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Journal of Mathematical Physics

Phase space path integrals, without limiting procedure

Maurice M. Mizrahi

Published Jan 1, 1978• 54 citations
Publisher

Abstract

This paper defines path integrals in phase space without using a time-division approach followed by a limiting process, thereby generalizing a similar procedure used in configuration space. This is useful since the path integral approach cannot always be formulated in configuration space (e.g., when the Hamiltonian is arbitrary) but can always be formulated in phase space. The most general Gaussian measure, absorbing the quadratic portion of the functional to be integrated, is constructed, and large classes of path integrals are evaluated with respect to it. Applications are given to the perturbation expansion and the semiclassical (WKB) expansion for arbitrary Hamiltonians.

DOI
10.1063/1.523504

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This paper defines path integrals in phase space without using a time-division approach followed by a limiting process, thereby generalizing a similar procedure used in configuration space. This is useful since the path integral approach cannot always be formulated in configuration space (e.g., when the Hamiltonian is arbitrary) but can always be formulated in phase space. The most general Gaussian measure, absorbing the quadratic portion of the functional to be integrated, is constructed, and large classes of path integrals are evaluated with respect to it. Applications are given to the perturbation expansion and the semiclassical (WKB) expansion for arbitrary Hamiltonians.
Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.
Techniques for integration in function spaces which are not necessarily vector spaces are presented in the light of DeWitt-Morette’s recent redefinition of path integrals, which does not involve the usual limiting process, and is therefore free from the ambiguities inherent in this approach. General translated Gaussian measures defined by their Fourier transforms are introduced on various path spaces and used to derive generalized moments formulas, and what is essentially the Feynman–Kac formula for the expression of solutions of the Schrödinger equation as functional integrals.
Chapter 2 constructed a path integral representation of the matrix elements of the statistical operator e-βH. This chapter extends the construction to hamiltonians which are general functions of phase space variables. This results in integrals over trajectories or paths in phase space. When the hamiltonian is quadratic in the momentum variables, the integral over momenta is gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is somewhat ambiguous, reflecting the problem of order between quantum operators. Hamiltonians which are general quadratic functions of momentum variables provide other important examples, and these are analyzed thoroughly. Such hamiltonians arise in the quantization of the motion on Riemannian manifolds. The analysis is illustrated by the quantization of the free motion on the sphere SN-1.
The coherent-state representation of quantum-mechanical propagators as well-defined phase-space path integrals involving Wiener measure on continuous phase-space paths in the limit that the diffusion constant diverges is formulated and proved. This construction covers a wide class of self-adjoint Hamiltonians, including all those which are polynomials in the Heisenberg operators; in fact, this method also applies to maximal symmetric Hamiltonians that do not possess a self-adjoint extension. This construction also leads to a natural covariance of the path integral under canonical transformations. An entirely parallel discussion for spin variables leads to the representation of the propagator for an arbitrary spin-operator Hamiltonian as well-defined path integrals involving Wiener measure on the unit sphere, again in the limit that the diffusion constant diverges.
The Wigner function W ( q , p ) is formulated as a phase-space path integral, whereby its sign oscillations can be seen to follow from interference between the geometrical phases of the paths. The approach has similarities to the path-centroid method in the configuration-space path integral. Paths can be classified by the midpoint of their ends; short paths where the midpoint is close to ( q , p ) and which lie in regions of low energy (low P function of the Hamiltonian) will dominate, and the enclosed area will determine the sign of the Wigner function. As a demonstration, the method is applied to a sequence of density matrices interpolating between a Poissonian number distribution and a number state, each member of which can be represented exactly by a discretized path integral with a finite number of vertices. Saddle-point evaluation of these integrals recovers (up to a constant factor) the WKB approximation to the Wigner function of a number state.