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Physical Review A

Phase-space anomalies and canonical transformations

Mark S. Swanson

Published Apr 1, 1993• 5 citations
Publisher

Abstract

Anomalies associated with canonical transformations in quantum-mechanical systems are examined in the context of the path-integral formalism, where they appear as nontrivial Jacobians. The quantum-mechanical anomaly, for the case of a transformation to cyclic coordinates, is shown to be equivalent to a term in the action, similar to a gauge field configuration, whose presence maintains self-adjointness of the Hamiltonian or corrects the phase-space measure of the path integral to reflect the bound-state spectrum of the original problem. Various nonperturbative aspects of quantum-mechanical transition amplitudes are calculated by using classical solutions to Hamilton's equations of motion combined with the anomaly generated by the canonical transformation.

DOI
10.1103/physreva.47.r2431

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Anomalies associated with canonical transformations in quantum-mechanical systems are examined in the context of the path-integral formalism, where they appear as nontrivial Jacobians. The quantum-mechanical anomaly, for the case of a transformation to cyclic coordinates, is shown to be equivalent to a term in the action, similar to a gauge field configuration, whose presence maintains self-adjointness of the Hamiltonian or corrects the phase-space measure of the path integral to reflect the bound-state spectrum of the original problem. Various nonperturbative aspects of quantum-mechanical transition amplitudes are calculated by using classical solutions to Hamilton's equations of motion combined with the anomaly generated by the canonical transformation.
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.
The nontrivial transformation of the phase space path integral measure under certain discretized analogs of canonical transformations is computed. This Jacobian is used to derive a quantum analog of the Hamilton-Jacobi equation for the generating function of a canonical transformation that maps any quantum system to a system with a vanishing Hamiltonian. A perturbative solution of the quantum Hamilton-Jacobi equation is given. This solution gives a new way to compute quantum corrections for any soliton equation for which action-angle variables are known.
The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.
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.
We discuss how the Hamiltonian changes in quantum canonical transformations. To the operator $\stackrel{^}{\mathcal{H}}(\stackrel{^}{p},\stackrel{^}{q})$ one can associate (in a given ordering rule) a ccc-number function H(p,q)\mathcal{H}(p,q)H(p,q). It is this function that appears in the action of the phase-space path integral. A quantum canonical transformation $\stackrel{^}{\mathcal{H}}\ensuremath{\rightarrow}{\stackrel{^}{\mathcal{H}}}^{\ensuremath{'}}$ can now be expressed as an integral transformation ${\mathcal{H}}^{\ensuremath{'}}(\overline{p},\overline{q})=\ensuremath{\int}\mathrm{dp}\mathrm{dq}\mathcal{T}(\overline{p},\overline{q};p,q)\mathcal{H}(p,q)$. The kernel T\mathcal{T}T is constructed explicitly for point transformations and for the $p=\ensuremath{-}\overline{q}$, q=p‾q=\overline{p}q=p​ reflection by studying changes of variables in the path integral. The ordering dependence of T\mathcal{T}T is displayed. The invariance of commutation rules is also discussed.
The canonical functional action in the path integral in phase space is discretized by linking each pair of consecutive vertebral points - and or and - through the invariant complete solution of the Hamilton-Jacobi equation associated with the classical path defined by these extremes. When the measure is chosen to reflect the geometrical character of the propagator (it must behave as a density of weight in both of its arguments), the resulting infinitesimal propagator is cast in the form of an expansion in a basis of short-time solutions of the wave equation, associated with the eigenfunctions of the initial momenta canonically conjugated to a set of normal coordinates. The operator ordering induced by this prescription is a combination of a symmetrization rule coming from the phase, and a derivative term coming from the measure.