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European Journal of Physics

The Wigner function. I. The physical interpretation

Stig Stenholm

Published Oct 1, 1980• 26 citations
Publisher

Abstract

It is being argued that even in non-relativistic quantum mechanics the coordinate and momentum variables cannot be interpreted directly as observables. Only by defining these by suitable test bodies can one obtain verifiable predictions. When these arguments are implemented on a Wigner distribution the author shows that positive phase space probabilities always ensue, and hence this function can be used as a quantum mechanical phase space function.

DOI
10.1088/0143-0807/1/4/012

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It is being argued that even in non-relativistic quantum mechanics the coordinate and momentum variables cannot be interpreted directly as observables. Only by defining these by suitable test bodies can one obtain verifiable predictions. When these arguments are implemented on a Wigner distribution the author shows that positive phase space probabilities always ensue, and hence this function can be used as a quantum mechanical phase space function.
We show how the causal interpretation provides a physical basis for the Wigner phase-space formulation of quantum mechanics by analyzing the case of a relativistic spin-zero particle in an external electromagnetic field. The Wigner function can be interpreted as both a generalized transformation and a field which defines a quantum potential in phase space. We also propose a Lorentz-vector distribution function which always associates positive probability densities with the real motions of particles and antiparticles in phase space.
The Wigner function of quantum systems is an effective instrument for constructing an approximate classical description of the systems for which the classical approximation is possible. In recent years, the Wigner function formalism has been applied as well to seek indications of specific quantum properties of quantum systems leading to the impossibility of constructing the classical approximation. Used most of all as such an indication is the existence of negative values in the Wigner function for specific states of the quantum system being studied. The existence of such values itself prejudices the probabilistic interpretation of the Wigner function, though for an arbitrary observable depending jointly on the coordinate and the momentum of the quantum system just the Wigner function provides an effective instrument for calculating the average value and the other statistical characteristics. In this paper, a probabilistic interpretation of the Wigner function based on coordinating a theoretical probabilistic definition of the probability density, with restrictions to a physically small domain of phase space due to the uncertainty principle, is proposed.
This article outlines a novel interpretation of quantum theory: the Q-based interpretation.The core idea underlying this interpretation, recently suggested for quantum field theoriesby Drummond and Reid ([2020]), is to interpret the phase space functionQ—a transform ofthe better known Wigner function—as a proper probability distribution, roughly analogousto the probability distributionρin classical statistical mechanics.Here I motivate the Q-based interpretation, investigate whether it is empirically ad-equate, and outline some of its key conceptual features. I argue that the Q-based inter-pretation is attractive in that it promises having no measurement problem, is conceptuallyparsimonious and has the potential to apply elegantly to relativistic and field-theoretic con-texts.