Poisson Brackets / Algebra / Structures |
In General > s.a. hamiltonian;
lagrangian dynamics; symplectic manifold.
$ Pre-Poisson structure: A manifold M
and a Lie algebra structure on C∞(M) with
Leibniz identity.
$ Poisson structure: A pre-Poisson structure
satisfying the Jacobi identity, i.e., a pair (M, { , }), such that
∃ f, g ∈ C∞(M) , {f, g} = Ωij (∂f/∂xi) (∂g/∂xj)
is a bilinear, skew-symmetric form satisfying the Jacobi identity.
* Relationships: A
Poisson structure is canonically defined by a Lie groupoid.
* Example: A symplectic manifold,
where Ωij is non-degenerate and
is the inverse of the symplectic structure, which is closed by the Jacobi identity.
@ General references:
Laurent-Gengoux et al 13;
Moshayedi a2012-ln
[Poisson geometry and deformation quantization].
@ And other structures: Balinsky & Burman JPA(94) [compatible with algebraic structure];
Landsman RVMP(97)qp/96 [transition probability];
Boucetta CRAS(01) [and pseudo-Riemannian metric],
DG&A(04) [and pseudo-Riemannian Lie algebras];
Petalidou JPA(02) [and Jacobi structure];
Cattaneo et al CMP(05)m.SG/03,
Dherin LMP(06)m.SG/05 [generating functions];
Cortese & García PLA(06)ht [compatibility with equations of motion];
Morchio & Strocchi LMP(08)-a0805 [Lie-Rinehart algebra of a manifold, and dynamics];
Machon a2008
[Poisson bracket on the space of Poisson structures];
> s.a. non-commutative geometry.
@ Special cases: García-Naranjo et al LMP(15)-a1406 [on smooth four-manifolds];
Benini & Schenkel AHP(17)-a1602 [for non-linear scalar field theories, based on the Cahiers topos];
Jordan AJP(16)nov
[for generators of the Galilei and Poincaré groups of spacetime transformations];
Díaz-Marín a1812 [Yang-Mills fields on manifolds with boundary].
@ Quantum:
Racanière JGP(06);
Esposito a1502-proc;
Khorasani EJTP-a1411 [derivation];
Liebrich a2103 [in field theory, regularization].
@ Related topics: Grabowski et al MPLA(93) [classification];
Hojman JPA(96) [from symmetry and conservation law];
Bering PLB(00) [boundary Poisson brackets];
Cattaneo LMP(04)m.SG/03 [integration]:
Ortega & Ratiu LMP(04) [symmetry reduction];
McLachlan JPA(09) [vector fields];
Gürses et al JMP(09) [finding, for a given dynamical system];
Leclerc a1211 [symmetric, for fermion fields];
Pavelka et al PhyD(16)-a1512 [change of description and hierarchy of Poisson brackets];
Cattaneo et al a1811-en [graded Poisson algebras];
> s.a. non-equilibrium thermodynamics;
riemann tensor.
> Online resources:
see Wikipedia page.
Jacobi Bracket / Structure on a Manifold
* Idea: A
generalization of the Poisson bracket / structure, which
represents a weakening the Leibniz rule.
* Generalization –
Jacobi algebroid: A graded Lie bracket on the Grassmann algebra
associated with a vector bundle which satisfies a property similar
to that of the Jacobi brackets.
@ General references: de León et al JMP(97) [geometric quantization];
Berceanu RVMP(06) [holomorphic representation].
@ Generalizations: Pérez Bueno JPA(97)ht;
Grabowski & Marmo JPA(01)m.DG,
JPA(03) [algebroid].
@ Physics examples: Asorey et al MPLA(17)-a1706 [test particles].
Moyal Algebra / Brackets > s.a. algebra;
deformation quantization; Dirac
Bracket; Peierls Bracket; wigner
functions; Wigner-Weyl-Moyal Formalism.
* Idea: A deformation of the
Poisson algebra/bracket, obtained by introducing higher-derivative terms in it.
* Freedom: The Jacobi identity
fixes it almost uniquely, but it depends on a parameter κ, with
{ , }κ → { , }PB
as κ → 0.
@ General references: Moyal PCPS(49);
Fletcher PLB(90) [uniqueness];
Gozzi & Reuter MPLA(93),
IJMPA(94)ht/03;
Strachan JPA(95);
Tzanakis & Dimakis JPA(97) [uniqueness];
Merkulov mp/00;
Dias & Prata JMP(07)qp/06 [and evolution];
Hiley a1211
[and the von Neumann operator algebra].
@ For spin: Amiet & Weigert PRA(01) [spin and particle];
Heiss & Weigert PRA(01) [discrete].
@ For other theories: Fairlie MPLA(98) [in M-theory];
Finkelstein ht/99 [gauge theory].
@ Variations: Masuda & Saito MPLA(99)ht [supersymmetric];
Dimakis & Müller-Hoissen LMP(00)ht [covariant, and Seiberg-Witten maps];
Gouba et al MPLA(12)-a1106
[generalization of the Moyal and Voros products, and physical interpretation].
> Online resources:
see Wikipedia pages on Moyal
bracket and Moyal product.
Nambu Brackets
> s.a. deformation quantization; Nambu Dynamics;
phase space; Ternary Operations.
* Idea:
A generalization of Poisson brackets of the form
{f1, ..., fn} = ηi1, ..., in ∂i1 f1 ... ∂in fn ,
where η is the Nambu tensor; It is used in a modified form
of classical dynamics; Its quantization is still not understood.
@ General references: Takhtajan CMP(94);
Hietarinta JPA(97);
Gautheron CMP(98);
Pandit & Gangal JPA(98) [geometric];
Grabowski & Marmo JPA(99) [inductive definition];
Ogawa & Sagae IJTP(00) [Lagrangian formalism];
Dufour & Zhitomirskii LMP(03) [and singularities of integrable 1-forms];
Tegmen IJMPA(06)mp [with constraint functionals];
Dereli et al IJMPA(09) [3D phase space, canonical transformations].
@ Examples, systems: Yamaleev AP(00) [relativistic particle];
Guha JMP(02) [hydrodynamic models];
Salazar & Kurgansky a1011 [electromagnetic field];
Horikoshi & Kawamura a1304 [from a variant formulation of Hamiltonian dynamics].
Other Generalizations
@ Covariant: Marsden et al AP(86) [classical fields and electromagnetism];
Pol'shin IJGMP(08)-a0801 [Leibniz bracket];
D'Avignon a1510
[non-canonical bracket and physical consequences];
> s.a. symplectic structures.
@ Non-local: Mokhov FAA(03)m.DG/02,
TMP(02)m.DG,
TMP(04)m.DG/02 [hydrodynamic];
Casati et al a1903 [weakly non-local].
@ Related topics:
de Azcárraga et al JPA(96),
ht/96-proc,
JPA(97);
Grabowski & Marmo MPLA(98) [based on 2k-forms];
Severa & Weinstein PTPS(01)m.SG [closed 3-form background];
Lavagno et al EPJC(06)qp [q-deformed];
Golmankhaneh TJP(08)-a0807 [fractional];
Khudaverdian & Voronov AIP(08)-a0808 [higher-order];
Mokhov a1001 [deformation];
Bruce JoM-a1301 [Loday-Poisson brackets];
Beltiţă et al JGP(18)-a1710 [on Banach manifolds].
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