Deformation
Quantization| Deformation
Quantization |
In General > s.a. geometric quantization; Orbifold; quantum
group; schrödinger equation [generalizations].
* 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], a0911 [basic theory].
@ 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-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 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.
@ 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 SIGMA(08)-a0804 [and
Hochschild cohomology]; Bieliavsky et al a0806 [space
of invariant deformation
quantizations on the hyperbolic plane]; de Gosson & Luef a0901 [use
of Feichtinger's modulation
spaces].
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 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 [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 simple
harmonic oscillator because it gives
En = 
n 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.
@ 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 a0907; > s.a. symplectic
manifolds.
@ Fedosov formalism, generalizations: Bering Sigma(09)-a0804; > 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]; > s.a. Quantum
Algebra.
@ Related topics: Kontsevich LMP(03)
[Poisson manifolds]; Khrennikov qp/04 [hyperbolic
analog of Moyal]; Gerstenhaber m.QA/05 [criterion];
Sontz a0809-in
[deformed Segal-Bargmann space]; Govaerts & Mattelaer a0812-in [and phase-space
path integral].
Types of Systems > s.a. composite
quantum systems; quantum
oscillators;
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 PLB(08)-a0805 [3D
models and physical meaning of deformation]; Hawkins CMP(08)
[S2,
obstruction]; Becher et al a0908 [open
systems]; Waldmann a0909-in [Rieffel's deformation quantization].
@ 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; modified quantum field
theories.
@ Strings: García-Compeán et al JPA(00) [bosonic].
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