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[[Image:Wignerfunction catstate even 2.0.png|thumb|Wigner function of a so-called cat state.]]
{{other uses|Wigner distribution (disambiguation)}}
The '''Wigner quasiprobability distribution''' (also called the '''Wigner function''' or the '''Wigner–Ville distribution''' after [[Eugene Wigner]] and [[:fr:Jean Ville|Jean-André Ville]])  is a [[quasiprobability distribution]]. It was introduced<ref>
{{cite journal
 |author=E. P. Wigner
 |year=1932
 |title=On the quantum correction for thermodynamic equilibrium
 |journal=Phys. Rev.
 |volume=40 |issue=5 |pages=749–759
 |doi=10.1103/PhysRev.40.749|bibcode=1932PhRv...40..749W}}
</ref> 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 {{math|''ψ''(''x'')}}.
Thus, it maps<ref>
{{cite journal
 |author=H. J. Groenewold
 |year=1946
 |title=On the principles of elementary quantum mechanics
 |journal=Physica
 |volume=12 |issue=7 |pages=405–460
 |doi=10.1016/S0031-8914(46)80059-4|bibcode=1946Phy....12..405G}}
</ref> on the quantum [[density matrix]] in the map between real phase-space functions and [[Hermitian]] operators introduced by [[Hermann Weyl]] in 1927,<ref>
{{cite journal
 |author=H. Weyl
 |title=Quantenmechanik und gruppentheorie
 |year=1927
 |journal=Z. Phys.
 |volume=46 |page=1
 |doi=10.1007/BF02055756|bibcode=1927ZPhy...46....1W}};  H. Weyl, ''Gruppentheorie und Quantenmechanik'' (Leipzig: Hirzel) (1928); H. Weyl, ''The Theory of Groups and Quantum Mechanics''   (Dover, New York, 1931).</ref> in a context related to [[Group representation|representation theory]] in mathematics (cf. [[Weyl quantization]] in physics). 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) [[Wigner distribution function|representation of the local time-frequency energy of a signal]],<ref>J. Ville, "Théorie et Applications de la Notion de Signal Analytique", ''Câbles et Transmission'', '''2''', 61–74 (1948).</ref> effectively a [[spectrogram]].

In 1949, [[José Enrique Moyal]], who had derived it independently, recognized it as the [[quantum]] moment-generating functional,<ref>[[José Enrique Moyal|J.E. Moyal]], "Quantum mechanics as a statistical theory", ''Proceedings of the Cambridge Philosophical Society'', '''45''', 99–124 (1949). {{DOI|10.1017/S0305004100000487}}</ref> and thus as the basis of an elegant encoding of all quantum expectation values, and hence quantum mechanics, in phase space (cf. [[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]], [[spectrogram]]s in [[biology]] and speech processing, and [[Waterfall plot|engine design]].

== Relation to classical mechanics ==

A classical particle has a definite position and momentum, and hence it is represented by a point in phase space. Given a collection ([[Statistical ensemble (mathematical physics)|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 {{mvar|ħ}} (e.g., convolving with a
phase-space Gaussian, a [[Weierstrass transform]], to yield the [[#Other related quasiprobability distributions|Husimi representation]], below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one.{{efn|Specifically, since this convolution is invertible, in fact, no information has been sacrificed, and the full quantum entropy has not increased, yet.  However, if this resulting Husimi distribution is then used as a plain measure in a phase-space integral evaluation of expectation values ''without the requisite star product of the Husimi representation'', then, at that stage,  quantum information ''has been forfeited'' and the distribution ''is a semi-classical one'', effectively. That is, depending on its usage in evaluating expectation values, the very '''same distribution may serve as a quantum or a classical distribution function'''.}}

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  {{mvar|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 {{mvar|ħ}}, and thus renders such "[[negative probability|negative probabilities]]" less paradoxical.

