Wigner quasiprobability distribution

Wigner function of a so-called cat state

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[1] 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[2] on the quantum density matrix in the map between real phase-space functions and Hermitian operators introduced by Hermann Weyl in 1927,[3] 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,[4] effectively a spectrogram.

In 1949, José Enrique Moyal, who had derived it independently, recognized it as the quantum moment-generating functional,[5] 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.

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 (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.[a]

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.

Definition and meaning

The Wigner distribution W(x,p) of a pure state is defined as

${\displaystyle W(x,p)~{\stackrel {\text{def}}{=}}~{\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\psi ^{*}(x+y)\psi (x-y)e^{2ipy/\hbar }\,dy,}$

where ψ is the wavefunction, and x and 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 x even in regions where ψ has no support in x ("beats").

It is symmetric in x and p:

${\displaystyle W(x,p)={\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\varphi ^{*}(p+q)\varphi (p-q)e^{-2ixq/\hbar }\,dq,}$

where φ is the normalized momentum-space wave function, proportional to the Fourier transform of ψ.

In 3D,

${\displaystyle W({\vec {r}},{\vec {p}})={\frac {1}{(2\pi )^{3}}}\int \psi ^{*}({\vec {r}}+\hbar {\vec {s}}/2)\psi ({\vec {r}}-\hbar {\vec {s}}/2)e^{i{\vec {p}}\cdot {\vec {s}}}\,d^{3}s.}$

In the general case, which includes mixed states, it is the Wigner transform of the density matrix:

${\displaystyle W(x,p)={\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\langle x-y|{\hat {\rho }}|x+y\rangle e^{-2ipy/\hbar }\,dy,}$

where ⟨x|ψ⟩ = ψ(x). 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 (analogous to a probability density function) in phase space, to yield expectation values from phase-space c-number functions g(xp) uniquely associated to suitably ordered operators Ĝ through Weyl's transform (see Wigner–Weyl transform and property 7 below), in a manner evocative of classical probability theory.

Specifically, an operator's Ĝ expectation value is a "phase-space average" of the Wigner transform of that operator:

${\displaystyle \langle {\hat {G}}\rangle =\int dx\,dp\,W(x,p)g(x,p).}$

Mathematical properties

The Wigner quasiprobability distribution for different energy eigenstates of the quantum harmonic oscillator: a) n = 0 (ground state), b) n = 1, c) n = 5

1. W(xp) is a real-valued function.

2. The x and p probability distributions are given by the marginals:

${\displaystyle \int _{-\infty }^{\infty }dp\,W(x,p)=\langle x|{\hat {\rho }}|x\rangle .}$ If the system can be described by a pure state, one gets ${\displaystyle \int _{-\infty }^{\infty }dp\,W(x,p)=|\psi (x)|^{2}.}$
${\displaystyle \int _{-\infty }^{\infty }dx\,W(x,p)=\langle p|{\hat {\rho }}|p\rangle .}$ If the system can be described by a pure state, one has ${\displaystyle \int _{-\infty }^{\infty }dx\,W(x,p)=|\varphi (p)|^{2}.}$
${\displaystyle \int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dp\,W(x,p)=\operatorname {Tr} ({\hat {\rho }}).}$
Typically the trace of the density matrix ${\displaystyle {\hat {\rho }}}$ is equal to 1.

3. W(x, p) has the following reflection symmetries:

• Time symmetry: ${\displaystyle \psi (x)\to \psi (x)^{*}\Rightarrow W(x,p)\to W(x,-p).}$
• Space symmetry: ${\displaystyle \psi (x)\to \psi (-x)\Rightarrow W(x,p)\to W(-x,-p).}$

4. W(x, p) is Galilei-covariant:

${\displaystyle \psi (x)\to \psi (x+y)\Rightarrow W(x,p)\to W(x+y,p).}$
It is not Lorentz-covariant.

