Non-Commutative Geometry| Non-Commutative Geometry |
In General
> s.a. manifolds / holonomy;
quantum group; Spectral Triple;
Star Product.
* Idea: Non-commutative spaces are
spaces with quantum group symmetry; They are based on (1) A non-commutative algebra
\(\cal A\) defined by a star product which replaces the Abelian one of functions
on a manifold, with a representation on a Hilbert space \(\cal H\); (2) An exterior
differential algebra on Ω( \(\cal A\)), (n+1)-forms; (3) Possibly some
additional structure, like a Dirac operator, which encodes the metric structure.
* History: A precursor was Snyder's
spacetime in which coordinates are operators with commutation relations of the form
[xμ, xν]
= i q2θμν;
> s.a. quantum spacetime.
Examples and Other Structures > s.a. non-commutative cosmology,
gravity [black holes], spacetime [including causality].
* Example: The commutation relations
between coordinates become [xm,
xn]
= i θmn,
where θmn
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];
Dąbrowski JGP(06)
[S2q
and S3q];
Govindarajan et al JPA(10)-a0906 [polynomial deformations of fuzzy spheres];
D'Andrea et al LMP(13);
Berenstein et al a1506 [rotating fuzzy spheres];
Ishiki & Matsumoto a1904 [diffeomorphisms of fuzzy spheres];
> s.a. spherical harmonics.
@ Moyal / Groenewold-Moyal plane:
Amelino-Camelia et al a0812 [distance observable];
Balachandran et al a0905-conf,
Balachandran & Padmanabhan a0908-proc,
Balachandran et al FP(10) [causality, statistics and other effects];
Cagnache et al JGP(11)-a0912 [geometry];
Acharyya & Vaidya JHEP(10)-a1005 [accelerated observers];
Isidro et al AMP(11)-a1007 [commutator algebra];
Martinetti & Tomassini CMP(13)-a1110 [and spectral distance between coherent states],
a1205-proc [length and distance];
> 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(02)ht/01 [physical];
Alexanian et al JGP(02) [\(\mathbb C\)P2];
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];
Burić & Madore ht/04-conf [2D, review],
PLB(05) [2D, example];
Gromov a1002 [quantum analogs of constant-curvature spaces];
D'Andrea et al Sigma(14)-a1406 [from deformations of canonical commutation relations];
> s.a. classical particles; deformed minkowski
space; rindler space; schwarzschild spacetime.
@ Manifolds with boundary: Iochum & Levy JFA(10)-a1001;
Belishev & Demchenko JGP(14)-a1306 [recovering the manifold from boundary data].
@ Tensor fields / calculus: Dubois-Violette qa/95 [derivations, connections];
Dvoeglazov S&S(02)mp,
in(03)mp [derivatives];
Dimitrijević et al JPA(04) [κ-deformed euclidean space];
Vassilevich CQG(10) [tensor calculus];
> s.a. exterior calculus.
@ 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 CMP(09)-a0704;
Banica & Goswami CMP(10) [new examples of non-commutative spheres];
Murray & Govaerts PRD(11)-a1008,
Burić & Madore EPJC(14)-a1401 [spherically symmetric spaces].
@ Related topics:
Breslav & Zapatrin IJTP(00) [quantum/Greechie logic];
Díaz & Pariguán JPA(07)mp/06 [measures and path integration];
Carey et al a0901,
Schenkel & Uhlemann Sigma(13)-a1308 [Dirac operators];
Berest et al a1202 [non-commutative Poisson structures];
> s.a. loop group; principal fiber bundles;
diffeomorphisms.
References > s.a. affine connections;
C*-algebras; differential geometry
[fuzzy]; models of topology change.
@ Texts and reviews: Manin 91;
Connes 94;
Madore 95 [connection and curvature],
gq/99-ln;
Landi LNP-ht/97;
Várilly phy/97-ln;
Bigatti ht/98;
Madore 99;
Connes JMP(00)ht,
m.QA/00;
Martinetti ht/06-proc;
Kar 08 [pedagogical, strings and quantum field theory];
Petitot a1505-in [rev];
Majid in Bullett et al 17;
Connes a1910 [developments];
Lupercio NAMS-a2008 [short intro].
@ 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);
Kisil in(99)fa/97 [approaches];
Dimakis & Madore JMP(96) [differential calculi];
Jackiw & Pi PRL(02)ht/01 [coordinate changes];
Rennie & Várilly m.OA/06 [manifold reconstruction];
Chaichian et al JMP(08)ht/06 [Riemannian, framework];
Piacitelli AIP(09)-a0901;
Lord et al JGP(12) [Riemannian manifolds];
Martinetti et al RVMP(12)-a1201 [minimal length];
Barrett et al JPA(19)-a1902 [extracting information from the spectrum of the Dirac operator];
Huggett et al a2006 [points cannot be defined].
@ Almost commutative geometry: Madore RPMP(99)gq/97 [Poisson structure and curvature];
Jureit & Stephan JMP(05),
Jureit et al JMP(05) [classification];
Kuntner & Steinacker JGP(12) [semiclassical limit, metric-compatible Poisson structures];
Boeijink & van den Dungen JMP(14)-a1405.
@ Categorical approach: Bertozzini et al a0801-proc,
JPCS(12)-a1409;
Bertozzini a1412-proc
[relational quantum theory and emergent spacetime].
@ Spectral distance: Wallet RVMP(12)-a1112 [examples];
D'Andrea & Martinetti a1807 [dual formula];
> s.a. spectral geometry;
Mathematical Garden page.
@ Curvature: Madore CJP(97)gq/96 [and connection];
Dubois-Violette et al JMP(96);
Floricel et al JNCG-a1612 [Ricci curvature];
Fathizadeh & Khalkhali a1901-fs [recent developments].
@ Related topics: Dimakis & Müller-Hoissen JMP(99)gq/98 [discrete];
Lord mp/00-wd;
Martinetti PhD(01)mp [distances];
Ponge LMP(08) ["lower-dimensional" volumes];
Wagner PLMS(13)-a1108 [smooth localization method];
D'Andrea & Martinetti LMP(12)-a1203,
D'Andrea a1507-proc [Pythagoras' theorem];
Barrett & Glaser JPA(16)-a1510,
Glaser JPA(17)-a1612 [random non-commutative geometries, simulations, critical behavior];
Glaser & Stern a1912
[effect of a spectral cut-off on Riemannian manifolds];
Khalkhali & Pagliaroli a2006 [phase transition].
@ Generalizations: Vacaru mp/02-ch,
mp/02-ch [Finsler, with local anisotropy];
Kalyanapuram a1806.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 15 dec 2020