Phase Space  

In General > s.a. hamiltonian dynamics; momentum; symplectic structures.
* Idea: Physically, the manifold M of all classical states of a system, usually specified in terms of configuration coordinates and canonical momenta (canonical phase space) or as histories (covariant phase space) with additional structure, these two being classically equivalent; Classically, a symplectic manifold (M, Ω) with a preferred function H, such that the evolution of a state is represented the flow of the Hamiltonian vector field of H; In the quantum theory, one needs in addition a complex structure J on M.
* Examples: In many cases, M has a cotangent bundle structure.
* Remark: A different point of view is to consider the cotangent bundle structure as essential and call phase space such a bundle even if covectors are not interpreted as momenta (see the approach of Kijowski and Tulczyjew).
@ General references: Nolte PT(10)apr [history]; Sławianowski et al a1404 [need, in physics].
@ In statistical mechanics: Gallavotti CMP(01) [counting cells in statistical mechanics]; Vesely EJP(05) [simple approach]; Sperling PRA(16)-a1605 [maximally singular phase-space distributions].

Special Topics and Results
* Phase curve: A curve representing the evolution of a system in phase space.
* Liouville theorem: The phase-space distribution function is constant along the trajectories of a system; I.e., time evolution preserves the phase space volume.
* Lagrange bracket: Given two functions u and v belonging to some set of 2n independent functions of the canonical qs and ps in phase space, their Lagrange bracket is

{u,v}q, p := {∂qi / ∂u} {∂pi / ∂v} – {∂pi / ∂u} {∂qi / ∂v} ;

It is a canonical invariant, but mostly of historical importance now.
@ Liouville theorem: in Tolman 38 [proof]; Momeni et al a0904-wd, Abadi et al IJMPB(09)-a0904 [rev]; Bravetti & Tapias JPA(15)-a1412 [for non-conservative systems]; > s.a. Wikipedia page.
@ Transformations: Luís PRA(04) [in phase space and Hilbert space].
@ Types of systems: Mann et al JPA(05) [finite phase space]; Tarasov JPA(05)m.DS/06 [non-Hamiltonian].
@ Fermions, Grassmann phase space methods: Dalton et al AP(16)-a1604 [for field theories]; Polyakov PRA(16)-a1609 [and probability distributions]; > s.a. hamiltonian systems.
@ Related topics: Friedman a0802 [relativistic, and representations of the Poincaré group].
> Related topics: see formalism of chaos [stochastic layer/web]; doubly special relativity; magnetism [momentum-space magnetic field]; Order [ordered and disordered states].

And Quantum Theory > s.a. canonical quantum mechanics; formulations of quantum theory; wigner functions; Wigner Transform.
* Quantum phase space: The complex projective space \({\mathbb C}{\rm P}^n\) with a Kähler structure given by the Fubini-Study metric and an associated symplectic form; The Schrödinger equation generates Hamiltonian dynamics on Γ.
* Approaches: Two approaches to the structure of quantum phase space are the Weyl-Wigner formalism and the theory of Coherent States.
* 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, or Dirac's quasiprobability distribution.
@ 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; Anastopoulos AP(03); Campos JPA(03); Isidro MPLA(05)qp/04 [complex structure and quantum]; de Gosson JPA(05)mp [irreducible representation of the 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]; Ranaivoson et al a1304; Burić et al PRA(12)-a1209, FP(13) [Hamiltonian formulation, and mixed states]; Karageorge & Makrakis a1402 [semiclassical initial-value problem]; Curtright et al 14 [intro]; Colomés et al JCE(15)-a1507 [comparing Wigner, Husimi and Bohmian distributions].
@ Reviews: Lee PRP(95) [distribution functions]; Brooke & Schroeck IJTP(05)qp/06; Lobo & Ribeiro a1212.
@ Related topics: Sala et al PLA(97) [equivalence, with singular kernel]; Ban JMP(98) [representation of vectors]; Brif & Mann PRA(99)qp/98 [with Lie symmetries]; de Gosson JPA(01) [and the symplectic camel]; Dragoman PiO(02)qp/04 [and classical optics]; Chruściński OSID(06)qp/04 [Berry's phase]; Andriambololona et al IJAMTP-a1503 [linear canonical transformations]; de Gosson & de Gosson a1510 [time-symmetric quantum mechanics].
@ Non-commutative phase space: Giunashvili mp/02; Li et al MPLA(05)ht/04 [oscillator]; Li & Dulat a0708 [and spacetime symmetries]; Bernardini & Bertolami PRA(13) [effects]; Liang et al PRA(14) [detection, and Aharonov-Bohm effect]; Chatzistavrakidis PRD(14); Beggs & Majid a1410 [quantum Riemannian geometry, and non-associativity]; > s.a. non-commutative theories.
@ Deformed phase space: Khosravi et al GRG(10) [equivalence with canonical quantization, in cosmological example]; Barcaroli et al PRD(15)-a1507 [phase-space geometry from modified dispersion relations]; Lukierski et al PLB(15)-a1507 [covariant, and Hopf algebroids]; Astuti & Freidel a1507 [Lorentz-invariant deformations]; Arzano & Nettel PRD(16)-a1602 [with group-valued momenta]; Meljanac et al PLB(17)-a1610 [and Poincaré symmetry].
@ Other proposals: García de Polavieja PLA(96) [causal]; Tsekov IJMS(01)-a1505; de Gosson JPA(05) [Torres-Vega & Frederick equation]; de Gosson FP(13)-a1106 [quantum blobs and squeezed states]; Watson & Bracken PRA(11) [phase-space amplitudes, for spinor systems]; Bolognesi IJMPD(14)-a1207 [generalized Fourier transform, and dark energy]; Costa Dias et al JPDOA(12)-a1209 [Schrödinger and Moyal representations]; Gneiting et al PRA(13)-a1309 [curved configuration space]; Bamber & Lundeen PRL(14) [Dirac's distribution, experimental observation]; López a1509 [extended phase space].
@ Discrete phase space: Marchiolli & Ruzzi AP(12), Marchiolli & Mendonça AP(13)-a1304 [discrete version of the Weyl-Wigner-Moyal formalism]; Das & DeBenedictis a1504.
> Related subjects: classical vs quantum states; path integrals.
> Specific topics: see Fermi Functions; geometric quantization [torus phase space]; Husimi Functions.


main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 9 jul 2017