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Communications in Theoretical Physics

Parametrization of Ordering Ambiguity and Its Representation in Path Integral Formalism

Dehai Bao, Zhongyuan Zhu

Published Sep 1, 1991• 2 citations
Publisher

Abstract

In this paper, we establish three different types (type I, type II and type III) of parametrization of ordering ambiguity in operator formalism of quantum mechanics and discuss their nontrivial relations. Some commonly occurred bewilderment about ordering problem is made clear. We also show that the ordering ambiguity exists in path intergral formalism and the representation of the ordering ambiguity in the path integral formalism (discrete schemes) has a one-to-one correspondence with type II parametrization of the ordering ambiguity in the operator formalism. As a result, we show that to select a physically a misled attempt. Instead, we have to work with type III parametrization in the operator formalism to reduce the ordering ambiguity by physical requirements.

DOI
10.1088/0253-6102/16/2/201

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In this paper, we establish three different types (type I, type II and type III) of parametrization of ordering ambiguity in operator formalism of quantum mechanics and discuss their nontrivial relations. Some commonly occurred bewilderment about ordering problem is made clear. We also show that the ordering ambiguity exists in path intergral formalism and the representation of the ordering ambiguity in the path integral formalism (discrete schemes) has a one-to-one correspondence with type II parametrization of the ordering ambiguity in the operator formalism. As a result, we show that to select a physically a misled attempt. Instead, we have to work with type III parametrization in the operator formalism to reduce the ordering ambiguity by physical requirements.
Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.
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.
Ambiguities arising in different approaches (canonical, quasiclassical, path integration) to quantization are discussed by an example of the mechanics of a point-like particle in the Riemannian space (the geodesic dynamics). A way to select a single rule of quantization is proposed by requirement of a consistency of the quantum mechanics' following from the canonical and quasiclassical approaches. This rule selects also a definition of the path integration. A geometric interpretation of the noncovariance of the canonical Hamilton operator of a particle with respect to the diffeomorphisms of the Riemannian configuration space is proposed.