Non-Commutative Geometry  

In General > s.a. manifolds / holonomy; quantum group; spectral geometry; 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.
> Spectral distance: see Mathematical Garden page.

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].
@ 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]; Wallet RVMP(12)-a1112 [examples of Connes spectral distance]; D'Andrea et al Sigma(14)-a1406 [from deformations of the 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; differentiable manifolds [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.
@ 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]; Vacaru mp/02-ch, mp/02-ch [Finsler, with local anisotropy]; 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].
@ 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].
@ Curvature: Madore CJP(97)gq/96 [and connection]; Dubois-Violette et al JMP(96); Floricel et al a1612 [Ricci curvature].
@ 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].


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