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 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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