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arXiv:quant-ph/0304110 · Acta Physica Hungarica B) Quantum Electronics

Non-Positivity of the Wigner Function and Bounds on Associated Integrals

Anthony J. Bracken, Demosthenes Ellinas, James Gareth Wood

Published Jan 1, 2004• 4 citations
arXivHTMLPDF
DOI
10.1556/aph.20.2004.1-2.24

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Wigner functions play a central role in the phase space formulation of quantum mechanics. Although closely related to classical Liouville densities, Wigner functions are not positive definite and may take negative values on subregions of phase space. We investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary Wigner function over noncompact subregions of the phase plane with hyperbolic boundaries. We show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly solvable eigenvalue problem and that the bounds differ from those on classical Liouville distributions. In particular, we show that the total “quasiprobability” on such a region can be greater than 1 or less than zero.
Wigner functions play a central role in the phase space formulation of quantum mechanics. Although closely related to classical Liouville densities, Wigner functions are not positive definite and may take negative values on subregions of phase space. We investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary Wigner function over noncompact subregions of the phase plane with hyperbolic boundaries. We show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly-solvable eigenvalue problem and that the bounds differ from those on classical Liouville distributions. In particular, we show that the total ``quasiprobability'' on such a region can be greater than 1 or less than zero.