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Physical Review A

General new-time formalism in the path integral

Namik Kemal Pak, İ. Sökmen

Published Oct 1, 1984• 71 citations
Publisher

Abstract

We describe a general method of applying point canonical transformations to the path integral followed by the corresponding new-time transformations aimed at reducing an arbitrary one-dimensional problem into an exactly solvable form. Our result is independent of operator-ordering ambiguities by construction. The method is used in getting the exact path-integral solutions of the Coulomb- and Morse-potential problems.

DOI
10.1103/physreva.30.1629

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We describe a general method of applying point canonical transformations to the path integral followed by the corresponding new-time transformations aimed at reducing an arbitrary one-dimensional problem into an exactly solvable form. Our result is independent of operator-ordering ambiguities by construction. The method is used in getting the exact path-integral solutions of the Coulomb- and Morse-potential problems.
The behavior of Hamiltonian path integrals under canonical transformations defined by a generator is investigated. The exact form of the kernel of the unitary operator which implements the quantum transformation is indicated. Equivalence rules are found (for the Hamiltonian formalism in the one-dimensional case) allowing one to get rid of the nonstandard terms in the action. It is shown that under canonical transformations the Hamiltonian path integral changes its form: in the transformed expression one cannot get along only with the corresponding classical Hamiltonian function.
The Feynman-Garrod path integral representation for time evolution is extended to arbitrary one-parameter continuous canonical transformations. One thereupon obtains a generalized Kerner-Sutcliffe formula for the unique quantum representation of the transformation generator, which can be an arbitrary classical dynamical variable. This closed-form quantization procedure is shown to be equivalent to a natural extension of the standard canonical quantization recipe -- an extension that resolves the operator-ordering ambiguity in favor of the Born-Jordan rule.
This paper is a generalization of previous work on the use of classical canonical transformations to evaluate Hamiltonian path integrals for quantum-mechanical systems. Relevant aspects of the Hamiltonian path integral and its measure are discussed, and used to show that the quantum-mechanical version of the classical transformation does not leave the measure of the path-integral invariant, instead inducing an anomaly. The relation to operator techniques and ordering problems is discussed, and special attention is paid to incorporation of the initial and final states of the transition element into the boundary conditions of the problem. Classical canonical transformations are developed to render an arbitrary power potential cyclic. The resulting Hamiltonian is analyzed as a quantum system to show its relation to known quantum-mechanical results. A perturbative argument is used to suppress ordering-related terms in the transformed Hamiltonian in the event that the classical canonical transformation leads to a nonquadratic cyclic Hamiltonian. The associated anomalies are analyzed to yield general methods to evaluate the path integral's prefactor for such systems. The methods are applied to several systems, including linear and quadratic potentials, the velocity-dependent potential, and the time-dependent harmonic oscillator.