== Definition and meaning ==

The Wigner distribution {{math|''W''(''x'',''p'')}} of a pure state is defined as:

{{Equation box 1
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|equation = <math> W(x,p)~\stackrel{\mathrm{def}}{=}~\frac{1}{\pi\hbar}\int_{-\infty}^\infty \psi^*(x+y)\psi(x-y)e^{2ipy/\hbar}\,dy\,</math>
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where {{mvar|ψ}}  is the wavefunction and {{mvar|x}} and {{mvar|p}} are position and momentum but could be any conjugate variable pair (e.g. real and imaginary parts of the electric field or frequency and time of a signal). Note that it may have support in {{mvar|x}}    even in regions where {{mvar|ψ}}  has no support in {{mvar|x}} ("beats").

It is symmetric in {{mvar|x}} and {{mvar|p}},
:<math> W(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^\infty \varphi^*(p+q)\varphi(p-q)e^{-2ixq/\hbar}\,dq</math>
where {{mvar|φ}} is the normalized momentum-space wave function, proportional to the [[Fourier transform]] of {{mvar|ψ}}.

In 3D,
:<math> W(\vec{r},\vec{p}) = \frac{1}{(2 \pi)^3} \int \psi^*(\vec{r} + \hbar\vec{s}/2) \psi (\vec{r} - \hbar\vec{s}/2) e^{i\vec{p} \cdot \vec{s}} \, d^3 s. </math>

In the general case, which includes mixed states, it is the Wigner transform of the  [[density matrix]],
:<math> W(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^\infty \langle x+y| \hat{\rho} |x-y \rangle e^{-2ipy/\hbar}\,dy,  </math>
where  ⟨''x''|''ψ''⟩ = {{math|''ψ(x)''}}.  This [[#The Wigner–Weyl transformation|Wigner transformation]] (or map) is the inverse of the [[Weyl quantization|Weyl transform]], which maps phase-space functions to [[Hilbert space|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 (analogous
to a [[probability density function]]) in phase space, to yield [[expected value|expectation values]] from phase-space [[c-number]] functions {{math|''g''(''x'',''p'')}} uniquely associated to suitably ordered operators {{math|''Ĝ''}} through Weyl's transform (cf. [[Wigner–Weyl transform]] and property 7 below), in a manner evocative of classical [[probability theory]].

Specifically, an operator's  {{math|''Ĝ''}}  expectation value is a "phase-space average" of the Wigner transform of that operator,
: <math>\langle  \hat{G}  \rangle =\int\! dx \,dp~ W(x,p)~ g(x,p)   ~.  </math>

== Mathematical properties ==

[[File:Wigner functions.jpg|thumb|280px|Figure 1:  The Wigner quasiprobability distribution for a) the vacuum b) An ''n'' =&nbsp;1 [[Fock state]] (e.g. a single photon) c) An ''n'' =&nbsp;5 Fock state.]]
1.  ''W''(''x'',&nbsp;''p'') is a real valued function.

2.  The ''x'' and ''p'' probability distributions are given by the [[Marginal distribution|marginals]]:
* <math>\int_{-\infty}^\infty dp\,W(x,p)=\langle x|\hat{\rho}|x \rangle.</math>  If the system can be described by a [[pure state]], one gets <math>\int_{-\infty}^\infty dp\,W(x,p)= |\psi(x)|^2</math>
* <math>\int_{-\infty}^\infty dx\,W(x,p)=\langle p|\hat{\rho}|p \rangle</math>. If the system can be described by a [[pure state]], one has <math>\int_{-\infty}^{\infty}dx\,W(x,p)=|\varphi(p)|^2</math>
* <math>\int_{-\infty}^\infty dx\int_{-\infty}^\infty dp\,W(x,p)=Tr(\hat{\rho})</math>
* Typically the trace of the density matrix  ''ρ̂'' is equal to 1.

