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Journal of Mathematical Physics

Quantum Operator Ordering and the Feynman Formulation

Francis Joseph Testa

Published Aug 1, 1971• 32 citations
Publisher

Abstract

A functional integral representation for the quantum Hamiltonian propagator is obtained from the Feynman formulation. It is shown that operator ordering arbitrariness in the quantum Hamiltonian is directly related to arbitrariness in the algorithmic definition of the functional integral. Examples are cited which demonstrate this fact.
DOI
10.1063/1.1665758

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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 a paper entitled ’’An Operator Calculus Having Applications in Quantum Electrodynamics’’ [Phys. Rev. 84, 108 (1951)], Feynman gives techniques and theorems for working with ordered operators. We believe that the entire development, because of its direct involvement in the transition from the Heisenberg–Schrödinger to Feynman’s path-integral formulation of quantum mechanics, is of general interest. We aim to make these less appreciated early techniques of Feynman more accessible to the general physicist.
Operator calculusThe operator calculus developed by Feynman [Fey51] makes it possible to represent functions of (noncommuting) operators as path integrals, with the integrand being the path-ordered exponential of operators, the order of which is controlled by a parameter that varies along the trajectory.This procedure is termed Feynman disentangling.It is also applicable to functions of matrices (say, γ-matrices which are associated with a spinor particle).When applied to the evolution operator, this procedure results in the standard path-integral representation of quantum mechanics.In this chapter we first demonstrate the general technique using the simplest example, a free propagator in Euclidean space, and then consider the path-integral representation of quantum mechanics, as well as propagators in an external electromagnetic field. Free propagatorLet us first consider the simplest propagator of a free scalar field which is given in the operator formalism by the vacuum expectation value of thewith ϕ being the field-operator.The T -product (1.1) obeys the equationwhere d = 4 is the dimension of space-time, however the formulas are applicable at any value of d.In the operator formalism, Eq. (1.2) is a * The ordered products of operators were introduced by Dyson [Dys49].This paper and other classical papers on quantum electrodynamics are collected in the book edited by Schwinger [Sch58].
Operator ordering for Fermi systems is formulated generally with the aid of the Grassmann numbers. The ordering schemes are restricted to a special class called s-ordering in contrast with those for Bose systems. A relation, similar to that for Bose systems, is shown to exist between different values of the equal-time contractions in the Dyson–Wick expansions of the S matrix and different operator orderings in the interaction Hamiltonian. The S matrix is independent of the ordering scheme chosen. In the functional integral context, different orderings correspond to different discretization prescriptions for the definition of the functional integrals in terms of coherent states for Fermi operators.