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 Ω that makes it a symplectic vector space or manifold Γ; Then the canonical transformations are those diffeomorphisms that preserve Ω, 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 Γ and imposing that time evolution be generated by the Hamiltonian vector field of H.
* Important result: Gromov's 1985 non-squeezing theorem, essentially a classical form of the uncertainty principle.
> Related topics: see classical limit of quantum mechanics; hamiltonian 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]; Müller mp/01-wd; Romero & Vergara NPB(03)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]; Crnković 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; Libermann & Marle 87.
@ General references: Golovnev & Ushakov JPA(08) [and variational principle]; de Gosson a1208, AJP(13)may [Symplectic Egg, Symplectic Camel]; de Gosson 17 [and metaplectic].
@ And field theory: Kijowski & Tulczyjew 79 [for field theory]; de León et al 15 [k-symplectic and k-cosymplectic geometry].
@ And quantization: Marmo & Vilasi MPLB(96)ht; Montesinos & Torres del Castillo PRA(04)qp, comment Latimer PRA(07) + reply PRA(07) [ambiguity]; de Micheli & Zanelli JMP(12)-a1203 [degenerate symplectic structures]; Ziegler a1310 [localized quantum states].
@ 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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