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Journal of Mathematical Physics

Sensitive terms in the path integral: Ordering and stochastic options

Bernard Gaveau, Lawrence S. Schulman

Published Sep 1, 1989• 7 citations
Publisher

Abstract

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.

DOI
10.1063/1.528239

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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.
Abstract Recently, doubts have been cast on the validity of the continuous-time coherent state path integral. This has led to controversies regarding the correct way of performing calculations with path integrals, and to several alternative definitions of what should be their continuous limit. Furthermore, the issue of a supposedly proper ordering of the Hamiltonian operator, entangled with the continuous-time limit, has led to considerable confusion in the literature. Since coherent state path integrals are at the basis of the modern formulation of many-body quantum theory, it should be laid on solid foundations. Here, we show that the issues raised above are coming from the illegitimate use of the (standard) rules of calculus, which are not necessarily valid in path integrals. This is well known in the context of stochastic equations, in particular in their path integral formulation. This insight allows for solving these issues and addressing the correspondence between the various orderings at the level of the path integral. We also use this opportunity to address the proper calculation of a functional determinant in the presence of a Hubbard–Stratonovich field, which shares in the controversies.
Recently, doubts have been cast on the validity of the continuous-time coherent state path integral. This has led to controversies regarding the correct way of performing calculations with path integrals, and to several alternative definitions of what should be their continuous limit. Furthermore, the issue of a supposedly proper ordering of the Hamiltonian operator, entangled with the continuous-time limit, has led to considerable confusion in the literature. Since coherent state path integrals are at the basis of the modern formulation of many-body quantum theory, it should be laid on solid foundations. Here, we show that the standard coherent state path integral is fine as it is, and does not need any modification. We show that the issues raised above are coming from illegitimate use of the (standard) rules of calculus, which are not necessarily valid in path integrals. This is well known in the context of stochastic equations, in particular in their path integral formulation. This insight allows for solving these issues, and addressing the correspondence between the various orderings at the level of the path integral. We also use this opportunity to address the proper calculation of a functional determinant in the presence of a Hubbard-Stratonovich field, which shares in the controversies.
The general theory of path integral propagators for the solution of linear quantum state diffusion (LQSD) stochastic Schr\"odinger equations describing open quantum systems is developed. Both Hamiltonian and, where possible, Lagrangian path integrals are derived and their connection established. The Hamiltonian version turns out to be more suitable. The results also show how the stochastic terms in the LQSD equation introduce a weight functional under the path integral, thus restricting the set of contributing paths. The center of this weight functional is determined by the stochastic processes governing the LQSD equation. In general, this picture holds in a semiclassical limit only. Some peculiarities of stochastic path integrals are pointed out. We evaluate the stochastic path integral in closed form for soluble models, gaining further insight into the behavior of the solutions of the LQSD equation. \textcopyright{} 1996 The American Physical Society.