5. The equation of motion for each point in the phase space is classical in the absence of forces:

${\displaystyle {\frac {\partial W(x,p)}{\partial t}}={\frac {-p}{m}}{\frac {\partial W(x,p)}{\partial x}}.}$
In fact, it is classical even in the presence of harmonic forces.

6. State overlap is calculated as

${\displaystyle |\langle \psi |\theta \rangle |^{2}=2\pi \hbar \int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dp\,W_{\psi }(x,p)W_{\theta }(x,p).}$

7. Operator expectation values (averages) are calculated as phase-space averages of the respective Wigner transforms:

${\displaystyle g(x,p)\equiv \int _{-\infty }^{\infty }dy\,\left\langle x-{\frac {y}{2}}\right|{\hat {G}}\left|x+{\frac {y}{2}}\right\rangle e^{ipy/\hbar },}$
${\displaystyle \langle \psi |{\hat {G}}|\psi \rangle =\operatorname {Tr} ({\hat {\rho }}{\hat {G}})=\int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dp\,W(x,p)g(x,p).}$

8. For W(x, p) to represent physical (positive) density matrices, it must satisfy

${\displaystyle \int _{-\infty }^{\infty }dx\,\int _{-\infty }^{\infty }dp\,W(x,p)W_{\theta }(x,p)\geq 0}$
for all pure states |θ⟩.

9. By virtue of the Cauchy–Schwarz inequality, for a pure state, it is constrained to be bounded:

${\displaystyle -{\frac {2}{h}}\leq W(x,p)\leq {\frac {2}{h}}.}$
This bound disappears in the classical limit, ħ → 0. In this limit, W(xp) 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 δ-function in phase space, as a reflection of the uncertainty principle.[6]

10. The Wigner transformation is simply the Fourier transform of the antidiagonals of the density matrix, when that matrix is expressed in a position basis.[7]

Examples

Let ${\displaystyle |m\rangle \equiv {\frac {a^{\dagger m}}{\sqrt {m!}}}|0\rangle }$ be the ${\displaystyle m}$-th Fock state of a quantum harmonic oscillator. Groenewold (1946) discovered its associated Wigner function, in dimensionless variables:

${\displaystyle W_{|m\rangle }(x,p)={\frac {(-1)^{m}}{\pi }}e^{-(x^{2}+p^{2})}L_{m}{\big (}2(p^{2}+x^{2}){\big )},}$

where ${\displaystyle L_{m}(x)}$ denotes the ${\displaystyle m}$-th Laguerre polynomial.

This may follow from the expression for the static eigenstate wavefunctions,

${\displaystyle u_{m}(x)=\pi ^{-1/4}H_{m}(x)e^{-x^{2}/2},}$

where ${\displaystyle H_{m}}$ is the ${\displaystyle m}$-th Hermite polynomial. From the above definition of the Wigner function, upon a change of integration variables,

${\displaystyle W_{|m\rangle }(x,p)={\frac {(-1)^{m}}{\pi ^{3/2}2^{m}m!}}e^{-x^{2}-p^{2}}\int _{-\infty }^{\infty }d\zeta \,e^{-\zeta ^{2}}H_{m}(\zeta -ip+x)H_{m}(\zeta -ip-x).}$

The expression then follows from the integral relation between Hermite and Laguerre polynomials.[8]

Evolution equation for Wigner function

The Wigner transformation is a general invertible transformation of an operator Ĝ on a Hilbert space to a function g(xp) on phase space and is given by

${\displaystyle g(x,p)=\int _{-\infty }^{\infty }ds\,e^{ips/\hbar }\left\langle x-{\frac {s}{2}}\right|{\hat {G}}\left|x+{\frac {s}{2}}\right\rangle .}$

Hermitian operators map to real functions. The inverse of this transformation, from phase space to Hilbert space, is called the Weyl transformation:

${\displaystyle \langle x|{\hat {G}}|y\rangle =\int _{-\infty }^{\infty }{\frac {dp}{h}}e^{ip(x-y)/\hbar }g\left({\frac {x+y}{2}},p\right)}$

(not to be confused with the distinct Weyl transformation in differential geometry).

