Canonical Transformations and Path Integral Measures

TL;DR

This paper investigates how classical canonical transformations affect the measure in quantum Hamiltonian path integrals, revealing anomalies and developing methods to evaluate these effects for various quantum systems.

Contribution

It generalizes previous work by analyzing the quantum measure's non-invariance under classical transformations and introduces techniques to handle associated anomalies in path integrals.

Findings

Canonical transformations induce anomalies in the path integral measure.

Methods to evaluate path integral prefactors for systems with anomalies.

Application of techniques to linear, quadratic, and time-dependent potentials.

Abstract

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.