Wigner phase space distribution as a wave function

TL;DR

This paper shows that the Wigner function can be viewed as a wave function in phase space, offering a new perspective on its negativity and its classical limit as a Koopman-von Neumann wave function.

Contribution

It introduces a wave function interpretation of the Wigner function and connects it to classical phase space formulations, providing novel insights into quantum-classical correspondence.

Findings

Wigner function acts as a probability amplitude in phase space.

In the classical limit, it becomes a Koopman-von Neumann wave function.

Provides an alternative explanation for Wigner function negativity.

Abstract

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.