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(xp) with the Wigner function.

The Wigner transform of the von Neumann evolution equation of the density matrix in the Schrödinger picture is Moyal's evolution equation for the Wigner function:

${\displaystyle {\frac {\partial W(x,p,t)}{\partial t}}=-\{\{W(x,p,t),H(x,p)\}\},}$

where H(x, p) is the 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 equation of classical statistical mechanics.

Formally, the classical Liouville equation can be solved in terms of the phase-space particle trajectories which are solutions of the classical Hamilton equations. This technique of solving partial differential equations is known as the method of characteristics. This method transfers to quantum systems, where the characteristics' "trajectories" now determine the evolution of Wigner functions. The solution of the Moyal evolution equation for the Wigner function is represented formally as

${\displaystyle W(x,p,t)=W{\big (}\star {\big (}x_{-t}(x,p),p_{-t}(x,p){\big )},0{\big )},}$

where ${\displaystyle x_{t}(x,p)}$ and ${\displaystyle p_{t}(x,p)}$ are the characteristic trajectories subject to the quantum Hamilton equations with initial conditions ${\displaystyle x_{t=0}(x,p)=x}$ and ${\displaystyle p_{t=0}(x,p)=p}$, and where ${\displaystyle \star }$-product composition is understood for all argument functions.

Since ${\displaystyle \star }$-composition of functions is thoroughly nonlocal (the "quantum probability fluid" diffuses, as observed by Moyal), vestiges of local trajectories in quantum systems are barely discernible in the evolution of the Wigner distribution function.[b] In the integral representation of ${\displaystyle \star }$-products, successive operations by them have been adapted to a phase-space path integral, to solve the evolution equation for the Wigner function[9] (see also [10][11][12]). This non-local feature of Moyal time evolution[13] is illustrated in the gallery below, for Hamiltonians more complex than the harmonic oscillator. In the classical limit, the trajectory nature of the time evolution of Wigner functions becomes more and more distinct. At ħ = 0, the characteristics' trajectories reduce to the classical trajectories of particles in phase space.

Harmonic-oscillator time evolution

In the special case of the quantum harmonic oscillator, however, the evolution is simple and appears identical to the classical motion: a rigid rotation in phase space with a frequency given by the oscillator frequency. This is illustrated in the gallery below. This same time evolution occurs with quantum states of light modes, which are harmonic oscillators.

Classical limit

The Wigner function allows one to study the classical limit, offering a comparison of the classical and quantum dynamics in phase space.[15][16]

It has 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.[17]

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 ${\displaystyle W(x,p)\geq 0}$ for all ${\displaystyle x}$ and ${\displaystyle p}$, then the wave function must have the form

${\displaystyle \psi (x)=e^{-ax^{2}+bx+c}}$

for some complex numbers ${\displaystyle a,b,c}$ with ${\displaystyle \operatorname {Re} (a)>0}$ (Hudson's theorem[18]). Note that ${\displaystyle a}$ is allowed to be complex, so that ${\displaystyle \psi }$ 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 Heisenberg uncertainty formula; rather, they give equality in the 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

${\displaystyle \psi (x)=e^{-(x,Ax)+b\cdot x+c},}$

where ${\displaystyle A}$ is a symmetric complex matrix whose real part is positive-definite, ${\displaystyle b}$ is a complex vector, and c is a complex number.[19] The Wigner function of any such state is a Gaussian distribution on phase space.

