Special Types of Symplectic Structures| 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:= Sac ∇c Sbm − Sbc ∇c Sam − Scm (∇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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