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Journal of Physics A Mathematical and General

Hamiltonian path integrals in n dimensions

J C Martinez

Published Aug 1, 1986• 2 citations
Publisher

Abstract

Using Hamiltonian path integral the author obtains the spurious potential term that arise in the action when a path integral is transformed to general n-dimensional coordinates.
DOI
10.1088/0305-4470/19/11/001

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The path integral evaluation of the hydrogen atom by Duru and Kleinert is studied by recognizing it as a special case of the general treatment of the separable Hamiltonian of Liouville-type. The basic dynamical principle involved is identified as the Jacobi’s principle of least action for given energy which is reparametrization invariant, and thus the appearance of a gauge freedom is naturally understood. Unlike the conventional Feynman path integral,which deals with a space-time picture of particle motion, the path integral on the basis of the Jacobi’s principle sums over orbits in space. We illustrate these properties by evaluating an exact path integral of the hydrogen atom in parabolic coordinates. In 3-dimensions the parabolic coordinates do not quite lead to a Liouville-type Hamiltonian, but we can devise asimple trick which converts it to a Liouville-type Hamiltonian and thus avoiding the use of the Kustaanheimo-Stiefel transformation. In the present formulation, the Hamiltonian for Stark effect is converted to the one for anharmonic oscillators with an unstable quartic coupling. We also study the hydrogen atom path integral from a view point of one-dimensional quantum gravity coupled to matter fields representing the electron coordinates. A simple BRST analysis of the problem is presented. 1 1
The behavior of Hamiltonian path integrals under canonical transformations defined by a generator is investigated. The exact form of the kernel of the unitary operator which implements the quantum transformation is indicated. Equivalence rules are found (for the Hamiltonian formalism in the one-dimensional case) allowing one to get rid of the nonstandard terms in the action. It is shown that under canonical transformations the Hamiltonian path integral changes its form: in the transformed expression one cannot get along only with the corresponding classical Hamiltonian function.