3.  ''W''(''x'', ''p'') has the following reflection symmetries:
* Time symmetry: <math>\psi(x) \rightarrow \psi(x)^* \Rightarrow W(x,p) \rightarrow W(x,-p)</math>
* Space symmetry: <math>\psi(x) \rightarrow \psi(-x) \Rightarrow W(x,p) \rightarrow W(-x,-p)</math>

4.  ''W''(''x'', ''p'') is Galilei-covariant:
* <math>\psi(x) \rightarrow \psi(x+y) \Rightarrow W(x,p) \rightarrow W(x+y,p)</math>
* It is not [[Lorentz covariant]].

5.  The equation of motion for each point in the phase space is classical in the absence of forces:
* <math>\frac{\partial W(x,p)}{\partial t}=\frac{-p}{m}\frac{\partial W(x,p)}{\partial x}</math>
In fact, it is classical even in the presence of harmonic forces.

6.  State overlap is calculated as:
* <math>|\langle \psi|\theta \rangle|^2=2\pi\hbar\int_{-\infty}^{\infty}dx\,\int_{-\infty}^\infty dp\,W_{\psi}(x,p)W_{\theta}(x,p)</math>

7.  Operator expectation values (averages) are calculated as
phase-space averages of the respective Wigner transforms:
* <math>g(x,p)\equiv\int_{-\infty}^\infty dy\, \left\langle x-\frac{y}{2} \right| \hat{G} \left| x+\frac{y}{2} \right\rangle e^{ipy/\hbar},</math>
* <math>\langle \psi|\hat{G}|\psi\rangle=Tr(\hat{\rho}\hat{G})=\int_{-\infty}^\infty dx\, \int_{-\infty}^\infty dp\, W(x,p)g(x,p).
</math>

8.  In order that ''W''(''x'', ''p'') represent physical (positive) density matrices:
* <math>\int_{-\infty}^\infty dx\, \int_{-\infty}^\infty dp\, W(x,p)W_\theta(x,p) \ge 0 ~,</math>
for all pure states |θ〉.

9. By virtue of the [[Cauchy–Schwarz inequality]], for a pure state, it is constrained to be  bounded,
* <math> -\frac 2 h \leq W(x,p)  \leq \frac 2 h.</math>

This bound disappears in the classical limit, ''ħ'' → 0.   In this limit, ''W''(''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.<ref>{{cite journal | last1 = Curtright | first1 = T. L. | last2 = Zachos | first2 = C. K. | doi = 10.1142/S2251158X12000069 | title = Quantum Mechanics in Phase Space | journal = Asia Pacific Physics Newsletter | volume = 01 | page = 37 | year = 2012 | pmid =  | pmc = | arxiv = 1104.5269 }};  [[Cosmas Zachos|C. Zachos]], D. Fairlie, and [[Thomas Curtright|T. Curtright]], ''Quantum  Mechanics in Phase Space'' ( World Scientific, Singapore, 2005) {{isbn|978-981-238-384-6}}.</ref>

== Evolution equation for Wigner function ==
{{Main|Wigner–Weyl transform|Phase space formulation}}

[[File:SmallDisplacedGaussianWF.gif|220px|left|thumb|Figure 2: Wigner function for the [[quantum harmonic oscillator]] ground state, displaced from the origin of phase space (i.e., a [[coherent state]]). (Click to animate.) Note the rigid rotation, identical to classical motion: this is a special feature of the SHO. From the general pedagogy web-site.<ref>Curtright, T.L., [http://www.physics.miami.edu/~curtright/TimeDependentWignerFunctions.html Time-dependent Wigner Functions]</ref>]]

The '''Wigner transformation''' is a general invertible transformation of an operator  {{math|''Ĝ''}} on a [[Hilbert space]] to a function ''g(x,p)''   on [[phase space]], and is given by
:<math>g(x,p)=\int_{-\infty}^\infty ds ~e^{ips/\hbar} \left.\left\langle x-\frac s2\right. \right|\ \hat G\ \left.\left|x+\frac s2 \right. \right\rangle.</math>

Hermitian operators map to real functions. The inverse of this transformation,
so from phase space to Hilbert space, is called the '''[[Wigner–Weyl transform|Weyl transformation]]''',
:<math>  \langle x |\ \hat G\ | y \rangle= \int_{-\infty}^\infty {dp \over h} ~
e^{ip(x-y)/\hbar}  g\left({x+y\over 2},p\right),</math>
(not to be confused with the distinct [[Weyl transformation|Weyl transformation in differential geometry]]).

