Deformation Quantization| 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];
Bishop et al PLB(20)-a1911 [modified operators vs modified commutators, and minimal length];
Chan et al a2009 [quantization on Kähler manifolds];
Moshayedi a2012-ln [and Poisson geometry].
@ 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];
Berra-Montiel & Molgado CQG(20)-a1911 [group-averaging approach];
> 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(17)-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; Wigner-Weyl-Moyal Formalism.
@ 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{\(\hbar\)∂a∂a*}
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];
Rudolph & Schmidt a2009
[symplectification of the complete lift of a Levi-Civita connection];
> 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];
Vassilevich & Oliveira LMP(18)-a1802 [based on non-associative algebras].
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 proc(10)-a0909 [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];
Wu & Zhou a1807;
Berra Montiel & Molgado a2005 [holomorphic representation].
@ 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; loop quantum cosmology; 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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