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Canadian Journal of Physics

Coherent-state path integral operator ordering, and regularization: A simple example

L. C. Lew Yan Voon

Published Oct 1, 1991• 0 citations
Publisher

Abstract

For a restricted class of potential problems, the Feynman path integral in the coordinate representation circumvents operator-ordering ambiguities. We will show that, in the coherent-state representation, the traditional continuum time path integral cannot resolve this issue. Using a simple solvable model, we carry out a careful analysis of the continuum limit with regularization. The latter is found to be necessary to address the operator-ordering issue.

DOI
10.1139/p91-183

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For a restricted class of potential problems, the Feynman path integral in the coordinate representation circumvents operator-ordering ambiguities. We will show that, in the coherent-state representation, the traditional continuum time path integral cannot resolve this issue. Using a simple solvable model, we carry out a careful analysis of the continuum limit with regularization. The latter is found to be necessary to address the operator-ordering issue.
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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.
Infinitesimal propagators have been used in quantum-mechanical problems to establish ordering rules for classical quantities. Ambiguities arise in operator ordering as a result of the freedom of choice in evaluating the classical action over a short-time interval. The primitive quadrature rule employed in a short-time propagator establishes the quantum system associated with the classical Hamiltonian. Normal propagators are constructed from products of infinitesimal propagators and represent a compound quadrature in which the path is broken into line segments and a separate quadrature rule is applied to each line segment. It is demonstrated for Hamiltonians involving the vector potential that higher-order quadratures of the infinitesimal propagator all give a unique and proper ordering. Similarly, Hamiltonians that are products of powers of position x and momentum p, i.e., pl{\mathit{p}}^{\mathit{l}}plxm{\mathit{x}}^{\mathit{m}}xm, show some preference for Born-Jordon ordering as higher quadrature rules are applied. Unique orderings can be obtained in some cases by solving for the normal propagator. This is done for the Hamiltonian pl{\mathit{p}}^{\mathit{l}}plxm{\mathit{x}}^{\mathit{m}}xm by using a Fourier representation for the phase-space propagator. This formalism requires that both the momentum and position coordinates be represented by Fourier series. These results show that care must be taken when generating quantum-mechanical operators by using primitive quadrature rules in conjunction with a short-time propagator. These ambiguities are eliminated when the compound quadrature is performed to generate the normal propagator.