Special Types of Symplectic Structures  

In General > s.a. hamiltonian dynamics [pataplectic]; poisson structure; symplectic structures [generalizations].
@ Using boundary values: Soloviev JMP(93)ht, TMP(97)gq [for general relativity]; Bering JMP(00)ht/98, PLB(00)ht/99.
@ On null surfaces: Nagarajan & Goldberg PRD(85).

Covariant > s.a. hamiltonian dynamics; modified uncertainty relations; Peierls Brackets; symplectic structures in physics.
* Idea: Introduce a symplectic structure on the space of histories for a field theory satisfying the field equations.
@ General references: Kugo & Ojima PLB(78); in Woodhouse 81; Kuchař PRD(86); Zuckerman pr(86); Grishchuk & Petrov JETP(87); Barnich et al PRD(91); Hannibal IJTP(91); Torre JMP(92); Marolf AP(94)ht/93; Gotay et al phy/98 [GIMMsy 1]; Mashkour IJTP(98), IJTP(01) [fields]; Bérard et al IJTP(00); Fernández et al JMP(00) [gauge-invariant]; Ozaki ht/00, ht/00; Cartas-Fuentevilla JMP(02)mp; Julia & Silva ht/02; Basu PRD(05)gq/04 [perturbative expansion and observables]; Basini & Capozziello MPLA(05) [from conservation laws]; Capozziello et al PiP(05)gq [hydrodynamic form]; Montesinos JPCS(05)gq/06 [rev]; Vitagliano JGP(09)-a0809; Khavkine IJMPA(14)-a1402 [rev]; Sharapov a1412-conf; Kaminaga a1703 [field theory]; Harlow & Wu a1906, Margalef-Bentabol & Villaseñor a2008 [with boundaries]; Shi et al a2008 [with null boundaries].
@ And quantization: Amelino-Camelia et al PAN(98)ht/97 [κ-deformed, and quantum gravity]; Mostafazadeh CQG(03)mp/02 [inner product, Klein-Gordon fields]; Benini MS(11)-a1111 [spin-1 fields on curved spacetimes]; > s.a. geometric quantum mechanics; hilbert space; path integrals in quantum field theory.
@ Relationships: Giachetta et al JPA(99)ht [and BRST]; Rovelli LNP(03)gq/02 [and Hamilton-Jacobi equation]; Mondragón & Montesinos IJMPA(04)gq/03 [parametrized, and observables]; Forger & Romero CMP(05)mp/04, Hélein a1106-conf, Forger & Salles a1501 [and multisymplectic].
@ For brane dynamics: Cartas-Fuentevilla PLB(02)ht [p-branes in curved spacetime], PLB(02)ht [extendons]; Carter IJTP(03)ht-conf; Escalante IJMPA(06)mp/04 [Dirac-Nambu-Goto p-branes].
@ Other systems: Nutku PLA(00)ht [Monge-Ampère], ht/00-in [Korteweg-de Vries]; Kouletsis & Kuchař PRD(02)gq/01 [strings]; Cartas-Fuentevilla CQG(02)ht [topological defects in curved spacetime]; Schreiber ht/03 [supersymmetric theories]; Grant et al JHEP(05)ht ["bubbling anti-de Sitter"]; Piña a0907 [charges]; Nazaroglu et al PRD(11)-a1104 [topologically massive gravity]; Alkac & Devecioglu PRD(12)-a1202 [new massive gravity]; > s.a. BF theory; geometric quantization [Klein-Gordon theory]; modified canonical general relativity; quantum particle and spinning particle models.
@ Related topics: Guo et al PRD(03)gq/02 [diffeomorphism algebra]; Reyes IJTP(04) [variational bicomplex, for Monge-Ampère equation]; Ciaglia et al MPLA(20)-a2005 [in terms of contact geometry]; > s.a. Alexandrov Sets; observables.

