Symplectic Structures in Physics

In General > s.a. [symplectic structure]; formulations of classical mechanics; higher-order lagrangian theories; Peierls Brackets.
* Idea: In the symplectic formulation of a physical theory, the phase space is given a symplectic structure that makes it a symplectic vector space or manifold ; Then the canonical transformations are those diffeomorphisms that preserve , and they are often generated by the Hamiltonian vector fields of some observables, whose Lie algebra structure coincides with the Poisson bracket structure; In particular, the equations of motion are recovered by giving a function H on and imposing that time evolution be generated by the Hamiltonian vector field of H.
> Related topics: see classical limit of quantum mechanics; poisson brackets; quantum spacetime; semiclassical quantum mechanics.

Examples > s.a. loop space; geometric quantization; hamiltonian systems; supergravity.
@ Particles: van Drie mp/00 [electromagnetic and gravitational force from connection]; Isidro IJGMP(07) [in magnetic fields].
@ Fields: DeWitt JMP(61), JMP(62) [with infinite-dimensional invariance groups]; Kijowski & Szczyrba CMP(76); Günter JDG(87) [scalar]; Mueller mp/01; Romero & Vergara ht/02 [boundary conditions as constraints]; Rey et al mp/06 [k-cosymplectic formalism]; Torres del Castillo & López-Villanueva IJMPA(06) [symplectic currents and symmetries]; Amorim et al PLA(07) [and representations of the Poincaré group]; Sibold NPB(09) [from variables conjugate to energy-momentum operator].
@ Space of projective and affine curves: Guieu & Ovsienko JGP(95).
@ Space of connections: King & Sengupta JMP(94) [explicit description], CMP(96) [with boundary]; Leung CMP(98).
@ Space of G-monopoles: Finkelberg et al CMP(99).
@ Other examples: Ellis JPA(75) [acceleration-dependent lagrangians]; Crnkovic NPB(87), CQG(88) [strings]; Cariñena & López IJMPA(91) [space of geodesics]; Ramos & Soloviev ht/93 [space of quantum field theories]; Mokhov ht/95 [loop space].
> Specific theories: see ADM formulation of canonical general relativity; klein-gordon field theory; oscillators.

Other References > s.a. boundaries in field theory; complex structure [Kähler structure].
@ Books: Abraham & Marsden 78; Kijowski & Tulczyjew 79 [for field theory]; Libermann & Marle 87.
@ General references: Golovnev & Ushakov JPA(08) [and variational principle].
@ And quantization: Marmo & Vilasi MPLB(96)ht; Montesinos & Torres del Castillo PRA(04)qp, comment Latimer PRA(07) + reply PRA(07) [ambiguity].
@ Covariant: D'Adda et al AP(85) [group manifold]; Nelson & Regge AP(86) [gravity]; > s.a. modified symplectic and poisson structures.
@ Forms and diffeomorphisms: Baulieu & Henneaux PLB(87); Henneaux & Teitelboim PLB(88).
@ Related topics: Souriau 64; Kijowski CMP(73); in Madore PRP(81); Brown & Henneaux JMP(86).

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