Deformation Quantization  

In General > s.a. geometric quantization; quantum group; schrödinger equation [generalizations]; Star Product.
* Idea: An approach to quantization in which the classical algebra of observables for a physical systems is replaced by a deformed algebra, with multiplication replaced by a (non-commutative but associative) star product; The best known example is the Moyal star product.
@ Founding papers: Sternheimer, Flato, Lichnerowicz [precursors]; Bayen et al AP(78), AP(78).
@ General references: 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 mathematics and physics]; Lavagno JPA(08)-a0808 [and q-Hermitian operators], RPMP(09)-a0911 [basic theory]; Curtright & Zachos APPN(12)-a1104 [history]; Dey et al JPA(13)-a1302 [different types of representations]; Velhinho EJTP-a1412 [strict quantization, intro]; Waldmann a1502-proc [recent developments].
@ 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 constraints.
@ Polarized: Donin JGP(03) [classification].
@ Books and intros: Fedosov 96; Carroll 00; Hirshfeld & Henselder AJP(02)may [and teaching]; Dito & Sternheimer m.QA/02-proc [development]; Waldmann ht/03-ln, RVMP(05)m.QA/04 [representation theory]; Tillman JPA(07)gq/06-proc [and the Klein-Gordon equation]; in Todorov BulgJP(12)-a1206 [and geometric quantization].

Special Topics and Consequences > s.a. coherent states; GNS construction; Orbifold; uncertainty relations; 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.
@ General references: Matos-Abiague JPA(01)qp [and fractional dimensions]; González et al JOB(03)qp [on S1 × \(\mathbb R\)]; Andersen m.DG/06 [Hitchin's connection, Toeplitz operators]; Bieliavsky et al CMP(09)-a0806 [space of invariant deformation quantizations on the hyperbolic plane]; Much JMP-a1608 [curving flat spacetime]; Domański & Błaszak a1706 [with minimal length, complete theory].
@ Techniques: Waldmann CMP(00) [locality and GNS construction]; Periwal ht/00 [non-perturbative corrections]; Butin Sigma(08)-a0804 [and Hochschild cohomology]; de Gosson & Luef JPA(09)-a0901 [use of Feichtinger's modulation spaces]; Tosiek et al JMP(16)-a1502 [WKB method]; Tosiek a1609 [shortcomings of formal series calculus].

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]; Sato PRD(12) [Zariski quantization].
@ 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 mechanics.
@ 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]; Berra-Montiel et al IJGMP(16)-a1408 [Poisson structure and deformation quantization]; > s.a. Liouville Theory.
@ And coherent states: Daoud & El Kinani JPA(02)mp/03; Tan JPA(06)ht, Ghorashi et al IJMPA(12) [coherent-state-based path integral].
@ Phase-space formulation: Hakioglu & Dragt JPA(01)qp; Zachos IJMPA(02)ht/01.
@ With constraints: Antonsen gq/97, gq/97-proc; Chapline & Granik ht/98; Hori et al PTP(02)ht; Krivoruchenko ht/06-conf; > s.a. dirac quantization.
@ For fermions: Galaviz et al AP(08)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, JPA(10) [membrane]; > s.a. minisuperspace quantum gravity; quantum oscillators.

Other Deformations > s.a. fock space; non-commutative theories; poisson brackets.
* Normal star product: Can use the non-commutative star product f *g := exp{\(\hbar\)∂aa*} f(a) g(a*), where a and a* are holomorphic coordinates on phase space; Does not work for the simple harmonic oscillator because it gives En = \(\hbar\)ωn without the 1/2 (Moyal brackets are better in this sense), but may be desirable in quantum field theory.
* Fedosov formalism: A generalization of the Moyal star product for an arbitrary symplectic manifold, based on a symplectic torsion-free affine (Fedosov) connection.
@ 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 CPC(08)-a0801 [*-product with Mathematica]; Vacaru a0801 [and lqg], Tosiek JMP(11)-a0907; Tosiek PLA(12) [1D, solution method for eigenvalue equation]; > s.a. symplectic manifolds.
@ Fedosov formalism, generalizations: Bering Sigma(09)-a0804; Dobrski IJGMP(15)-a1411 [generalized Fedosov algebras]; > s.a. manifolds [supermanifolds].
@ 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]; Ribeiro-Silva & Oliveira-Neto IJMPA(08) [in quantum field theory]; Dorsch & Nogueira IJMPA(12)-a1106; Masłowski et al JPA(12)-a1201 [and minimal length]; Gavrilik & Kachurik MPLA(12)-a1204 [3-parameter deformation]; Faizal IJGMP-a1404 [maximal momentum]; Pramanik et al AP(15)-a1411 [and path-integral quantization]; > s.a. dirac equation; Quantum Algebra.
@ Related topics: Pflaum ht/96 [normal-order quantization on cotangent bundles]; Kontsevich LMP(03) [Poisson manifolds]; Khrennikov IDAQP(07)qp/04 [hyperbolic analog of Moyal bracket]; Gerstenhaber m.QA/05 [criterion]; Sontz a0809-proc [deformed Segal-Bargmann space]; Govaerts & Mattelaer a0812-proc [and phase-space path integral]; Duval & Gotay RPMP(12)-a1108 [deformation of prequantization]; Esposito a1207-ln [Poisson manifolds]; Garay et al AP(14)-a1309 [based on endless analytic continuation].

Types of Systems > s.a. composite quantum systems; integrable quantum systems; quantum oscillators; quantum particles; sigma model.
@ Phenomenology: Maziashvili & Megrelidze PTEP(13)-a1212 [issues and problems].
@ 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 PLB(08)-a0805 [3D models and physical meaning of deformation]; Hawkins CMP(08) [S2, obstruction]; Becher et al LMP(10)-a0908 [open systems]; Waldmann a0909-proc [Rieffel's deformation quantization]; Buisseret PRA(10)-a1011 [N-body systems]; Berger & Maziashvili PRD(11)-a1010 [free particle, wave function].
@ 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) [κ-deformation and UV-IR mixing]; Maziashvili PRD(12)-a1104 [free massless field on de Sitter spacetime].
@ 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]; Arzano AIP(09)-a0909 [pedagogical, κ-Fock space]; Morales PhD(12)-a1202 [free Dirac field in curved spacetime]; > s.a. 2D quantum gravity; modified approaches to quantum gravity; gravitation; klein-gordon quantum field theory; minisuperspace quantization [Kantowski-Sachs]; modified quantum field theories; non-commutative gauge theory; non-commutative gravity; Pais-Uhlenbeck Model.
@ Strings: García-Compeán et al JPA(00) [bosonic].

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