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 (cf. Weyl quantization in physics). In effect, it is the Weyl-Wigner transform of the density matrix, so the realization of that operator in phase space. It was later rederived by J. Ville in 1948 as a quadratic (in signal) representation of the local time-frequency energy of a signal.
In 1949, José Enrique Moyal, who had also rederived it independently, recognized it as the quantum moment-generating functional, and thus as the basis of an elegant encoding of all quantum expectation values, and hence quantum mechanics, in phase space (cf Weyl quantization). It has applications in statistical mechanics, quantum chemistry, quantum optics, classical optics and signal analysis in diverse fields such as electrical engineering, seismology, biology, speech processing, and engine design.
A classical particle has a definite position and momentum, and hence it is represented by a point in phase space. Given a collection (ensemble) of particles, the probability of finding a particle at a certain position in phase space is specified by a probability distribution, the Liouville density. This strict interpretation fails for a quantum particle, due to the uncertainty principle. Instead, the above quasi-probability Wigner distribution plays an analogous role, but does not satisfy all the properties of a conventional probability distribution; and, conversely, satisfies boundedness properties unavailable to classical distributions.
For instance, the Wigner distribution can and normally does go negative for states which have no classical model---and is a convenient indicator of quantum mechanical interference. Smoothing the Wigner distribution through a filter of size larger than (e.g., convolving with a phase-space Gaussian to yield the Hussimi representation, below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one.
Regions of such negative value are provable (by convolving them with a small Gaussian) to be "small": they cannot extend to compact regions larger than a few , and hence disappear in the classical limit. They are shielded by the uncertainty principle, which does not allow precise location within phase-space regions smaller than , and thus renders such "negative probabilities" less paradoxical.
The Wigner distribution P(x, p) is defined as:
where ψ is the wavefunction and x and p are position and momentum but could be any conjugate variable pair. (ie. real and imaginary parts of the electric field or frequency and time of a signal). It is symmetric in x and p:
where Φ is the Fourier transform of ψ.
In the general case, which includes mixed states, it is the Wigner transform of the density matrix:
This Wigner transformation (or map) is the inverse of the Weyl transform, which maps phase-space functions to Hilbert-space operators, in Weyl quantization. Thus, the Wigner function is the cornerstone of quantum mechanics in phase space. In 1949, José Enrique Moyal elucidated how the Wigner function provides the integration measure in phase space to yield expectation values from phase-space c-number functions g(x,p) (uniquely associated to suitably ordered operators through Weyl's transform (cf. Weyl quantization and property 7 below), in a manner evocative of classical probability theory.
Specifically, an operator's expectation value is a "phase-space average" of the Wigner transform of that operator,
1. P(x, p) is real
2. The x and p probability distributions are given by the marginals:
3. P(x, p) has the following reflection symmetries:
4. P(x, p) is Galilei-invariant:
5. The equation of motion for each point in the phase space is classical in the absence of forces:
In fact, it is classical even in the presence of harmonic forces.
6. State overlap is calculated as:
7. Operator expectation values (averages) are calculated as phase-space averages of the respective Wigner transforms:
8. In order that P(x, p) represent physical (positive) density matrices:
where |θ> is a pure state.
9. By virtue of the Schwarz inequality, for a pure state, it is constrained to be bounded,
This bound disappears in the classical limit, . In this limit, P(x, p) reduces to the probability density in coordinate space x, usually highly localized, multiplied by δ-functions in momentum: the classical limit is "spiky". Thus, this quantum-mechanical bound precludes a Wigner function which is a perfectly localized delta function in phase space, as a reflection of the uncertainty principle.
The inverse of this transformation is called the Weyl transformation, not to be confused with another definition of the Weyl transformation. The Wigner function is the Weyl-Wigner Transform of the density matrix.
The Wigner distribution was the first quasi-probability distribution, but many more followed, formally equivalent and transformable to and from it. As in the case of coordinate systems, on account of varying properties, several such have with various advantages for specific applications:
As indicated, the formula for the Wigner function was independently derived several times in different contexts. In fact, apparently, Wigner was unaware that even within the context of quantum theory, it had been introduced previously by Heisenberg and Dirac, albeit purely formally: these two missed its significance, and that of its negative values, as they merely considered it as an approximation to the full quantum description of a system such as the atom. Incidentally, Dirac would later become Wigner's brother-in-law. Symmetrically, in most of his legendary 18-month correspondence with Moyal in the mid 1940s, Dirac was unaware that Moyal's quantum-moment generating function was effectively the Wigner function, and it was Moyal who finally brought it to his attention.