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Physics Letters A

Uniqueness of Wigners phase space distribution resulting from linear conditions

W.R. Theis

Published Jun 1, 1999• 0 citations
Publisher
DOI
10.1016/s0375-9601(99)00227-3

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A quantum condition on phase-space distribution functions is introduced based on the probability for the quasimomentum k and quasicoordinate q. This condition is complementary to the conditions on the momentum p and the coordinate x. It is shown that the Wigner distribution does not satisfy this condition, and it is the Rihaczek function that obeys it. An equality is established for sums on the Wigner and the Rihaczek distributions. Linear dependences of phase-space distribution functions are derived based on the zeros of the kq functions.
The Wigner distribution is a milestone of Time–frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kinds of modifications have been proposed. This contribution fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As a main result, we provide a characterization for the Cohen’s class [L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7 (1996) 781–786; Time–frequency Analysis (Prentice Hall, New Jersey, 1995)]. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role.