Soto and Claverie[19] give an elegant proof of this characterization, using the Segal–Bargmann transform. The reasoning is as follows. The Husimi Q function of ${\displaystyle \psi }$ may be computed as the squared magnitude of the Segal–Bargmann transform of ${\displaystyle \psi }$, multiplied by a Gaussian. Meanwhile, the Husimi Q function is the convolution of the Wigner function with a Gaussian. If the Wigner function of ${\displaystyle \psi }$ 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 ${\displaystyle F(x+ip)}$ of ${\displaystyle \psi }$ will be nowhere zero. Thus, by a standard result from complex analysis, we have

${\displaystyle F(x+ip)=e^{g(x+ip)}}$

for some holomorphic function ${\displaystyle g}$. But in order for ${\displaystyle F}$ to belong to the Segal–Bargmann space—that is, for ${\displaystyle F}$ to be square-integrable with respect to a Gaussian measure—${\displaystyle g}$ must have at most quadratic growth at infinity. From this, elementary complex analysis can be used to show that ${\displaystyle g}$ 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 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 ħ-deformation of another phase-space distribution function that describes an ensemble of de Broglie–Bohm causal trajectories.[20] 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".[21][22]

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.[23]

Uses of the Wigner function outside quantum mechanics

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 and the full wave analysis of the system. Here p/ħ is replaced with 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 x and angle θ while still including the effects of interference.[24] 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).

Other related quasiprobability distributions

The Wigner distribution was the first quasiprobability distribution to be formulated, but many more followed, formally equivalent and transformable to and from it (see 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:

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, Wigner was unaware that even within the context of quantum theory, it had been introduced previously by Heisenberg and Dirac,[25][26] 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.[27]

Footnotes

1. ^ 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.
2. ^ Quantum characteristics should not be confused with trajectories of the Feynman path integral, or trajectories of the 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 theories.

