The Wigner function is a function that represents the state of a quantum-mechanical system via a real-valued quasiprobability distribution on phase space. It was introduced by Wigner [1] in 1932 in an article entitled “On the quantum correction for thermodynamic equilibrium”. This title already hints at the original context in which this formalism was developed, namely statistical mechanics. Consequently, although presented in this entry mainly for the single-particle case (which is relevant in many applications), the Wigner function was from the beginning also a many-body formalism. Specifically, Wigner’s aim was to be able to systematically obtain quantum corrections to classical statistical mechanics [1]. Wigner was not the only one who worked on reformulating quantum mechanics in a way more akin to classical mechanics. For instance, Husimi [2] introduced a different phase space distribution in 1940, and Madelung [3] showed in 1927 that the Schrödinger equation for a one-particle system can be re-written as a hydrodynamic equation resembling compressible Euler equations. The most important approach of this type is probably the formalism developed by David Bohm ^[4][5][4,5], which involves, in addition to the quantum-mechanical wavefunction, classical particle trajectories as hidden variables and constitutes the basis for a possible solution of the quantum-mechanical measurement problem [6] (see below). Major contributions to the quantum-mechanical phase space formalism used today were made by Groenewold [7] and Moyal [8].

Wigner functions have developed into a widely used tool in quantum information theory and quantum technology. A key advantage of Wigner functions is that they allow for an easier visualization of quantum states and for a study of the classical limit. A disadvantage is that the Schrödinger equation for the quantum-mechanical wavefunctions (the main alternative) is in many cases easier to solve [9], for example, due to the availability of many well-developed solution techniques and due to the fact that the wavefunction depends on fewer coordinates. (More specifically, for a particle in one spatial dimension, the wavefunction is a function of position x, while the Wigner function is a function of position x and momentum p. Higher-dimensional partial differential equations are harder to solve. This advantage is relevant only for systems that can be attributed to a wavefunction and therefore not generally for systems entangled with their environment.) There are many excellent reviews on this topic ^{[9][10][11][12][13][14]}[9,10,11,12,13,14], which this entry does not intend to compete with—the author will, here, in the style of a dictionary entry, provide a brief glance at the topic for the impatient reader. The presentation of the formalism here mostly follows that in Refs. ^[15][16][15,16]. For historical aspects, the author mostly follows Ref. [11]. A focus will be the use of Wigner functions in studying the relation between classical and quantum mechanics.