[[File:TwoStateWF.gif|230px|thumb|Figure 3: Quantum flip-flop: Time evolution of combined ground and 1st excited state Wigner function for the simple harmonic oscillator. Note the rigid motion in phase space corresponding to the conventional oscillations in coordinate space. (Click to animate.)]]

The '''Wigner function''' ''W''(''x,p'') discussed here is thus seen to be the Wigner transform  of the [[density matrix]] operator ''ρ̂''. Thus,  the trace of an operator with the density matrix Wigner-transforms to the equivalent phase-space integral overlap of ''g''(''x'',&nbsp;''p'') with the Wigner function.

The Wigner transform of the [[Density matrix|von Neumann evolution equation]] of the density matrix in the [[Schrödinger picture]] is
;''Moyal's evolution equation'' for the Wigner function,
::<math>{\partial W(x,p,t) \over \partial t} = - \{\{ W(x,p ,t) ~,~ H(x,p)\}\}~,</math>
where H(x,p) is Hamiltonian and { {•, •} } is the [[Moyal bracket]]. In the classical limit ''ħ'' → 0, the Moyal bracket reduces to the Poisson bracket, while this evolution equation reduces to the [[Liouville's theorem (Hamiltonian)#Poisson bracket|Liouville equation]] of classical statistical mechanics.

Strictly formally, in terms of [[Method of quantum characteristics|quantum characteristics]],  the solution of
this evolution equation reads, <math> W(x,p,t) = W(\star (x_{-t}(x,p),p_{-t}(x,p)),0)</math>,
where <math>x_{t}(x,p)</math> and <math>p_{t}(x,p)</math> are solutions of
so-called [[Method of quantum characteristics|quantum Hamilton's equations]], subject to initial conditions
<math>x_{t=0}(x,p) = x</math> and <math>p_{t=0}(x,p)=p</math>, and where [[Moyal product|<math>\star</math>-product]]
composition is understood for all argument functions.

