Non-Commutative Geometry| Non-Commutative Geometry |
In General > s.a. [manifolds];
holonomy; quantum
group; spectral geometry; Star
Product.
* Idea: Non-commutative
spaces are spaces with quantum group symmetry; They are based on (1) A non-commutative
algebra 
defined by
a star product which replaces the Abelian one of functions on a manifold,
with a representation on a Hilbert
space 
; (2) An exterior
differential algebra on 
(),
with d taking n-forms into (n+1)-forms; (3) Possibly some
additional structure,
like a Dirac operator, which encodes the metric structure.
* History: A precursor
was Snyder's spacetime with non-commuting coordinates; > s.a. quantum
spacetime.
Non-Commutative Spacetime > s.a. linear
connections; Hopf
Algebra; matrix [models]; quantum
geometry; quantum
spacetime; uncertainty.
* Semi-Riemannian: The
spectral triple can be generalized; One replaces Hilbert spaces with Krein
spaces, and Dirac operators are Krein-selfadjoint.
@ Intros and reviews: Majid LNP(00)ht,
JMP(00)ht [and
quantum groups]; Saito G&C(00);
Balachandran Pra(02)ht [fuzzy
spacetime]; Masson mp/06-ln;
Perini IJMPA(08)-in;
Lizzi a0811-in;
Chamseddine & Connes a0812 [introduction
and overview]; Chamseddine a0901-in.
@ General references: Rosen AP(62)
[complex combinations of x and p]; Chamseddine & Fröhlich ht/93-in;
Heller & Sasin
JMP(96) [singularities];
Madore & Mourad JMP(98)gq/96 [differential
calculus and Minkowski space]; Mangano JMP(98)gq/97 [even D,
path integral]; Strohmaier JGP(06)mp/01;
Moretti RVMP(03)gq/02 [C*-algebra
approach]; Chaichian et al PLB(04)ht,
Kosinski & Maslanka ht/04 [interpretation];
Agostini JMP(05)ht/05 [operators
and integration]; Martinetti IJMPA(09)-a0904-in
[notion of line element].
@ Locally non-commutative spacetime: Bahns & Waldmann RVMP(07)m.QA/06;
Heller et al LMP(07)
[C*-algebraic model].
@ And causality: Seiberg
et al JHEP(00)ht;
Chu et al IJMPA(06)ht/05.
@ Non-commutative coordinates: Doplicher et al CMP(95)ht/03;
Toller qp/97,
PRA(99)qp/98,
IJTP(99)qp/98;
Dzhunushaliev GRG(02)ht/01 [interpretation];
Jarvis & Morgan FPL(06)
[Born reciprocity]; Bander PRD(07)ht [Lorentz-invariant].
@ Non-commutative lattices: Bimonte et al PLB(94)
[distances]; Balachandran et al JGP(96)ht/95 [from
posets as finite topologies]; 't Hooft gq/96-in;
Landi & Lizzi mp/98-in
[projective systems];
> s.a. lattice
field theories.
@ Fuzzy, quantum spacetime: Madore CQG(92);
Balachandran et al NPPS(94)ht;
Mack & Schomerus ht/94;
Kehagias et al JMP(95)ht [modified
Kaluza-Klein]; Sladkowski
ht/96; Demaret
et al gq/97;
Francis phy/99;
Requardt & Roy
CQG(01)
[fuzzy lumps]; de Albuquerque et al PRL(03)ht [Euclidean,
from spectral action], MPLA(03)ht; > s.a. non-commutative
gravity, [quantum
spacetime].
@ Particle propagation: Gamboa et al ht/01,
Falomir et al PRD(02)ht [Aharonov-Bohm
effect]; Amelino-Camelia et al PRD(03)
[string-inspired]; Elias & Steele IJMPE(07)ht/06 [Snyder
spacetime, massless scalar propagator]; > s.a. non-commutative
physics.
@ Effective non-commutativity: Das & Gegenberg GRG(08)-ht/04 [geodesics
in Gödel-like
spacetimes]; Corichi & Zapata IJMPD(08)-a0705 [from
1D, polymeric quantum geometry]; > s.a. 3D
quantum gravity.
@ Area
quantization / discreteness: Romero et al PRD(03)ht; Amelino-Camelia
et al PLB(09).
@ Related topics: Masson qa/95 [submanifolds,
quotients]; Coquereaux phy/96 [higher-order
differentials]; Jaffe phy/97 [invariants];
Perrot LMP(99)
[BRS cohomology & Chern
character]; Hawkins CMP(04)m.QA/02 [obstructions],
MPLA(03)
[compatibility]; Schwarz NPB(03)
[supergeometry]; Sardanashvily mp/03, mp/03,
m.QA/07 [differential
operators]; Licht ht/05-in
[star product for Snyder's approach].
> Related topics: see higher-dimensional
gravity; thermal
radiation; types
of distances; unimodular relativity.
Examples and Other Structures > s.a. principal
fiber bundles; diffeomorphisms; velocity.
