Wigner Functions and Quantum Theory in Phase Space

Phase Space Approach to Quantum Theory > s.a. formulations of quantum theory; Husimi Functions; path integrals.
* Distribution functions: Different ones are used, such as the Wigner distribution function, the Glauber-Sudarshan P and Q functions, the Kirkwood distribution function and the Husimi distribution function.
@ General references: Flandrin et al PLA(84) [properties]; Wang & O'Connell FP(88); Kim & Wigner AJP(90)may; Fairlie & Manogue JPA(91); Kim & Noz 91; Schroeck IJTP(94) [advantages], 96; Stulpe 97-qp/06; Sala et al PLA(97) [equivalence, with singular kernel]; Ban JMP(98) [representation of vectors]; Brif & Mann qp/98 [with Lie symmetries]; de Gosson JPA(01) [and the symplectic camel]; Dragoman PiO(02)qp/04 [and classical optics]; Anastopoulos AP(03); Campos JPA(03); Chruscinski OSID(06)qp/04 [Berry's phase]; de Gosson JPA(05)mp [irrep of Heisenberg algebra]; Chaturvedi et al JPA(06)qp/05, qp/05/JPA [new approach]; Smith JPA(06); Nasiri et al JMP(06)qp [general approach]; Nha PRA(08)-a0804 [conditions for physical realizability].
@ Reviews: Lee PRP(95) [distribution functions]; Brooke & Schroeck IJTP(05)qp/06.
@ Proposals: García de Polavieja PLA(96) [causal]; de Gosson JPA(05) [Torres-Vega & Frederick equation].

Wigner Function > s.a. [quantum mechanics]; entropy; experiments in quantum mechanics; quantum states.
* Idea: The distribution function or density matrix in phase space quantization.
$ Def: If is a solution of the Schrödinger equation, it is the real function

W(x, p, t):= ()^–1 dy *(x+y, t) (x–y, t) exp{2i py/} .

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

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

* Hudson's theorem: For 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); Royer PRL(85); Wootters AP(87); Dragt & Habib qp/98-in [and symplectic maps]; Li et al PRA(04), Revzen FP(06) [and phase space probability density]; Khademi qp/06; Frieden & Soffer PRA(06) [and information theory]; Johansen a0804; Case AJP(08)oct [and Weyl transforms, for pedestrians].
@ Intros, reviews: Zachos IJMPA(02)ht/01 [and deformation]; Dias & Prata MPLA(05)ht-in [and deformation]; Nassimi a0706.
@ Hudson's theorem: Gross JMP(06), APB(07)qp-in [for finite-dimensional system]; Mandilara et al PRA-a0808 [extension to 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]; > s.a. quantum chaos.

Specific Types of Systems > s.a. Kerr State; photon; quantum oscillator; quantum systems; spinning quantum particle.
@ Discrete systems: Leonhardt PRL(95), PRA(96) [and tomography]; Atakishiyev et al JMP(98) [finite data sets]; Takami et al PRA(01)hl/00 [lattice]; Bianucci et al PLA(02) [e.g., quantum computers]; Luís PRA(04) [states on single phase space points]; Gibbons et al PRA(04)qp [based on finite fields]; Galvão PRA(05)qp/04 [and computation]; Argüelles & Dittrich PhyA(05)qp; Cormick et al PRA(06)qp/05 [classicality]; Chaturvedi et al Pra(05)qp [algebraic]; Franco & Penna JPA(06)qp/05 [2 qubits, entanglement]; Ruzzi et al JPA(05) [Cahill-Glauber formalism]; Gross JMP(06)qp [non-negative]; Klimov et al qp/06-ln; Cormick & Paz PRA(06)qp [interference]; Wootters & Sussman a0704 [and minimum uncertainty states]; Gross & Eisert QIC(08)-a0710 [quantum Margulis expander map]; Klimov et al AP(09)-a0806 [n qubits]; Barry & Drummond a0808 [qubits, SU(n) coherent states]; Casaccino et al PRA(08) [extrema]; Livine a0906.
@ Discrete phase space: Gudder JMP(86); Bang & Berger PRA(09) [wave packets]; Hadhrami & Vourdas PRA(09) [fourth-order interference].
@ H atom, Coulomb potential: Holstein AJP(95)aug; Nouri PRA(98); Praxmeyer et al JPA(06)qp/05.
@ Infinite potential walls: Belloni et al AJP(04)sep, Dong et al PS(04) [square well]; Walton AP(07)qp/06.
@ Damped systems: Chruscinski mp/02; Belchev & Walton AP(09)-a0810 [and deformation quantization].
@ Other systems: Lee PLA(90) [potential step, quantum potential]; Dias & Prata JMP(02)qp/00 [with boundaries]; Curtright et al PLA(02)ht/01 [N-body]; Diósi & Kiefer JPA(02)qp/01 [Markovian open system]; Horvat & Prosen JPA(03), qp/06-in [classically chaotic]; O'Connell JOB(03)qp [dissipative, decoherence]; Genovese et al qp/05 [open boundary, failure].
@ Field theory: Ochs & Heinz AP(98) [covariant]; Amorim et al PhyA(09); > s.a. quantum cosmology; quantum field theory in curved spacetime.

Related Topics > s.a. huygens principle; pilot-wave interpretation; quantum correlations; quantum measurement; classical limit.
@ Semiclassical states: Rios & Ozorio de Almeida JPA(02)mp/01; Veble et al JPA(02); Dittrich et al PRL(06)qp/05 [propagator]; de Gosson & de Gosson qp/06 [squeezed]; Pulvirenti JMP(06); de Gosson JPA(08) [and Feichtinger algebra]; > s.a. decoherence; quantum states.
@ And foundations: Banaszek & Wódkiewicz PRA(98) [EPR]; Dias & Prata PLA(01), PLA(02)qp [and pilot-wave interpretation]; 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].
@ 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].
@ Entanglement: Hardy et al PS(04); Narnhofer JPA(06); Ozorio de Almeida LNP-qp/06 [in phase space].
@ Other topics: Mehta JMP(64); Wlodarz 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]; Dias & Prata AP(04) [admissible states]; Ozorio de Almeida & Brodier AP(06) [propagators for operators]; Tegmen NCB(07)mp [simple states]; Klauder & Skagerstam JPA(07) [generalized representations of operators]; Scott & Caves AP(08)-a0801 [sub-Planck structure]; Kube et al JCP(09) [Monte Carlo sampling].

Variations, Generalizations > s.a. deformation quantization [including constrained systems]; Fermi Function.
@ Relativistic: Sonego PRA(91); Morgan PLA(04)qp/03; Dragoman a0803 [and localization].
@ In curved spaces: Antonsen gq/97-in; Alonso et al JMP(02)qp [hyperboloids], JMP(03) [spheres].
@ Other variations: Cohen JMP(66) [more general quasi-probability distribution functions]; Fairlie CSF(99)ht/98 [Wigner-Weyl-Moyal approach to quantum mechanics]; Hakioglu & Tepedelenlioglu JPA(00) [action-angle Wigner function]; Curtright & Veitia JMP(07)qp [quasi-hermitian]; Ibort et al JPA(09) [on the space of irreducible representations of the Weyl-Heisenberg group]; Hu & Fan PRA(09) [generalized phase-space representation]; Wang et al a0908 [in non-commutative phase space].

main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 17 oct 2009