductionFor a surprisingly long time the problem of evaluating the path integral for the Coulomb propagator remained unsolved. Only in 1979, Duru/Kleinert [1] presented a solution by reducing the Coulomb path integral to gaussian ones. This reduction was achieved by using just the same two transformations of paths which in the mid-sixties had been invented by Kustaanheimo and Stiefel [2] in order to develop a linear and regular celestial mechanics. One of these transformations, the now called Kustaanheimo -Stiefel transformation consists in a rather special change of variables by which the Kepler trajectories are transferred to a four-dimensional configuration space. The other one works by a pathwise time substitution causing a kind of slow motion in the vicinity of the central-body.In the sequel this second kind of transformations, now called (new -)time transformations, became a general tool for the evaluation of path integrals. Employed for the first time by Duru and Kleinert for solving the Coulomb path integral, Pak/Sokmen [3] developed a general new time formalism for path integrals, and Inomata and collaborators [4] showed in a series of papers the applicability of the new-time technique to several relevant potentials other than the Coulomb one. But also the old attempt of solving the Coulomb path integral by exploiting explicitly its radial symmetry -a general partial wave expansion of path integrals using polar coordinates had been provided by Peak/Inomata [5] -finally succeeded by employing the new-time method, see Inomata [6] and Steiner [7], and also [8]. Soon after the solution of the Coulomb problem by Duru and Kleinert it became clear that the new-time transformation has a more fundamental meaning than to be a mere integration technique. Blanchard/Sirugue [9], Blanchard Quantum Probability and Related Topics Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 08/27/15. For personal use only.

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