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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Comment on "Discretization problems of functional integrals in phase space"

Maurice M. Mizrahi

Published Oct 15, 1980• 1 citation
Publisher

Abstract

It is shown that the definition of path integrals without limiting procedure does not give rise to unresolved factor-ordering ambiguities. Any inference from the recent work of Langouche et al. [Phys. Rev. D 20, 419 (1979)] that one is forced to resort to "discretization" (time slicing) to resolve these so-called "ambiguities" is thereby shown to be incorrect.

DOI
10.1103/physrevd.22.2096

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It is shown that the definition of path integrals without limiting procedure does not give rise to unresolved factor-ordering ambiguities. Any inference from the recent work of Langouche et al. [Phys. Rev. D 20, 419 (1979)] that one is forced to resort to "discretization" (time slicing) to resolve these so-called "ambiguities" is thereby shown to be incorrect.
In a recent letter [PRL 86, 1 (2001)], Gollisch and Wetterich show that a careful treatment of discretization errors in a phase-space path integral formulation of quantum mechanics leads to a correction term as compared to the standard form based on coherent states. We point out in this comment that their approach is not unique and that the coherent state path integral formalism, without correction term, yields the same result. It does this as long as known (but sometimes neglected) exponential convergence factors, which control discretization errors in this approach, are taken into account.
In a previous publication [Phys. Rev. D 36, 1135 (1987)] it was shown how the stochastic process associated with the solution of the Schrödinger or diffusion equation can be derived by an infinity of ‘‘Gaussian tricks.’’ In this article the method is extended to differential operators of the form (p−eA/c)2, ∂c2(x)∂, and Δ =(1/(g)1/2)∂μ((g)1/2gμν∂ν). In this formulation the relation between operator ordering ambiguities and time labeling in the functional integral is immediate. In particular, it is clear where the choice between Ito and Stratonovich integrals enters.
Techniques for integration in function spaces which are not necessarily vector spaces are presented in the light of DeWitt-Morette’s recent redefinition of path integrals, which does not involve the usual limiting process, and is therefore free from the ambiguities inherent in this approach. General translated Gaussian measures defined by their Fourier transforms are introduced on various path spaces and used to derive generalized moments formulas, and what is essentially the Feynman–Kac formula for the expression of solutions of the Schrödinger equation as functional integrals.
This paper defines path integrals in phase space without using a time-division approach followed by a limiting process, thereby generalizing a similar procedure used in configuration space. This is useful since the path integral approach cannot always be formulated in configuration space (e.g., when the Hamiltonian is arbitrary) but can always be formulated in phase space. The most general Gaussian measure, absorbing the quadratic portion of the functional to be integrated, is constructed, and large classes of path integrals are evaluated with respect to it. Applications are given to the perturbation expansion and the semiclassical (WKB) expansion for arbitrary Hamiltonians.