Multisymplectic and Polysymplectic Formalism > s.a. constrained systems; hamiltonian dynamics; symmetries.
@ General references: Gotay in(91); Giachetta et al NCB(99)ht [BRST-extended], JPA(99); Echeverría-Enríquez et al JMP(00)mp; Hélein & Kouneiher ATMP(04)mp/02; Sardanashvily mp/02 [field theory, no brackets]; Vey a1203-proc [notion of observable]; Marrero et al JPA(15)-a1306 [reduction]; Ryvkin & Wurzbacher a1804; Román-Roy a1807-ln [rev].
@ Related structures: Mangiarotti & Sardanashvily MPLA(99)ht [Koszul-Tate cohomology]; Paufler RPMP(01)mp/00 [vertical exterior derivative], RPMP(01)mp [Gerstenhaber structures]; Forger & Römer RPMP(01)mp/00, et al RVMP(03)mp/02, RPMP(03)mp/02 [Poisson brackets]; Chen LMP(05) [variational formulation]; Forger & Salles a1010 [Hamiltonian vector fields].
@ De Donder-Weyl: Kanatchikov RPMP(98)ht/97, IJTP(98)qp/97 [general], gq/98-proc, RPMP(00)ht/99, IJTP(01)gq/00 [quantum gravity]; Paufler & Römer RPMP(02)mp/01-in; Hélein mp/02-conf [and generalizations]; Hélein & Kouneiher mp/04 [vs Lepage-Dedecker]; Román-Roy Sigma(09)mp/05-conf [first-order field theories]; Kanatchikov a0807-proc [generalized Dirac bracket]; Kanatchikov JPCS(13)-a1302 [for vielbein gravity]; Riahi & Pietrzyk a1912 [for general relativity, and canonical Hamilton-Jacobi equation]; > s.a. approaches to quantum field theory.
@ For Yang-Mills theories: Vey a1303 [Maxwell theory]; Hélein a1406; Ibort & Spivak a1506, a1511 [and constrained theories, with boundaries].
@ Gravity: Ibort & Spivak a1605 [Palatini gravity, with boundaries]; Gaset & Román-Roy JMP(18)-a1705 [Einstein-Hilbert gravity]; Capriotti et al a1911 [Lovelock gravity]; Klusoň & Matouš a2008 [f(R) gravity]; McClain a2012-conf [polysymplectic approach and geometric quantization].
@ Other classical field theories: Günther JDG(87); Binz et al RPMP(02)mp, de León et al IJGMP(08)-a0803 [non-holonomic constraints]; Munteanu et al JMP(04); Román-Roy et al JGM(11)-a0705 [k-symplectic, k-cosymplectic and multisymplectic, relationships]; Prieto-Martínez & Román-Roy JGM(15)-a1402 [second-order field theories]; de León et al a1409-book [k-symplectic and k-cosymplectic approaches]; Sardanashvily a1505; > s.a. field theory [geometry].
@ For quantum field theory: Kanatchikov IJTP(98)qp/97, RPMP(98)ht/98, ht/01; Bashkirov IJGMP(04)ht [BV quantization]; Giachetta et al ht/04-proc.
@ Other theories: Marsden et al JGP(01) [continuum mechanics]; Hydon PRS(05) [for differential-difference equations]; > s.a. BF theory.

Bi-Hamiltonian Structures > s.a. integrable systems; quantum systems.
* Idea: (M, Ω, H, Ω', H'), such that (Ω, H) and (Ω', H') induce the same Hamiltonian vector fields (equations of motion).
* Useful tensor: Can define the 1-1 tensor Sab:= Ω'acΩbc; satisfies \(\cal L\)XH Sab = 0.
* Conserved quantities: They can be obtained by K0:= ln |det S|; Kn:= (1/n) tr Sn.
* Nijenhuis tensor: Defined using S, by

Nabm:= Sacc SbmSbcc SamScm (∇a Sbc − ∇b Sac) ,

which is Lie-derived by XH; The system is integrable if N = 0.
@ Bi-Hamiltonian vector fields: Magri JMP(78); Fuchssteiner PTP(82); Marmo et al NCB(87); in Das & Okubo AP(89).
@ Nijenhuis tensors: Bogoyavlenskij DG&A(06) [algebraic identities].

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