Symplectic Structures in Physics| 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].
@ Gravitational field:
Alessio & Arzano PRD(19)-a1906 [asymptotically flat, and BMS symmetries];
> s.a. ADM formulation of canonical general relativity.
@ Other 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 MPAG(12)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];
> s.a. klein-gordon field theory.
@ 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];
> s.a. 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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