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Progress of Theoretical Physics

Operator Ordering and Perturbation Expansion in the Path Integral Formalism

M.-a. Sato

Published Oct 1, 1977• 23 citations
Publisher

Abstract

The operator ordering problem in the path integral formalism is studied in the context of the perturbation expansion. For a given ordered Hamiltonian, there exists a path integral formula which reproduces the corresponding quantum mechanics, i.e., the same Feynman­ Wick rules as the operator formalism. The Weyl ordering is shown to play an important role in connecting the discrete path integral (defined on a finite mesh) with the continuous one.

DOI
10.1143/ptp.58.1262

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The operator ordering problem in the path integral formalism is studied in the context of the perturbation expansion. For a given ordered Hamiltonian, there exists a path integral formula which reproduces the corresponding quantum mechanics, i.e., the same Feynman­ Wick rules as the operator formalism. The Weyl ordering is shown to play an important role in connecting the discrete path integral (defined on a finite mesh) with the continuous one.
The integrals in the path integral are strongly oscillating and can only be defined by analytic continuation. It is important to note that the continuous expression is just a notation for the discrete version of the path integral, but formal manipulations will be much easier to perform in this continuous formulation. Furthermore, the integral is only defined through the analytic continuation in time. As long as the Hamiltonian is quadratic in the momenta, the stationary phase approximation for the momentum integral is exact. In practical situations one splits from the action the quadratic part in the coordinates and velocities and considers the rest as a perturbation.
The old problem of how to represent uniquely a prescribed classical Hamiltonian H as a well-defined quantal operator Ĥ is shown to have a clear answer within Feynman's path-integral scheme (as expanded by Garrod) for quantum mechanics. The computation of Ĥ involves the momentum Fourier transform of a coordinate average of H. A differential equation for a reduced form of the Feynman propagator giving Ĥ from H is found; and the example of polynomial H worked out to give the Born-Jordan ordering rule for Ĥ in this case.
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.
Feynman path integrals in a phase space are analyzed in detail. The analysis is based on the theory of operator symbols, in contrast with the traditional approach based on the direct use of canonical commutation relations. Particular attention is paid to the Weyl and Wick symbols, which are the most important in applications. The set of paths on which the integral is concentrated is studied. It is found that these paths are always discontinuous. This discontinuity is responsible for errors in certain papers on path integrals in a phase space. The most important properties of the Weyl and Wick symbols are reviewed.