Since, however,  <math>\star</math>-composition is thoroughly nonlocal (the "quantum probability fluid" diffuses, as observed by Moyal),  vestiges of local trajectories
are normally barely discernible  in the evolution of  the Wigner distribution function.{{efn|Quantum characteristics should not be confused with
trajectories of the Feynman path integral, or trajectories of the
[[de Broglie-Bohm theory|de Broglie – Bohm theory]].
This three-fold ambiguity  allows better understanding of  the position of [[Niels Bohr]],
who, vigorously but counterproductively, opposed the notion of trajectory in the atomic physics.
At the 1948 Pocono Conference, e.g., he said to [[Richard Feynman]]: "… one could not talk about the trajectory of an electron in the atom,
because it was something not observable."
("The Beat of a Different Drum: The Life and Science of Richard Feynman", by Jagdish Mehra (Oxford, 1994, pp. 245-248)).
Arguments of this kind were widely used in the past by [[Ernst Mach]] in his criticism of an atomic theory of physics and later, in the 1960s, by [[Geoffrey Chew]], [[Tullio Regge]] and others to motivate replacing the local quantum field theory by the [[S-matrix]] theory.
Today, statistical physics entirely based on atomistic concepts is included in standard courses, the S-matrix theory went out of fashion, while the Feynman path integral method has been recognized as the most efficient method in [[Gauge theory|gauge theories]].}}
In the integral representation of <small>★</small>-products, successive operations by them have been adapted to a phase-space path-integral, to  solve this evolution equation for the Wigner function <ref>
{{cite journal
 |author=B. Leaf
 |year=1968
 |title=Weyl transform in nonrelativistic quantum dynamics
 |journal=J. Math. Phys.
 |volume=9 |issue=5 |pages=769–781
 |doi=10.1063/1.1664640|bibcode=1968JMP.....9..769L}}
</ref> (see also <ref>
{{cite journal
 |author=P. Sharan
 |year=1979
 |title=Star-product representation of path integrals
 |journal=Phys. Rev. D
 |volume=20 |issue=2 |pages=414–418
 |bibcode=1979PhRvD..20..414S
 |doi=10.1103/PhysRevD.20.414}}
</ref><ref>
{{cite journal
 |author=M. S. Marinov
 |year=1991
 |title=A new type of phase-space path integral
 |journal=Phys. Lett. A
 |volume=153 |issue=1 |pages=5–11
 |doi=10.1016/0375-9601(91)90352-9|bibcode=1991PhLA..153....5M}}
</ref><ref name="OlshanetskyVainshtein2002">B. Segev: ''Evolution kernels for phase space distributions''. In: {{cite book|author1=M. A. Olshanetsky (ed.)|author2=Arkady Vainshtein (ed.)|title=Multiple Facets of Quantization and Supersymmetry: Michael Marinov Memorial Volume|url=https://books.google.com/books?id=_pwmuPF1DY4C&pg=PA68|accessdate=26 October 2012|year=2002|publisher=World Scientific|isbn=978-981-238-072-2|pages=68–90}} see especially  section 5. "Path integral for the propagator" on [https://books.google.com/books?id=_pwmuPF1DY4C&pg=PA86 pages 86-89]
[http://homepages.spa.umn.edu/~vainshte/Marinov/Segev/segev.pdf online]</ref>).
This non-trajectoral feature of Moyal time evolution<ref> {{cite journal|author=M. Oliva, D. Kakofengitis, and O. Steuernagel |year=2018 |title=Anharmonic quantum mechanical systems do not feature phase space trajectories |journal=Physica A | volume=502 |pages=201-210 |doi=10.1016/j.physa.2017.10.047  |arxiv=1611.03303 }}</ref> is illustrated in Figs 5,6,7,  below, for Hamiltonians more complex than the oscillator (in some contrast to the quasi-classical oscillator evolution of Fig 4.)
{{Gallery
|title=Examples of Wigner function time-evolutions
|align=center
|lines=6
|File:Wigner_function_of_a_Schrödinger_cat_state.gif|Figure 4:Schrödinger [[cat state]] of two interfering oscillator [[coherent state]]s; the [[Marginal distribution|marginals]] are plotted on the right (''p'') and underneath (''x'').
|File:Wigner function propagation for morse potential.ogv|Figure 5: The [[Morse potential]]: <math>U (x) = 20 ( 1 - e^{-0.16x} )^2 </math> in [[atomic units]] (a.u.). The green dashed lines represent [[level set]] of the [[Hamiltonian mechanics|Hamiltonian]].
|File:Wigner function for quartic potential.ogv|Figure 6: The quartic potential: <math> U (x) = 0.1 x^4 </math> in atomic units (a.u.). The solid lines represent the [[level set]] of the [[Hamiltonian mechanics|Hamiltonian]].
|File:Wigner function for tunnelling.ogv|Figure 7: [[Quantum tunnelling]] through the potential barrier: <math>U(x) = 8e^{-0.25 x^2} </math> in atomic units (a.u.). The solid lines represent the [[level set]] of the [[Hamiltonian mechanics|Hamiltonian]].
}}

== Classical limit ==
The Wigner function allows one to study the [[classical limit]], offering a comparison of the classical and quantum dynamics in phase space.<ref>See for example: [[Wojciech H. Zurek]]: Decoherence and the transition from quantum to classical – revisited, Los Alamos Science, 27, 2002, [https://arxiv.org/abs/quant-ph/0306072 arXiv:quant-ph/0306072], pp. 15 ff.</ref><ref>See for example: C. Zachos, D. Fairlie, T. Curtright: ''Quantum mechanics in phase space: an overview with selected papers, World Scientific, 2005   {{isbn|978-981-4520-43-0}}</ref>

