The Wigner Distribution

TL;DR

The paper discusses the Wigner distribution, a quantum phase-space distribution function that bridges classical and quantum mechanics by representing quantum states in a quasi-classical framework.

Contribution

It revisits Wigner's original formulation of a quantum distribution function that allows quantum expectations to be computed as phase-space averages, connecting quantum and classical descriptions.

Findings

Provides a phase-space formulation of quantum mechanics

Highlights the Wigner distribution's role in quantum-classical correspondence

Discusses the properties and applications of the Wigner distribution

Abstract

In contrast to classical physics, the language of quantum mechanics involves operators and wave functions (or, more generally, density operators). However, in 1932, Wigner formulated quantum mechanics in terms of a distribution function $W(q,p)$, the marginals of which yield the correct quantum probabilities for $q$ and $p$ separately \cite{wigner}. Its usefulness stems from the fact that it provides a re-expression of quantum mechanics in terms of classical concepts so that quantum mechanical expectation values are now expressed as averages over phase-space distribution functions. In other words, statistical information is transferred from the density operator to a quasi-classical (distribution) function.