Deformation Quantization  

In General > s.a. geometric quantization; Orbifold; quantum group; schrödinger equation [generalizations].
* Fedosov formalism: A generalization of the Moyal star product for an arbitrary symplectic manifold.
@ General references: Bayen & al AP(78), AP(78); Bakas & Kakas pr(85); Fairlie & Nuyts JPA(91); Ubriaco MPLA(93); Lledó & García-Sucre JMP(96); Weaver CMP(97); Farkas LMP(00) [and affine symplectic algebras]; Landsman CMP(03)mp/02 [Baum-Connes conjecture]; de Gosson m.SG/05 [phase space formulation]; Fialowski IJTP(08) [in math and physics].
@ Fedosov formalism: Fedosov JDG(94); Gadella et al JGP(05)ht/04 [in fiber bundle terms]; Tillman & Sparking JMP(06) [particle on S2, observables]; Tillman gq/06-wd [Fedosov star in some spacetimes]; Tosiek APPB(07)mp/06 [abelian connection]; Vacaru JMP(07)-a0707 [for Finsler and Lagrange spaces]; Tosiek a0801 [*-product with Mathematica]; Vacaru a0801 [and lqg], > s.a. symplectic manifolds.
@ Fedosov formalism, generalizations: Bering a0804; > s.a. manifolds [supermanifolds].
@ With constraints: Shabanov MPLA(95)qp/96; Bordemann et al CMP(00) [BRST cohomology]; Grigoriev & Lyakhovich CMP(01)ht/00 [Fedosov quantization as BRST theory]; Dias & Prata JMP(05)mp [deformation, phase space quantization]; > s.a. second class.
@ Polarized: Donin JGP(03) [classification].
@ Books and intros: Fedosov 96; Carroll 00; Hirshfeld & Henselder AJP(02) [and teaching]; Dito & Sternheimer m.QA/02-in [development]; Waldmann ht/03-ln, RVMP(05)m.QA/04 [representation theory]; Tillman JPA(07)gq/06-in [and Klein-Gordon].

Special Topics and Consequences > s.a. coherent states; uncertainty; wigner functions.
* Idea: A deformation of the Heisenberg algebra (commutators) leads to modified uncertainty relations, which could correspond to the existence of a minimal length.
@ References: Waldmann CMP(00) [locality and GNS]; Periwal ht/00 [non-perturbative corrections]; Matos-Abiague JPA(01)qp [and fractional dimensions]; González et al JOB(03)qp [on S1 R]; Andersen m.DG/06 [Hitchin's connection, Toeplitz operators]; Butin a0804 [and Hochschild cohomology]; Bieliavsky et al a0806 [space of invariant deformation quantizations on the hyperbolic plane].

Based on Nambu Algebras / Brackets > s.a. poisson brackets [classical Nambu brackets].
@ General references: Xiong PLB(00); Frønsdal LMP(01) [quantization, and QCD]; Curtright & Zachos PRD(03) [and classical mechanics].
@ Examples, systems: Curtright & Zachos NJP(02)mp [superintegrable systems]; Nutku JPA(03)qp [harmonic oscillator]; Zachos & Curtright CzJP(04)mp [H atom].

Based on Moyal Brackets > s.a. poisson brackets; formulations of quantum mechanic; quantum oscillators.
@ General references: Braunss JMP(94); Osborn & Molzahn AP(95); Gracia-Bondía & Várilly JMP(95) [geometric]; Ovsienko JDG(97); Finkelstein LMP(99)ht [observable properties]; Koikawa PTP(01)ht, PTP(02)ht/01 [oscillator, vacuum], PTP(01)ht [Im, and Toda lattice]; Dias & Prata JMP(07) [Moyal trajectories and classical motion]; > s.a. Liouville Theory.
@ And coherent states: Daoud & El Kinani JPA(02)mp/03; Tan JPA(06)ht [coherent-state-based path integral].
@ Phase space formulation: Hakioglu & Dragt JPA(01)qp; Zachos IJMPA(02)ht/01.
@ For fermions: Galaviz et al ht/06, AP(08)ht/07; Odendahl & Henselder PLA(08) [and path integral].
@ Other types of systems: Castro PLB(97)ht [continuous Toda field, geometric], ht/97 [membrane].
@ With constraints: Antonsen gq/97, gq/97-in; Chapline & Granik ht/98; Hori et al PTP(02)ht; Krivoruchenko ht/06-in; > s.a. dirac quantization.

Other Deformations > s.a. fock space; non-commutative theories; poisson brackets.
* Normal star product: Can use the non-commutative star product f *g := exp{aa*} f(a) g(a*), where a and a* are holomorphic coordinates on phase space; Does not work for the sho because it gives En = n without the 1/2 (Moyal brackets are better in this sense), but may be desirable in quantum field theory.
@ Deformed Heisenberg algebra: Iorio & Vitiello MPLB(94)mp/00; Schmüdgen JMP(99) [operator representations]; El Kinani IJTP(00) [R-deformed]; Lubo ht/00 [thermodynamic implications]; > s.a. Quantum Algebra.
@ Related topics: Kontsevich LMP(03) [Poisson manifolds]; Khrennikov qp/04 [hyperbolic analog of Moyal]; Gerstenhaber m.QA/05 [criterion].

Types of Systems > s.a. oscillator; quantum particles; sigma model.
@ General references: Hirshfeld & Henselder AP(02) [with fermions]; García-Compeán et al JPA(02) [geometric quantum mechanics]; Benczik et al PRA(05)ht [H atom with minimal length]; Kupriyanov et al JPA(05)qp [linear dissipative]; Frønsdal & Kontsevich LMP(07)mp/05 [on varieties with singularities]; Gamboa et al a0805 [3D models and physical meaning of deformation].
@ 1D infinite wall / well: Kryukov & Walton AP(05)qp/04, comment Dias & Prata AP(05)qp/05; Kryukov & Walton CJP(06)qp/05.
@ Scalar quantum field theory: García-Compeán et al IJMPA(01) [and abelian gauge theory]; Hirshfeld & Henselder AP(02); Grosse & Wohlgenannt NPB(06) [-deformation and UV/IR mixing].
@ Other quantum field theory: Ferrara & Lledó JHEP(00) [supersymmetric theories]; Hirshfeld & Henselder AP(02) [fermions]; Finkelstein ht/03 [pairs of dual algebras]; Harrivel mp/06 [covariant]; > s.a. 2D quantum gravity; approaches to quantum gravity; gravitation; klein-gordon quantum field theory; types of quantum field theories.
@ Strings: García-Compeán et al JPA(00) [bosonic].

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