A Family of Generalized Wigner Operators and Their Physical Meaning as Bivariate Normal Distribution

By extending the usual Wigner operator to the s−parameterized one, we find that in the process of the generalized Weyl quantization the s parameter plays the role of correlation between two quadratures Q and P. This can be exposed by comparing the normally ordered form of Ω_s with the standard form of the Gaussian bivariate normal distribution of random variables in statistics. Three different expressions of Ω_s and the quantization scheme with use of it are presented.