Wigner Functions

In General > s.a. quantum mechanics; formulations of quantum theory; phase space / entropy; experiments in quantum mechanics; quantum states.
* Idea: The distribution function or density matrix in phase space quantization; Its time evolution is governed by Moyal's equation.
\$ Def: For a solution ψ of the Schrödinger equation, the Wigner function is the real function

W(x, p, t):= (π$$\hbar$$)−1 dy ψ*(x+y, t) ψ(xy, t) exp{2i py/$$\hbar$$} .

* Properties: It is not directly a probability distribution function, but it is useful, and satisfies

dp W(x, p, t) = |ψ(x, t)|2,   dx W(x, p, t) = |ψ(p, t)|2.

* Hudson's theorem: For non-relativistic continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian.
@ General references: Wigner PR(32) [proposal]; Tatarskii SPU(83); Narcowich & Fulling ed; Royer PRL(85); Wootters AP(87); Dragt & Habib qp/98-proc [and symplectic maps]; Li et al PRA(04), Revzen FP(06) [and phase-space probability density]; Khademi qp/06; Nassimi a0706; Johansen a0804; Case AJP(08)oct [and Weyl transforms, for pedestrians]; Surhone et al ed-10; O'Connell in(09)-a1009 [rev]; Bauke & Itzhak a1101; Bednorz & Belzig PRA(11), Bondar et al PRA(13)-a1202 [negativity]; Schroeck JPA(12) [probability]; Steuernagel et al PRD(12) [flow and topological order in quantum dynamics]; Bondar et al PRA(13) [as a wave function]; Giese et al proc(14)-a1402 [intro]; Blass & Gurevich a1502 [and marginal distributions of x and p]; Rakotoson et al a1707 [phase-space representations]; Schwonnek & Werner a1802 [for arbitrary sets of observables]; Perepelkin et al a1904 [new representation, and universal density matrix].
@ Propagator: Dittrich et al PRL(06)qp/05 [semiclassical]; Ozorio de Almeida & Brodier AP(06); Sels et al JPA(13)-a1207 [path-integral approach]; Cabrera et al PRA(15)-a1212 [effective numerical propagation].
@ And information theory: Frieden & Soffer PRA(06); Bernardini & Bertolami a1901-conf [continuity equations for quantum information flux].
@ Hudson's theorem: Gross JMP(06), APB(07)qp-proc [for finite-dimensional system]; Mandilara et al PRA(09)-a0808 [for mixed states].
@ Relationships: Leavens & Sala Mayato PLA(01) [and wave function]; Bracken RPMP(06)qp/05 [vs Hilbert space, and superposition]; Praxmeyer & Wodkiewicz LP(05)phy [and spectrum, for light]; Isidro IJGMP(08)-a0710 [and symplectic connection]; Parisio JPA(08)-a0712 [Bargmann representation]; Lieb & Ostrover JMP(10)-a1007 [Gaussians and localization in phase space]; > s.a. quantum chaos.

Related topics: see specific systems and generalizations.

Related Topics > s.a. huygens principle; pilot-wave interpretation; quantum correlations; quantum measurement; classical limit; Wigner Transform.
@ Semiclassical states: Rios & Ozorio de Almeida JPA(02)mp/01; Veble et al JPA(02); de Gosson & de Gosson qp/06 [squeezed]; Pulvirenti JMP(06); de Gosson JPA(08) [and Feichtinger algebra]; Dechoum et al PRA(10)-a1107 [two-mode entangled state]; Song & Fan IJTP(12) [squeezed]; Kalligiannaki & Makrakis a1402 [perturbative analysis]; Giannopoulou & Makrakis a1705 [approximate series solution]; > s.a. decoherence; quantum states.
@ Other states: Tegmen NCB(07)mp [simple states]; Vanbever a2104 [vacuum, as majorizing mixtures of Fock states].
@ And pilot-wave interpretation: Dias & Prata PLA(01)qp, PLA(02)qp; Hiley FP(10).
@ And foundations: Banaszek & Wódkiewicz PRA(98) [EPR]; Franco qp/07 [EPR].
@ Calculation: Hug et al JPA(98); Samson JPA(00) [coherent state path integral], JPA(03)qp [phase-space path integral]; Curtright et al JMP(01) [generating functions]; Sels et al PhyA(13) [propagator for complex dynamical systems, path integral approach]; Kakofengitis et al PRA(17)-a1611 [integral form]; Gozzi et al a2004 [Marinov path interal and response field].
@ Time evolution: Schleich et al FP(88) [and transition probabilities]; Moshinsky & Sharma AP(00) [and canonical transformations]; Hashimoto et al JPA(07)qp/06 [and Markov process]; Zueco & Calvo JPA(07)qp/06 [Bopp operators for dynamics]; Lewis-Swan et al a1503 [interpretation of individual phase-space trajectories].
@ Entanglement: Hardy et al PS(04); Narnhofer JPA(06); Ozorio de Almeida LNP(09)qp/06 [in phase space].
@ Other topics: Mehta JMP(64); Włodarz PLA(88) [averaging and positivity]; Zavialov & Malokostov TMP(99)ht/98; Lougovski et al PRL(03)qp/02 [operational def]; Dias & Prata JMP(04)ht/03 [t-dependent transformations]; Oliveira et al AP(04) [from star product]; Klauder & Skagerstam JPA(07) [generalized representations of operators]; Scott & Caves AP(08)-a0801 [sub-Planck structure]; Kube et al JCP(09) [Monte Carlo sampling]; Pennini & Plastino PLA(10) [thermal effects]; Mari & Eisert PRL(12)-a1208 [Wigner-function positivity and efficient classical simulation]; Przanowski & Brzykcy a1512 [number-phase Wigner function]; Hernández & Riedel a2103 [rapidly decaying Wigner functions]; > s.a. complexity; non-extensive statistical mechanics.