Non-Commutative Field Theory  

In General > s.a. non-commutative geometry; energy-momentum tensor; lattice field theory; types of field theories.
* Idea: In principle, quantize the manifold underlying a field theory (spacetime or space) by replacing it with a non-commutative matrix model or a "fuzzy manifold"; In practice, replace products of fields in the action / Hamiltonian by star products, then integrate as usual.
* Motivation: Non-commutative spaces (spacetimes) naturally arise in some approaches to quantum gravity; Field theories on non-commutative spacetimes come with natural regularization parameters, while preserving their symmetries and topological features and altogether overcoming the fermion-doubling problem.
* Particle physics: A nice result is that, once fermions are fixed, there is no arbitrariness in the Higgs sector; An issue is that models generally suffer from a manifestly non-Wilsonian coupling of infrared and ultraviolet degrees of freedom known as the "IR/UV problem".
* Result: Free κ-Minkowski space field theory is equivalent to a relativistically invariant, non local, free field theory on Minkowski spacetime.
> Related topics: see algebraic quantum field theory; anomalies; boundaries in field theory; CPT symmetry; fock space; twistors.

Types of Theories > s.a. gauge theories [including standard model]; GUTs; non-commutative physics; supersymmetry and supersymmetric theories.
* Different frameworks: One can use different star products, for example the Moyal and Wick-Voros (or normally ordered) ones.
@ Scalar fields: Gubser & Sondhi NPB(01) [phase structure]; Moffat PLB(01)ht/00; Habara PTP(02)ht/01; Bietenholz et al NPPS(04)ht/03, APPB(03)ht-conf [λφ4, numerical]; Bertolami & Guisado PRD(03) [coupled to gravity]; Daszkiewicz et al IJMPA(05) [and DRS]; Steinacker JHEP(05) [non-perturbative], ht/05-conf [eigenvalue distribution]; Grosse & Steinacker JHEP(06)ht [4D self-dual φ3]; Panero JHEP(07) [numerical]; Freidel et al IJMPA(08)-a0706 [free scalar in κ-Minkowski]; Galluccio et al a0807, a0807, PRD(08)-a0810 [with Wick-Voros product]; Bertolami & Zarro PLB(09)-a0812 [coupled to gravity, stability conditions]; Balachandran et al JHEP(11); Bietenholz et al JPCS(15)-a1402; Rea & Sämann JHEP(15)-a1507 [scalar field theory on the fuzzy disc, phase diagram].
@ Scalar fields, Snyder spacetime: Battisti & Meljanac PRD(10)-a1003; Girelli & Livine JHEP(11)-a1004; > s.a. modified quantum field theories..
@ Fermion fields: Gracia-Bondía et al PLB(98)ht/97, Balachandran et al MPLA(00) [fermion doubling]; Bourouaine & Benslama MPLA(05)ht [Dirac, and gravity], JPA(05) [in electromagnetic field]; Bertolami & Queiroz PLA(11)-a1105 [phase-space non-commutativity]; Verch APP-a1106-proc [Dirac field on Moyal-Minkowski spacetime]; Williams & Scholtz a1512 [manifestly Lorentz covariant, interacting and non-commutative Dirac equation].
@ In curved non-commutative spacetime: Schenkel & Uhlemann Sigma(03)-a1003; Jafari a1011; Schenkel PoS-a1101.
@ String theory: Seiberg & Witten JHEP(99)ht; Witten CQG(00); Tezuka MSc-ht/01; Barbosa JHEP(03)ht [interpretation]; Wang ht/05 [free, bosonic]; > s.a. String Field Theory.
@ Non-commutative target space: Balachandran et al PRD(08)-a0706 [scalar field], Sigma(10)-a1003.
@ Other: Jack & Jones PLB(01) [ultraviolet-finite]; Heckman & Verlinde NPB(15)-a1401 [covariant non-commutative deformation of 4D cft]; Zois a1401-conf [non-commutative topological quantum field theory and non-commutative Floer homology]; > s.a. 3D quantum gravity; Born-Infeld Theory; gravity; Gross-Neveu; Wess-Zumino Theory.
> Related topics: see dirac procedure; Haag's Theorem; minkowski space [deformed]; particle statistics; spin-statistics theorem; statistical mechanics.

