Функция Вигнера: от квантовой механики к обработке сигналов

Wigner distribution function in physics as opposed to in signal processing

The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in Schrödinger's equation to a probability distribution in phase space. 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 (see Weyl quantization). In effect, it is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space. It was later rederived by Jean Ville in 1948 as a quadratic (in signal) representation of the local time frequency energy of a signal, effectively a spectrogram. In 1949, José Enrique Moyal, who had derived 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 (see Phase space formulation). It has applications in statistical mechanics, quantum chemistry, quantum optics, classical optics and signal analysis in diverse fields, such as electrical engineering, seismology, time–frequency analysis for music signals, spectrograms in biology and 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 quasiprobability 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 take on negative values for states which have no classical model—and is a convenient indicator of quantum mechanical interference. (See below for a characterization of pure states whose Wigner functions are non negative.) Smoothing the Wigner distribution through a filter of size larger than ħ (e. g., convolving with a
phase space Gaussian, a Weierstrass transform, to yield the Husimi 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 ħ, 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 ħ, and thus renders such "negative probabilities" less paradoxical.

In the modelling of optical systems such as telescopes or fibre telecommunications devices, the Wigner function is used to bridge the gap between simple ray tracing and the full wave analysis of the system. Here p/ħ is replaced with in the small angle (paraxial) approximation. In this context, the Wigner function is the closest one can get to describing the system in terms of rays at position x and angle θ while still including the effects of interference. If it becomes negative at any point, then simple ray tracing will not suffice to model the system. That is to say, negative values of this function are a symptom of the Gabor limit of the classical light signal and not of quantum features of light associated with ħ. In signal analysis, a time varying electrical signal, mechanical vibration, or sound wave are represented by a Wigner function. Here, x is replaced with the time, and p/ħ is replaced with the angular frequency , where f is the regular frequency. In ultrafast optics, short laser pulses are characterized with the Wigner function using the same f and t substitutions as above. Pulse defects such as chirp (the change in frequency with time) can be visualized with the Wigner function. See adjacent figure. In quantum optics, x and p/ħ are replaced with the X and P quadratures, the real and imaginary components of the electric field (see coherent state).

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, marrying his sister Manci.) 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.