It has recently been suggested that the Wigner function approach can be viewed as a quantum analogy to the operatorial formulation of classical mechanics introduced in 1932 by [[Bernard Koopman]] and [[John von Neumann]]: the time evolution of the Wigner function approaches, in the limit ''ħ'' → 0, the time evolution of the [[Koopman–von Neumann wavefunction]] of a classical particle.<ref>Denys I. Bondar, Renan Cabrera, Dmitry V. Zhdanov, Herschel A. Rabitz: Wigner Function's Negativity Demystified [https://arxiv.org/abs/1202.3628v2 arXiv:1202.3628v2 (quant-phys)] (submitted February 2012, version of 3 November 2012)</ref><ref>Renan Cabrera, Denys I. Bondar, Herschel A. Rabitz: Relativistic Wigner function and consistent classical limit for spin 1/2 particles, [https://arxiv.org/abs/1107.5139v2 arXiv:1107.5139v2] (submitted on 26 July 2011, version of 22 August 2012)</ref>


The [[truncated Wigner approximation]] is a semiclassical approximation to the dynamics obtained by replacing Moyal's equation with the classical [[Liouville's theorem (Hamiltonian)|Liouville's equation]].

==Positivity of the Wigner function==
As already noted, the Wigner function of quantum state typically takes some negative values. Indeed, for a pure state in one variable, if <math>W(x,p)\ge 0</math> for all <math>x</math> and <math>p</math>, then the wave function must have the form
:<math>\psi(x)=e^{-ax^2+bx+c}</math>
for some complex numbers <math>a,b,c</math> with <math>\mathrm{Re}(a)>0</math> (Hudson's theorem<ref name="Hudson1974">{{cite journal
 | author = Hudson, Robin L.
 | title = When is the Wigner quasi-probability density non-negative?
 | journal = Reports on Mathematical Physics
 | volume = 6
 | issue = 2
 | pages = 249-252
 | year = 1974
 | doi = 10.1016/0034-4877(74)90007-X
}}</ref>). Note that <math>a</math> is allowed to be complex, so that <math>\psi</math> is not necessarily a Gaussian wave packet in the usual sense. Thus, pure states with non-negative Wigner functions are not necessarily minimum uncertainty states in the sense of the [[Uncertainty principle|Heisenberg uncertainty formula]]; rather, they give equality in the [[Robertson-Schrodinger relation|Schrödinger uncertainty formula]], which includes an anticommutator term in addition to the commutator term.  (With careful definition of the respective variances,  all pure state Wigner functions lead to Heisenberg's inequality all the same.)

In higher dimensions, the characterization of pure states with non-negative Wigner functions is similar; the wave function must have the form
:<math>\psi(x)=e^{-(x,Ax)+b\cdot x+c}</math>
where <math>A</math> is a symmetric complex matrix whose real part is positive definite, <math>b</math> is a complex vector, and {{mvar|c}} is a complex number.<ref>F. Soto and P. Claverie, "When is the Wigner function of multidimensional systems nonnegative?", ''Journal of Mathematical Physics'' '''24''' (1983) 97-100</ref>  The Wigner function of any  such  state is a Gaussian distribution on phase space.

The cited paper of Soto and Claverie gives an elegant proof of this characterization, using the [[Segal–Bargmann space#The Segal–Bargmann transform|Segal–Bargmann transform]]. The reasoning is as follows. The [[Husimi Q representation|Husimi Q function]] of <math>\psi</math> may be computed as the squared magnitude of the Segal–Bargmann transform of <math>\psi</math>, multiplied by a Gaussian. Meanwhile, the Husimi Q function is the convolution of the Wigner function with a Gaussian. If the Wigner function of <math>\psi</math> is non-negative everywhere on phase space, then the Husimi Q function will be strictly positive everywhere on phase space. Thus, the Segal–Bargmann transform <math>F(x+ip)</math> of <math>\psi</math> will be nowhere zero. Thus, by a standard result from complex analysis, we have
:<math>F(x+ip)=e^{g(x+ip)}</math>
for some holomorphic function <math>g</math>. But in order for <math>F</math> to belong to the [[Segal–Bargmann space]]—that is, for <math>F</math> to be square-integrable with respect to a Gaussian measure—<math>g</math> must have at most quadratic growth at infinity. From this, elementary complex analysis can be used to show that <math>g</math> must actually be a quadratic polynomial. Thus, we obtain an explicit form of the Segal–Bargmann transform of any pure state whose Wigner function is non-negative. We can then invert the Segal–Bargmann transform to obtain the claimed form of the position wave function.

