Phase Space

In General > s.a. hamiltonian systems [including deformed phase space]; symplectic structures.
* Idea: Physically, the manifold M of all classical states or histories of the system (these are classically equivalent), usually specified in terms of configuration coordinates and canonical momenta, with additional structure; Classically, a symplectic manifold (M, ) 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 approach of Kijowski and Tulczyjew).
@ In statistical mechanics: Gallavotti CMP(01) [counting cells in statistical mechanics]; Vesely EJP(05) [simple approach]

Special Topics and Results > s.a. doubly special relativity; Order [ordered and disordered states].
* Phase curve: A curve representing the evolution of a system in phase space.
* Liouville theorem: The time evolution of a system preserves the phase space volume.
* Lagrange bracket: Given two functions u and v belonging to some set of 2n independent functions of the canonical q's and p's in phase space, their Lagrange bracket is

{u,v}_{q, p} := {q_i / u} {p_i / v} – {p_i / u} {q_i / v} .

It is a canonical invariant, but mostly of historical importance now.
@ Liouville theorem: in Tolman 38 [proof].
@ Width of stochastic layer: Tsiganis et al JPA(99) [driven pendulum]; Shevchenko PLA(08) [new estimation method].
@ 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]; > s.a. hamiltonian systems.
@ Related topics: Friedman a0802 [relativistic, and representations of the Poincaré group].

And Quantum Theory > s.a. canonical quantum mechanics; quantum mechanics in phase space [in particular, Wigner functions].
* Quantum phase space: The complex projective space CPⁿ with a Kähler structure given by the Fubini-Study metric and an associated symplectic form; The Schrödinger equation generates Hamiltonian dynamics on .
@ References: Isidro MPLA(05)qp/04 [complex structure and quantum].

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