Historical Note
As indicated, the formula for the Wigner function was independently derived several times in different contexts. In fact, apparently, Wigner was unaware that even within the context of quantum theory, it had been introduced previously by Heisenberg and Dirac, albeit purely formally: these two missed its significance, and that of its negative values, as they merely considered it as an approximation to the full quantum description of a system such as the atom. (Incidentally, Dirac would later become Wigner's brother-in-law.) Symmetrically, in most of his legendary 18-month correspondence with Moyal in the mid 1940s, Dirac was unaware that Moyal's quantum-moment generating function was effectively the Wigner function, and it was Moyal who finally brought it to his attention.
Read more about this topic: Wigner Quasiprobability Distribution
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