References > s.a. path integrals; Schwinger-Dyson Equation; non-commutative physics [Hamiltonian, time].
@ Intros, reviews: Kerner LNP(00)mp; Douglas & Nekrasov RMP(01)ht; Szabo PRP(03)ht/01-ln; Ydri PhD(01)ht; Gracia-Bondía AdP(02)ht; Dito & Sternheimer in(02)m.QA [history]; Girotti AJP(04)ht/03-ln; Schaposnik ht/04-ln [including solitons and instantons]; Wulkenhaar JGP(06); Bal & Qureshi Sigma(06)ht-proc, Akofor et al IJMPA(08)-a0803-ln [fuzzy physics and quantum field theory on Groenwald-Moyal plane]; Doplicher JPCS(06)ht; Rivelles JPCS(11)-a1101.
@ Unitarity: Gomis & Mehen NPB(00)ht; Bahns et al PLB(02)ht.
@ Hamiltonian, symplectic formalism: Neves et al JPA(04)ht/03, PRD(04)ht/03; Vassilevich ht/04; Abreu et al IJMPA(06)ht/04.
@ And causality: Greenberg PRD(06)ht/05, Soloviev PRD(08)-a0802 [failure of microcausality]; Haque & Joglekar JPA(08)ht/07; Balachandran et al a0905-conf [on the Groenewold-Moyal plane]; Soloviev TMP(10)-a1012 [locality and causality].
@ Renormalizable: Bieliavsky et al JNCG(09) [possibly finite]; Grosse & Wulkenhaar GRG(11); > s.a. renormalization.
@ Properties of quantum field theories: Álvarez-Gaumé & Vázquez-Mozo NPB(03)ht; Bahns FdP(04)ht-conf [UV]; Smailagic & Spallucci JPA(04) [Lorentz, unitarity, UV]; Kobayashi & Sasaki IJMPA(05) [supersymmetric interpretation]; Panero Sigma(06)ht-proc [rev]; Chaichian et al ht/06 [theorems, rev]; Gangopadhyay PhD(08)-a0806; Saxell PLB(08) [Lorentz-invariant, non-causality]; Bahns a1012 [IR/UV mixing problem]; van Suijlekom PLB(12)-a1204 [almost-commutative geometries, renormalizability].
@ Other formal aspects: Savvidy in(03)ht/02 [new type]; Bozkaya et al EPJC(03)ht/02 [amplitudes and path integrals]; Namsrai IJTP(03); Chaichian et al JMP(11)ht/04 [Wightman functions]; Paschke & Verch CQG(04)gq [covariant quantum field theory over spectral geometries]; Mandanici & Marcianò JHEP(04) [Heisenberg evolution]; Bahns et al PRD(05) [Wick products]; Gonera et al PLB(05)ht [deformed Poincaré symmetry]; Govindarajan et al MPLA(06) [regularization]; Freidel et al PLB(07)ht [equivalence to non-local field theory on Minkowski space]; Kersting & Yan MPLA(08)-a0901 [IR/UV problem and coupling to gravity]; Balachandran et al PRD(10)-a0910 [inequivalence of approaches]; Cortese & García IJMPA(10)-a1005 [Poincaré symmetry]; Akofor PhD-a1012 [symmetries, on the Moyal plane]; Basu et al JPA(11)-a1101 [relationship between the Moyal and Voros products]; Lukierski & Woronowicz JPA(12)-a1105 [braided tensor product and covariance]; > s.a. quantum field theory techniques [worldline approach].
@ Phenomenology: Mariz et al PRD(07) [dispersion relations]; Kurkov et al PLB(14)-a1312 [high-energy bosons do not propagate]; Vassilevich JPCS(16)-a1510 [bosonic fields at very high energies]; > s.a. cosmological consequences.
@ Spacetime symmetries: Iorio & Sykora IJMPA(02)ht/01 [gauge theories]; Szabo CQG(06)ht [and strings], AIP(07)ht [and renormalization].
@ Special backgrounds: Nakayama & Shimono PTP(04)ht [S4]; Nazaryan & Carlson PRD(05) [non-commutative superspace].
@ Quantization: Amorim & Barcelos-Neto JPA(01)ht, Acatrinei PRD(03)ht/02 [scalar]; Carmona et al JHEP(03); Abe IJMPA(07)ht/06 [non-commutative quantization]; Chaichian et al JHEP(08)-a0706 [space of test functions]; Daszkiewicz et al PRD(08)-a0708; Fiore JPA(10)-a0811 [with twisted symmetries].
@ Perturbative effects: Minwalla et al JHEP(00)ht/99; Kossow PRD(08)ht/06; Samary et al EPJC(14)-a1406 [pair production of Dirac particles].
@ At finite temperature: Strelchenko & Vassilevich PRD(07)-a0705; Fosco & Silva JHEP(08)-a0710 [2+1 scalar].
@ Braided quantum field theory: Oeckl CMP(01)ht/99; Sasai & Sasakura PTP(07)-a0704 [Hopf algebra symmetries and Ward-Takahashi identities].
@ Related topics: Grosse et al IJTP(96); Kempf JMP(97) [non-zero minimal uncertainties]; Cho et al IJMPD(00)ht/99 [propagator]; Ydri PRD(01) [as a regulator]; Chaichian et al NPB(01)ht [non-trivial topology]; Amelino-Camelia et al ht/02-conf; Kowalski-Glikman & Nowak PLB(02)ht, IJMPD(03)ht/02 [and DSR]; Barcelos-Neto ht/02 [in curved spacetime]; Pinzul & Stern NPB(05)ht [procedure for corrections]; Soloviev TMP(06) [axiomatic]; Bu et al PRD(06) [from twisted Fock space]; Aschieri et al PRD(08)-a0708 [from Drinfeld twist].
@ Particle physics: Chamseddine & Fröhlich PRD(94)ht/93 [SO(10)], PLB(93)ht [Higgs and top masses], ht/93-conf [rev]; Connes CMP(96)ht; Schücker ht/97-ln; Stephan JMP(07)ht/06 [massive neutrinos]; van den Dungen & van Suijlekom RVMP(12)-a1204 [in almost commutative spacetime, for physicists]; Gargiulo et al EPJC(14)-a1305 [algebra doubling and neutrino mixing].

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