Bivariate averaging and the Wigner distribution function

Abstract

In order to gain insight into the nature of the Wigner and related distribution functions, bivariate averaging functions of real unbounded variables with absolutely continuous marginals that are ordinary probabilities are considered. Accordingly variables are chosen to be phase space variables that are respectively eigenvalues of position and momentum operators. The impact of the condition that the marginals are squared magnitudes of amplitudes that are Fourier transforms of one another is emphasized by the delay of the introduction of this Fourier transform condition until after the form for a bivariate distribution with the given marginals is obtained. When the respective amplitudes are fourier transforms of one another, special cases of the bivariate averaging function correspond to generalized Wigner functions characterized by a parameterα. Such anα-Wigner function can be used as the basis of a consistent averaging procedure if an appropriate corresponding representation for underlying operators to be averaged is specified. Properties of theα-Wigner functions are summarized.