Wigner quasiprobability distribution

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The Wigner quasiprobability distribution, also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville, is a quasiprobability distribution, similar to the Margenau-Hill quasiprobability distribution and the Kirkwood–Dirac quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in the Schrödinger equation to a probability distribution in phase space. It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction ψ(x). Thus, it maps on the quantum density matrix in the map between real phase-space functions and Hermitian operators introduced by Hermann Weyl in 1927, in a context related to representation theory in mathematics (see Weyl quantization). In effect, it is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space...

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