page_title: Wigner quasiprobability distribution
section_title: Uses of the Wigner function outside quantum mechanics
hierarchical_section_title: Wigner quasiprobability distribution / Uses of the Wigner function outside quantum mechanics
caption_attribution_description: English: I created this myself. I created this myself.
context_page_description: The Wigner quasiprobability distribution 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. 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 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.
context_section_description: 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 k = |k|sinθ ≈ |k|θ 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 ω = 2πf, 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).
caption: A contour plot of the Wigner–Ville distribution for a chirped pulse of light. The plot makes it obvious that the frequency is a linear function of time.
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