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Physical Review A

Path-integral derivation of phase-space distributions

Sebastiano Sonego

Published Oct 1, 1990• 5 citations
Publisher

Abstract

A functional P[\ensuremath{\gamma}] is defined on the space of histories for a quantum particle, and its meaning as a quasiprobability is discussed. The method of obtaining the Wigner and the Margenau-Hill phase-space distributions is shown as particular path integrals over P[\ensuremath{\gamma}].
DOI
10.1103/physreva.42.3733

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The concepts and formalism of quasiprobabilities in the space of paths are extended to the relativistic domain. The definition of a relativistic wave function, needed in the construction of the quasiprobability functional, is discussed extensively, together with the formulation and the physical interpretation of an explicitly covariant quantum theory. The Wigner and the Margenau-Hill relativistic phase-space distribution functions are then obtained according to the standard procedures of the formalism.
We present the path-integral solutions to the distributions in classical (Gibbs) and quantum (Wigner) statistical mechanics. The kernel of the distributions are derived in two ways - one by time slicing and defining the appropriate short-time interval phase space matrix element and second by making use of the kernel in the path-integral approach to quantum mechanics. We show that the two approaches are perturbatively identical. We also present another computation for the Wigner kernel, which is also the Liouville kernel, for the harmonic oscillator and free particle. These kernels may be used as the starting point in the perturbative expansion of the Wigner kernel for any potential. With the kernel solved, we essentially solve also the distributions in classical and quantum statistical mechanics.
The Weyl-Wigner transformation enables us to construct a representation of quantum motion equations using functions in phase space. The states of the system are determined by the quasiprobability distribution in phase space and the motion is described by an orthogonal integral operator. This formalism is employed in the study of the classical meaning of the discrete symmetry groups, in the problem of defining the current probability and in the proving of the expression of cross section from the S-matrix.
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.