Special
Types of Symplectic Structures| Special
Types of Symplectic Structures |
In General > s.a. hamiltonian
dynamics [pataplectic]; poisson
structure.
@ On null surfaces: Nagarajan & Goldberg PRD(85).
@ Using boundary values: Soloviev JMP(93)ht,
TMP(97)gq [for
general relativity]; Bering
JMP(00)ht/98,
PLB(00)ht/99.
@ Generalizations: Zumino MPLA(91)
[Fermionic coordinates]; Nikolic qp/98 [non
equal-time];
Vanhecke BJP(06)mp/05 [non-commutative 
];
Sergyeyev SIGMA(07)mp/06 [weakly
non-local]; >
s.a. non-commutative physics; poisson
structures [Moyal and Nambu brackets]; Supermanifolds.
Covariant > s.a. geometric quantum
mechanics; hamiltonian
dynamics; hilbert
space; modified uncertainty relations; observables.
* 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; Kuchar 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;
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 gq/05 [hydrodynamic
form]; Montesinos JPCS(05)gq/06 [rev];
Vitagliano JGP(09)-a0809.
@ And quantization: Amelino-Camelia et al PAN(98)ht/97 [
-deformed,
and quantum gravity]; Mostafazadeh CQG(03)mp/02 [inner
product, Klein-Gordon fields]; > s.a.
path integrals in quantum field theory.
@ Relationships: Giachetta et al ht/99 [and
BRST]; Rovelli gq/02 [and
Hamilton-Jacobi equation]; Mondragón & Montesinos IJMPA(04)gq/03 [parametrized,
and observables]; Forger & Romero CMP(05)mp/04 [and
multisymplectic].
@ For brane dynamics: Cartas-Fuentevilla
PLB(02)ht [p-branes
in curved spacetime], PLB(02)ht [extendons];
Carter IJTP(03)ht-in;
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 & Kuchar
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]; > s.a. BF
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].
Multisymplectic and Polysymplectic Formalism > s.a. constrained
systems; hamiltonian dynamics; symmetries.
@
General references: Gotay in(91); Mangiarotti & Sardanashvily MPLA(99)ht [Koszul-Tate
cohomology]; 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].
@ DeDonder-Weyl: Kanatchikov RPMP(98)ht/97,
IJTP(98)qp/97 [general], gq/98-in,
RPMP(00)ht/99,
IJTP(01)gq/00 [quantum
gravity]; Paufler & Römer RPMP(02)mp/01-in;
Hélein
mp/02-in
[and generalizations]; Hélein & Kouneiher mp/04 [vs
Lepage-Dedecker];
Román-Roy mp/05-in
[first-order field theories]; Kanatchikov a0807-in [generalized Dirac bracket]; > s.a. approaches
to quantum field theory.
@ For classical field theories: Günther JDG(87);
Binz et al RPMP(02)mp,
de León et al a0803 [non-holonomic
constraints]; Munteanu et al JMP(04);
Román-Roy et al a0705 [k-symplectic, k-cosymplectic
and multisymplectic, relationships]; > 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-in.
@ Related topics: 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]; Marsden et al
JGP(01)
[continuum mechanics]; Chen LMP(05)
[variational formulation]; Hydon PRS(05)
[for differential-difference equations].
Bi-Hamiltonian Structures > s.a. integrable
systems; quantum systems.
* Idea: (M, 
, H, ', H'),
such that (, H)
and (', H')
induce the same Hamiltonian vector fields (equations of motion).
* Useful tensor: Can
define the 1-1 tensor Sab:= 'acbc;
satisfies 
X_H Sab
= 0.
* Conserved quantities: 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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aug 2009