There does not appear to be any simple characterization of [[density matrix|mixed states]] with non-negative Wigner functions.

== The Wigner function in relation to other interpretations of quantum mechanics ==
It has been shown that the Wigner quasiprobability distribution function can be regarded as an {{mvar|ħ}}-[[Deformation theory|deformation]] of another phase space distribution function that describes an ensemble of [[De Broglie–Bohm theory|de Broglie–Bohm]] causal trajectories.<ref>Nuno Costa Dias, Joao Nuno Prata, Bohmian trajectories and quantum phase space distributions, Physics Letters A vol.&nbsp;302 (2002) pp.&nbsp;261-272, {{DOI|10.1016/S0375-9601(02)01175-1}} [https://arxiv.org/abs/quant-ph/0208156v1 arXiv:quant-ph/0208156v1] (submitted 26 August 2002)</ref> [[Basil Hiley]] has shown that the quasi-probability distribution may be understood as the [[density matrix]] re-expressed in terms of a mean position and momentum of a "cell" in phase space, and the de Broglie–Bohm interpretation allows one to describe the dynamics of the centers of such "cells".<ref>B. J. Hiley: ''Phase space descriptions of quantum phenomena'', in: A. Khrennikov (ed.): ''Quantum Theory: Re-consideration of Foundations–2'', pp.&nbsp;267-286, Växjö University Press, Sweden, 2003 ([http://www.birkbeck.ac.uk/tpru/BasilHiley/ShadowPhaseVajxo03.pdf PDF])</ref><ref>B. Hiley: ''Moyal's characteristic function, the density matrix and von Neumann's idempotent'' ([http://www.bbk.ac.uk/tpru/BasilHiley/Moyal&DensityMatrix.pdf preprint])</ref>

There is a close connection between the description of quantum states in terms of the Wigner function and a method of quantum states reconstruction in terms of [[mutually unbiased bases]].<ref>F.C. Khanna, P.A. Mello, M. Revzen, Classical and Quantum Mechanical State Reconstruction, [https://arxiv.org/abs/1112.3164v1 arXiv:1112.3164v1] [quant-ph] (submitted December 14, 2011)</ref>

== Uses of the Wigner function outside quantum mechanics ==
[[File:Chirpedpulse.jpg|thumb|Figure 8: 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.]]
* 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 (physics)|ray tracing]] and the full wave analysis of the system.  Here {{math|''p/ħ''}}    is replaced with {{math|''k'' {{=}} {{!}}''k''{{!}}sin''θ'' ≈ {{!}}''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 {{mvar|x}} and angle {{mvar|θ}} while still including the effects of interference.<ref>I Bazarov, ''Phys Rev  ST Accel  Beams''  '''15''' (2012) 050703,  {{doi|10.1103/PhysRevSTAB.15.050703}}.</ref>  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 {{mvar|ħ}}.
* In [[signal analysis]], a time-varying electrical signal, mechanical vibration, or sound wave are represented by a [[Wigner distribution function|Wigner function]].  Here, {{mvar|x}} is replaced with the time and {{math|''p/ħ''}}   is replaced with the angular frequency {{math|''ω'' {{=}} 2π''f''}}, where {{mvar|f}}  is the regular frequency.
* In ultrafast optics, short laser pulses are characterized with the Wigner function using the same {{mvar|f}}  and {{mvar|t}} substitutions as above.  Pulse defects such as chirp (the change in frequency with time) can be visualized with the Wigner function. See Figure&nbsp;8.
* In quantum optics, {{mvar|x}}  and {{math|''p/ħ''}}   are replaced with the {{mvar|X}} and {{mvar|P}} quadratures, the real and imaginary components of the electric field (see [[coherent state]]). The plots in Figure 4 are of quantum states of light.

