Phase Space| Phase Space |
In General
> s.a. hamiltonian dynamics; momentum;
symplectic structures and types
of symplectic structures [covariant].
* 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 by the flow of its Hamiltonian vector field; 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];
Albert et al JPA(17)-a1709 [different types].
@ In statistical mechanics:
Gallavotti CMP(01) [counting cells];
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;
particle phenomenology in quantum gravity.
* 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];
Rundle & Everitt a2102 [rev].
@ Reviews: Lee PRP(95) [distribution functions];
Brooke & Schroeck IJTP(05)qp/06;
Lobo & Ribeiro a1212;
Rundle & Everitt a2102 [and applications].
@ 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(15)-a1503,
a1711 [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];
Jong et al a1710 [statistical formalism];
Budiyono PRA(19)-a2005 [epistemically restricted];
Sato a2012 [three-dimensional phase space, Nambu Bracket].
@ Discrete phase space:
Marchiolli & Ruzzi AP(12),
Marchiolli & Mendonça AP(13)-a1304 [discrete version of the Weyl-Wigner-Moyal formalism];
Das & DeBenedictis a1504;
Hashimoto et al a1802 [phase point operators].
> Related subjects: see
classical vs quantum states; logic; path
integrals; wigner functions; Wigner Transform.
> Specific topics: see Fermi Functions;
geometric quantization [torus phase space]; Husimi Functions.
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send feedback and suggestions to bombelli at olemiss.edu – modified 2 apr 2021