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arXiv:hep-th/9810164 · Journal of Physics A Mathematical and General

Wigner trajectory characteristics in phase space and field theory

Thomas Curtright, COSMAS K. ZACHOS

Published Jan 1, 1999• 43 citations
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Abstract

Exact characteristic trajectories are specified for the time-propagating Wigner phase-space distribution function. They are especially simple - indeed, classical - for the quantized simple harmonic oscillator, which 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. Applications to duality transformations in field theory are discussed.

DOI
10.1088/0305-4470/32/5/009

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Exact characteristic trajectories are specified for the time-propagating Wigner phase-space distribution function. They are especially simple - indeed, classical - for the quantized simple harmonic oscillator, which 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. Applications to duality transformations in field theory are discussed.
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).
We discuss the Heisenberg-Wigner phase-space formalism in quantum electrodynamics as well as scalar quantum electrodynamics with respect to transverse fields. In regard to the special characteristics of such field types we derive modified transport equations such that particle momenta perpendicular to the propagation direction of the waves show up as external parameters only. In case of spatially oscillating fields we further demonstrate how to transform momentum derivative operators of infinite order into simple coupling terms.
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 give a rigorous proof for the following result that combines those discussed earlier by Kr\"uger and Poffyn [Physica 85A, 84 (1976)] and by Springborg [J. Phys. A 16, 535 (1983)]: The Wigner correspondence of a quantum operator can be obtained by simple substitutions under an arbitrary linear phase-space transformation. This property cannot be generalized to any other transformations or rules of phase-space correspondence. As an application of this result, the Wigner distribution functions of an anisotropic harmonic-oscillator system in crossed magnetic and electric fields are obtained.
This paper deals with certain aspects of the phase-space formulation of quantum mechanics. Noting the ambiguities involved in quantizing the classical phase-space dynamics, a generalized quantum-mechanical phase space is introduced in terms of eigenvalues of two mutually incompatible complete sets of operators. General definitions are given for the phase-space distribution function corresponding to a given abstract state of the system and the phase-space function for an arbitrary operator. Our formulation goes over to the Wigner formulation when the complete sets correspond to position and momentum operators in Cartesian coordinates. Explicit examples are discussed with use of action-angle and polar coordinates for simple one-, two-, and three-dimensional systems. Symmetry changes in phase space are discussed by applying Gel'fand-Levitan transformations and it is shown that, in general, the degeneracies of motion in phase space are not reflected in the energy eigenvalues.