References

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2. ^ H. J. Groenewold (1946). "On the principles of elementary quantum mechanics". Physica. 12 (7): 405–460. Bibcode:1946Phy....12..405G. doi:10.1016/S0031-8914(46)80059-4.
3. ^ H. Weyl (1927). "Quantenmechanik und gruppentheorie". Zeitschrift für Physik. 46 (1–2): 1. Bibcode:1927ZPhy...46....1W. doi:10.1007/BF02055756. S2CID 121036548.; H. Weyl, Gruppentheorie und Quantenmechanik (Leipzig: Hirzel) (1928); H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931).
4. ^ J. Ville, "Théorie et Applications de la Notion de Signal Analytique", Câbles et Transmission, 2, 61–74 (1948).
5. ^ Moyal, J. E. (1949). "Quantum mechanics as a statistical theory". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press (CUP). 45 (1): 99–124. doi:10.1017/s0305004100000487. ISSN 0305-0041.
6. ^ Curtright, T. L.; Zachos, C. K. (2012). "Quantum Mechanics in Phase Space". Asia Pacific Physics Newsletter. 1: 37. arXiv:1104.5269. doi:10.1142/S2251158X12000069. S2CID 119230734.; C. Zachos, D. Fairlie, and T. Curtright, Quantum Mechanics in Phase Space (World Scientific, Singapore, 2005). ISBN 978-981-238-384-6.
7. ^ Hawkes, Peter W. (2018). Advances in Imaging and Electron Physics. Academic Press. p. 47. ISBN 9780128155424.
8. ^ Schleich, Wolfgang P. (2001-02-09). Quantum Optics in Phase Space (1st ed.). Wiley. p. 105. doi:10.1002/3527602976. ISBN 978-3-527-29435-0.
9. ^ B. Leaf (1968). "Weyl transform in nonrelativistic quantum dynamics". Journal of Mathematical Physics. 9 (5): 769–781. Bibcode:1968JMP.....9..769L. doi:10.1063/1.1664640.
10. ^ P. Sharan (1979). "Star-product representation of path integrals". Physical Review D. 20 (2): 414–418. Bibcode:1979PhRvD..20..414S. doi:10.1103/PhysRevD.20.414.
11. ^ M. S. Marinov (1991). "A new type of phase-space path integral". Physics Letters A. 153 (1): 5–11. Bibcode:1991PhLA..153....5M. doi:10.1016/0375-9601(91)90352-9.
12. ^ B. Segev: Evolution kernels for phase space distributions. In: M. A. Olshanetsky; Arkady Vainshtein (2002). Multiple Facets of Quantization and Supersymmetry: Michael Marinov Memorial Volume. World Scientific. pp. 68–90. ISBN 978-981-238-072-2. Retrieved 26 October 2012. See especially section 5. "Path integral for the propagator" on pages 86–89. Also online.
13. ^ M. Oliva, D. Kakofengitis, and O. Steuernagel (2018). "Anharmonic quantum mechanical systems do not feature phase space trajectories". Physica A. 502: 201–210. arXiv:1611.03303. Bibcode:2018PhyA..502..201O. doi:10.1016/j.physa.2017.10.047. S2CID 53691877.{{cite journal}}: CS1 maint: multiple names: authors list (link)
14. ^ a b Curtright, T. L., Time-dependent Wigner Functions.
15. ^ See, for example: Wojciech H. Zurek, Decoherence and the transition from quantum to classical – revisited, Los Alamos Science, 27, 2002, arXiv:quant-ph/0306072, pp. 15 ff.
16. ^ 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.
17. ^ Bondar, Denys I.; Cabrera, Renan; Zhdanov, Dmitry V.; Rabitz, Herschel A. (2013). "Wigner phase-space distribution as a wave function". Physical Review A. 88 (5): 052108. arXiv:1202.3628. doi:10.1103/PhysRevA.88.052108. ISSN 1050-2947. S2CID 119155284.
18. ^ Hudson, Robin L. (1974). "When is the Wigner quasi-probability density non-negative?". Reports on Mathematical Physics. 6 (2): 249–252. Bibcode:1974RpMP....6..249H. doi:10.1016/0034-4877(74)90007-X.
19. ^ a b F. Soto and P. Claverie, "When is the Wigner function of multidimensional systems nonnegative?", Journal of Mathematical Physics 24 (1983) 97–100.
20. ^ Dias, Nuno Costa; Prata, João Nuno (2002). "Bohmian trajectories and quantum phase space distributions". Physics Letters A. Elsevier BV. 302 (5–6): 261–272. arXiv:quant-ph/0208156v1. doi:10.1016/s0375-9601(02)01175-1. ISSN 0375-9601.
21. ^ B. J. Hiley: Phase space descriptions of quantum phenomena, in: A. Khrennikov (ed.): Quantum Theory: Re-consideration of Foundations–2, pp. 267–286, Växjö University Press, Sweden, 2003 (PDF).
22. ^ B. Hiley: Moyal's characteristic function, the density matrix and von Neumann's idempotent (preprint).
23. ^ F. C. Khanna, P. A. Mello, M. Revzen, Classical and Quantum Mechanical State Reconstruction, arXiv:1112.3164v1 [quant-ph] (submitted December 14, 2011).
24. ^ Bazarov, Ivan V. (2012-05-03). "Synchrotron radiation representation in phase space". Physical Review Special Topics - Accelerators and Beams. American Physical Society (APS). 15 (5): 050703. doi:10.1103/physrevstab.15.050703. ISSN 1098-4402.
25. ^ W. Heisenberg (1931). "Über die inkohärente Streuung von Röntgenstrahlen". Physikalische Zeitschrift. 32: 737–740.
26. ^ Dirac, P. A. M. (1930). "Note on Exchange Phenomena in the Thomas Atom". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press (CUP). 26 (3): 376–385. doi:10.1017/s0305004100016108. ISSN 0305-0041.
27. ^ Ann Moyal, (2006), "Maverick Mathematician: The Life and Science of J. E. Moyal", ANU E-press, 2006, ISBN 1-920942-59-9.