* Example: The commutation
relations between coordinates become [xm, xn]
= i 
mn,
where mn is
a (constant) real antisymmetric matrix.
* Symmetries: Lie algebra
symmetries are replaced by Hopf algebra symmetries.
@ Spheres: Madore CQG(97)gq;
Pinzul & Stern
PLB(01)ht [S2q,
Dirac operator]; Sitarz LMP(01)mp,
CMP(03)mp/01 [S4];
Freidel
& Krasnov JMP(02)
[star-product]; Connes & Dubois-Violette LMP(03),
CMP(08)m.QA/05 [S3];
Lizzi et al JMP(05)
[symmetries]; Dabrowski JGP(06)
[S2q
and S3q];
Govindarajan et al a0906 [polynomial deformations of fuzzy spheres].
@ Groenewold-Moyal plane: Balachandran et al a0905-in,
Balachandran & Padmanabhan a0908-in [causality, statistics and other effects]; > s.a. non-commutative
gauge theory [QED]; types
of quantum field theories.
@ Other examples: Dimakis & Müller-Hoissen phy/97;
Cerchiai et al EPJC(99)m.QA/98 [q-deformed
line]; Connes & Dubois-Violette
CMP(02)m.QA/01 [3D
spherical
manifolds]; Jackiw NPPS-ht/01
[physical];
Alexanian et al JGP(02)
[CP2];
Fiore et al JMP(02)
[real quantum plane]; van Suijlekom JMP(04)mp/03 [Lorentzian
cylinder];
Lubo
PRD(05)ht/04 [star
product
on fuzzy sphere from squeezed state]; Buric & Madore ht/04-in
[2D,
review], PLB(05)
[2D, example]; Amelino-Camelia et al a0812 [Moyal
plane, distance observable]; > s.a. classical
particles; minkowski spacetime; rindler
space; schwarzschild
spacetime.
@ Tensor fields: Dvoeglazov S&S(02)mp, mp/03-in
[derivatives]; Dimitrijevic et al JPA(04)
[
-deformed euclidean
space].
@ Symmetries: Agostini et al IJMPA(04)ht/03 [Hopf
algebra]; Calmet PRD(05);
Chaichian et al PRL(05)
[twisted Poincaré symmetry]; Gonera et al PRD(05)ht [global];
Gracia-Bondía et al PRD(06);
Szabo CQG(06)
[rev, and gravity]; Goswami a0704.
@ Related topics: Breslav & Zapatrin IJTP(00)
[quantum/Greechie logic]; Díaz & Pariguán JPA(07)mp/06 [measures
and path integration]; Carey et al a0901 [Dirac operators].
Phenomenology > s.a. non-commutative
physics.
@ And DSR: Kowalski-Glikman & Nowak IJMPD(03); Heuson gq/03.
@ And Lorentz invariance: Anisimov et al PRD(02) [violations]; Chaichian
et al PLB(04)ht.
@ Related topics: Tamaki et al PRD(02)
[-Minkowski
and astrophysics]; Vilela
Mendes EPJC(05)ht/04;
Hinchliffe et al IJMPA(04)
[rev]; Martinis & Mikuta-Martinis gq/05 [near-horizon
geometry]; Horvat & Trampetic PRD-a0901 [from
primordial nucleosynthesis]; Akofor et al PRD(09)
[from cmb].
References > s.a. models
of topology change.
@ Texts and reviews: Manin
91;
Connes 94; Madore 95, gq/96 [connection
and curvature], gq/99-in;
Landi ht/97-in;
Varilly phy/97-ln;
Bigatti ht/98;
Madore 99;
Connes
JMP(00)ht, m.QA/00;
Martinetti ht/06-in;
Kar 08.
@ General: Connes CRAS(80)ht/01;
Dubois-Violette CRAS(88), et al JMP(90)
[matrix algebras]; Coquereaux JGP(89),
JGP(93);
Connes LMP(95), JMP(95);
Dubois-Violette qa/95 [derivations,
connections],
et
al
JMP(96) [curvature];
Kisil in(99)fa/97 [approaches];
Dimakis & Madore JMP(96)
[differential calculi]; Jackiw & Pi PRL(02)ht/01 [coordinate
changes]; Vacaru mp/02, mp/02 [Finsler,
with
local anisotropy]; Rennie & Várilly m.OA/06 [manifold
reconstruction]; Chaichian et al JMP(08)ht/06 [Riemannian,
framework]; Bertozzini et al a0801-in
[categorical methods]; Piacitelli a0901.
@ Almost commutative: Madore RPMP(99)gq/97 [Poisson structure and curvature];
Jureit & Stephan JMP(05),
Jureit et al JMP(05)
[classification].
@ Related topics: Dimakis & Müller-Hoissen JMP(99)gq/98 [discrete];
Lord mp/00-wd;
Martinetti
mp/01-PhD
[distances]; Ponge LMP(08) ["lower-dimensional" volumes].
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 1
nov
2009