Non-Commutative Spacetime
blue bullet Description and properties of non-commutative spacetime; see also non-commutative gravity.

In General > s.a. non-commutative geometry [categorical approach]; quantum geometry; quantum spacetime; Spectral Triple.
* Rem: 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)-proc; Lizzi a0811-proc; Chamseddine & Connes CMP(10)-a0812 [introduction and overview]; Chamseddine a0901-proc; Chamseddine & Connes a1008-MG12; Zhang & Zhang a1108; Yang IJMPD(16)-a1610 [emergent spacetime and phenomenology].
@ General references: Rosen AP(62) [complex combinations of x and p]; Chamseddine & Fröhlich ht/93-proc; 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]; Agostini JMP(05)ht/05 [operators and integration]; Martinetti IJMPA(09)-a0904-proc [notion of line element]; Franco PhD(11)-a1108 [Lorentzian]; Bizi et al a1611 [and space and time dimensions of algebras]; Vilela IJTP(17)-a1710 [geometry].
@ Interpretation: Chaichian et al PLB(04)ht; Kosiński & Maślanka ht/04; Sakellariadou et al PRD(11)-a1106 [dissipation and quantization]; Bianchi & Rovelli PRD(11)-a1105.
@ 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]; Piacitelli Sigma(10)-a1004.
@ 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-ln; 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]; Sładkowski 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. quantum spacetime.
> Related topics: see linear connections; Hopf Algebra; matrix [models]; non-commutative gravity [singularities]; uncertainty; Very Special Relativity.

Special Cases, Other Structures and Spacetime Properties > s.a. deformed minkowski space; higher-dimensional gravity; kaluza-klein theories.
@ 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; Abreu & Neves a1108; Franco & Eckstein CQG(13)-a1212 [algebraic formulation], Sigma(14)-a1310 [almost-commutative geometries], a1409-in [rev]; Besnard & Bizi a1411 [disappearance of causality at small scales]; Franco & Wallet CM(16)-a1507 [on Moyal planes]; Besnard JPCS(15)-a1508 [isocone-based approach]; > s.a. non-commutative field theory.
@ 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).
@ Snyder proposal / spacetime: Snyder PR(47) [coordinates as operators]; Elias & Steele IJMPE(07)ht/06 [massless scalar propagator]; Guo et al FPC(07)ht/06, Guo ht/06-proc [and de Sitter invariance]; Romero & Zamora PLB(08)-a0802 [quantum of area]; Girelli Sigma(10)-a1009 [K-loop and Lie triple system]; Stern IJGMP(12) [particle dynamics and Poincaré invariance]; Lu & Stern NPB(12)-a1108, NPB(12)-a1110 [particle dynamics]; Gouba & Stern NPB(12)-a1111 [supersymmetric extension]; Valtancoli IJMPA(12)-a1209 [translational invariance and quantum field theory]; Ma & Zhao ChPC(14)-a1305 [some physical consequences]; Mignemi a1308 [deviations from special relativity]; Ivetić et al CQG(14) [curved, particle dynamics]; Mignemi & Strajn a1501 [quantum mechanics]; > s.a. non-commutative geometry and field theory; path integrals.
@ Other types of spacetimes: Tomassini & Viaggiu CQG(14)-a1308 [uncertainty relations for FLRW cosmologies]; Burić & Madore a1508 [de Sitter and FLRW spaces]; Fritz & Majid CQG(17)-a1611 [spherically symmetric spacetimes, semiclassical]; Ballesteros et al a1702-conf [(anti) de Sitter]; Burić et al a1709, Jurman a1710 [fuzzy de Sitter]; > s.a. gödel spacetimes.
@ Related topics: Masson JMP(96)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-conf [star product for Snyder's approach]; Perini & Tornetta RVMP(14)-a1211 [scale-covariant quantum spacetime]; Abreu et al a1501 [with variable non-commutativity parameter]; Anjana et al IJMPA(17)-a1704 [Hausdorff dimension]; > s.a. fractals [fractal spacetime]; types of distances.

Phenomenology > s.a. non-commutative geometry [including examples] and physics; quantum-gravity phenomenology [minimal length]; wormholes.
@ 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; Wang et al IJMPA(17)-a1702 [violation, superluminal electrons]; Much & Vergara a1704.
@ Particle propagation: Gamboa et al ht/01, Falomir et al PRD(02)ht [Aharonov-Bohm effect]; Amelino-Camelia et al PRD(03) [string-inspired]; > s.a. geodesics; non-commutative physics.
@ Constraints on non-commutativity: Akofor et al PRD(09) [from cmb]; Horvat & Trampetić PRD(09)-a0901 [from primordial nucleosynthesis], JHEP(11)-a1009 [from holography], PLB(12) [from the reheating phase after inflation]; Joby et al PRD(15)-a1412 [from Planck data]; Kobakhidze et al PRD(16)-a1607 [from GW150914].
@ Related topics: Vilela Mendes EPJC(05)ht/04; Hinchliffe et al IJMPA(04) [rev]; Martinis & Mikuta-Martinis gq/05 [near-horizon geometry]; Lukierski APPB(10)-a1012-conf [and deformed dynamical theories]; Saha & Gangopadhyay CQG(16)-a1508 [probe using gravitational-wave resonant-bar detectors].
> Related topics: see cmb polarization; Reference Frame; thermal radiation; tests of the equivalence principle; unimodular relativity; velocity.

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