== Measurements of the Wigner function ==
* [[Tomography]]
* [[Homodyne detection]]
* [[Frequency-resolved optical gating]]

== Other related quasiprobability distributions ==
{{Main|Quasiprobability distribution}}

The Wigner distribution was the first quasiprobability distribution to be formulated, but many more followed, formally equivalent and transformable to and from it (viz. [[Transformation between distributions in time–frequency analysis]]). As in the case of coordinate systems, on account of varying properties, several such have with various advantages for specific applications:
* [[Glauber P representation]]
* [[Husimi Q representation]]
Nevertheless, in some sense, the Wigner distribution holds  a privileged position among all these distributions, since it is the ''only one'' whose requisite star product  drops out (integrates out by parts to effective unity) in the evaluation of expectation values, as illustrated above, and so ''can'' be visualized as a quasiprobability measure analogous to the classical ones.

== Historical note ==

As indicated, the formula for the Wigner function was independently derived several times in different contexts.  In fact, apparently, [[Eugene Wigner|Wigner]] was unaware that even within the context of quantum theory, it had been introduced previously by [[Werner Heisenberg|Heisenberg]] and [[Paul Dirac|Dirac]],<ref>[[Werner Heisenberg|W. Heisenberg]], "Über die inkohärente Streuung von Röntgenstrahlen", ''Physik. Zeitschr.'' '''32''', 737–740 (1931); [[Paul Dirac|P.A.M. Dirac]], "Note on exchange phenomena in the Thomas atom", ''Proc. Camb. Phil. Soc.'' '''26''', 376–395 (1930). {{DOI|10.1017/S0305004100016108}}</ref> 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 [[Eugene Wigner#Middle years|Manci]].) Symmetrically, in most of his legendary 18-month correspondence with [[José Enrique Moyal|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.<ref>Ann Moyal, (2006), "Maverick Mathematician: The Life and Science of J.E. Moyal," ANU E-press, 2006, {{isbn|1-920942-59-9}},  accessed by  http://epress.anu.edu.au/maverick_citation.html</ref>

== See also ==
{{Col-begin}}
{{Col-1-of-2}}
* [[Heisenberg group]]
* [[Wigner–Weyl transform]]
* [[Phase space formulation]]
* [[Moyal bracket]]
* [[Negative probability]]
* [[Optical equivalence theorem]]
* [[Modified Wigner distribution function]]
{{Col-2-of-2}}
* [[Cohen's class distribution function]]
* [[Wigner distribution function]]
* [[Transformation between distributions in time–frequency analysis]]
* [[Squeezed coherent state]]
* [[Bilinear time–frequency distribution]]
* [[Continuous-variable quantum information]]
{{col-end}}


==Footnotes== 
{{notelist}}

== References ==
{{reflist}}

== Further reading ==
* M. Levanda and V Fleurov, "Wigner quasi-distribution function for charged particles in classical electromagnetic fields", ''Annals of Physics'', '''292''', 199–231 (2001). {{arxiv|cond-mat/0105137}}

== External links ==
* [http://iqis.org/quantech/wigner.php Wigner function tutorial] and [http://qis.ucalgary.ca/quantech/wiggalery.php Gallery of WFs], Institute for Quantum Information Science (IQIS), [[University of Calgary]]
* [http://qutip.org/docs/latest/guide/guide-visualization.html#wigner-function wigner] Wigner function implementation in QuTiP.
* [http://gerdbreitenbach.de/gallery/ Quantum Optics Gallery]

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