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arXiv:hep-th/9711183 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Features of time-independent Wigner functions

Thomas Curtright, D. B. Fairlie, COSMAS K. ZACHOS

Published Jun 8, 1998• 240 citations
arXivHTMLPDF

Abstract

The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional (``star'') eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux (``supersymmetric'') isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P\"oschl-Teller potential, and the Liouville potential.

DOI
10.1103/physrevd.58.025002

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The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional (star'') eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux (supersymmetric'') isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P"oschl-Teller potential, and the Liouville potential.
We consider the time-independent Wigner functions of phase-space quantum mechanics (also known as deformation quantization) for a Morse potential. First, we find them by solving the *-eigenvalue equations, using a method that can be applied to potentials that are polynomial in an exponential. A Mellin transform converts the *-eigenvalue equations to difference equations, and factorized solutions are found directly for all values of the parameters. The symbols found this way are of both diagonal and off-diagonal density operator elements in the energy basis. The Wigner transforms of the density matrices built from the known wavefunctions are then shown to confirm the solutions.
We consider the time-independent Wigner functions of phase-space quantum mechanics (a.k.a. deformation quantization) for a Morse potential. First, we find them by solving the ∗\ast∗-eigenvalue equations, using a method that can be applied to potentials that are polynomial in an exponential. A Mellin transform converts the ∗\ast∗-eigenvalue equations to difference equations, and factorized solutions are found directly for all values of the parameters. The symbols of both diagonal and off-diagonal density operator elements in the energy basis are found this way. The Wigner transforms of the density matrices built from the known wave functions are then shown to confirm the solutions.
In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasi-probability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space.
In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasiprobability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space.
In this lecture, a limited introduction of gauge invariance in phase-space is provided, predicated on canonical transformations in quantum phase-space. Exact characteristic trajectories are also specified for the time-propagating Wigner phase-space distribution function: they are especially simple – indeed, classical – for the quantized simple harmonic oscillator. This serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase-space. This is a pedagogical selection from work published in Refs. 1) and 2).
Recently a paper on the construction of consistent Wigner functions for cylindrical phase spaces ${\mathbb{S}}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}$, i.e., for the canonical pair angle and orbital angular momentum, was published [H. A. Kastrup, Phys. Rev. A 94, 062113 (2016)] in which the main properties of these functions are derived and discussed and their usefulness is illustrated with examples. The present paper is a continuation which compares the properties of the Wigner functions for cylindrical phase spaces with those of the well-known Wigner functions for planar phase spaces in more detail. Furthermore, the mutual (Weyl) correspondence between Hilbert space operators and their phase-space functions is discussed. The $\ensuremath{\star}$ product formalism is shown to be completely implementable. In addition, basic dynamical laws for Wigner and Moyal functions are derived as generalized Liouville and energy equations. They are very similar to those in the planar case but also show characteristic differences.