Wigner phase space distribution as a wave function
  We demonstrate that the Wigner function of a pure quantum state is a wave function in a specially tuned Dirac bra-ket formalism and argue that the Wigner function is in fact a probability amplitude for the quantum particle to be at a certain point of the classical phase space. Additionally, we establish that in the classical limit, the Wigner function transforms into a classical Koopman-von Neumann wave function rather than into a classical probability distribution. Since probability amplitude need not be positive, our findings provide an alternative outlook on the Wigner function's negativity. 
Denys I. Bondar, Renan Cabrera, Dmitry V. Zhdanov, Herschel A. Rabitz
quant-ph math-ph math.MP physics.atom-ph