Non-Commutative Theories in Physics  

In General > s.a. non-commutative geometry / lagrangian systems; quantum spacetime; symmetry breaking; symplectic structures.
* Motivation: The motivation can be traced to the quantum gravity idea that space(time) points become fuzzed out at the smallest scales, and the fact that one can regularize gauge theories by making the underlying manifold fuzzy; In practice, this tends to work perturbatively in terms of formal (possibly non-converging) power series in Planck's constant \(\hbar\), in an infinitesimal neighborhood of classical physics.
* History: The first appearance of non-commutative geometry in physics was in Witten's 1986 paper on string field theory; 1990, Connes' approach to the standard model; Finite-dimensional systems and quantum mechanics, which are simpler than field theory, have been studied as useful toy models.
* Remark: They are non-local, and can be considered examples of infinite-order Lagrangian theories.
@ Intros and general references: Chamseddine in(95); Dimakis & Müller-Hoissen phy/97 [intro]; Bigatti CQG(00)ht [intro]; Castellani CQG(00)ht-ln [rev]; Schücker ht/01-conf, LNP(05)ht/01 [forces]; Kauffman NJP(04)qp [origin of gauge theory, quantum mechanics]; Rosenbaum et al ht/06/CM, JPA(07) [from symplectic structure and Dirac procedure]; Balachandran et al 07; Szabo GRG(10)-a0906-conf [and quantum gravity]; Banerjee et al FP(09) [overview]; Samanta PhD(08)-a1006 [and deformed symmetries]; Bertozzini et al a0801-proc [rev]; Blumenhagen FdP(14)-a1403-proc [and string theory, pedagogical].
@ Relativistic particles: Deriglazov ht/02, PLB(03)ht/02; Malik IJMPA(07)ht/05 [in electromagnetic field, symmetries]; Wohlgenannt UJP(10)ht/06-talk [intro]; Balachandran et al JHEP(07)ht [discrete time, energy non-conservation]; Joseph PRD(09)-a0811; Abreu et al JHEP(11)-a1011 [curved spaces]; Falomir et al PRD(12)-a1208 [3D]; Pramanik et al AP(14)-a1212, Balasin et al Sigma(14)-a1403 [charged]; Abreu & Godinho a1701 [gauge-invariant actions]; > s.a. geodesics [quantum corrections]; quantum particles.
@ Spectral action: Chamseddine & Connes CMP(97)ht/96 [proposal]; Besnard JGP(07) [canonical quantization, example]; Iochum JPCS(08)-a1512 [example]; Chamseddine & Connes CMP(09); Iochum in-a1712 [fundamentals]; > s.a. inflationary universe; non-commutative geometry; non-commutative gravity; Spectral Dimension; higher-order gravity.
@ Hamiltonian formalism: Kalau JGP(96); Hawkins CMP(97)gq/96; Lukierski ht/98-proc [deformed phase space]; Gomis et al PRD(01)ht/00 [and path integral]; Malik MPLA(03) [and Lagrangian]; Cortese & García PLA(06) [equations of motion and symplectic form]; Cortese & García IJGMP(07)ht [variational formulation]; Burić & Madore GRG(11)-a1110; Sánchez-Santos & Vergara PLA(14) [and higher-order]; Valtancoli a1502 [relation between Lagrangian and Hamiltonian].
@ Related topics: Gruppuso JPA(05) [classical mechanics]; Kochan Sigma(08)ht/06 [classical Lagrangian mechanics]; Pinzul & Stern NPB(08)-a0705 [gauging the star product]; Acharyya & Vaidya JHEP(10)-a1005 [accelerated observers]; Castorina et al a1806 [and stability of the system].

Quantum Mechanics > s.a. deformation quantization; path integrals and specific theories; thermodynamic systems; wigner functions.
* Quantization: 1995, One does not know how to quantize non-commutative models directly, but each one can be rewritten as an equivalent field theory in the ordinary sense, which can be quantized.
* And quantum mechanics: It is natural to introduce a non-commutative structure in phase space, but one can do it in configuration space as well; A deformed Heisenberg algebra gives rise to discrete position and momentum spectra, → lattice structure.
@ General references: Madore PLB(91), AP(92); Dimakis & Müller-Hoissen JPA(92) [non-commutative symplectic geometry]; Vilela Mendes PLA(96); Adler ht/96-in; Heller & Sasin AIP(98)gq, gq/99; Rovelli PRL(99)gq; Wess mp/99-conf [Heisenberg algebra]; Nair & Polychronakos PLB(01)ht/00; Acatrinei JHEP(01) [path integral]; Espinosa & Gaete ht/02 [choice of H]; Barbosa & Pinto PRD(04)ht/03 [Bohmian]; Djemai & Smail CTP(04)ht/03; Girotti AJP(04)may; Riccardi JPA(06)ht/05 [observables]; Muthukumar JHEP(07)ht/06; Bastos et al JMP(08)ht/06 [phase-space formulation]; Dulat & Li MPLA(06) [commutator anomaly]; Bemfica & Girotti BJP(08)-a0709, PRD(08)-a0810; Giri IJMPA(09)-a0802; Jing & Lin PLA(08) [new representation]; Scholtz et al JPA(09)-a0812 [general formalism and applications]; Heydari IJTP(11)-a1007 [entangled quantum states]; Sinha et al JPA(12)-a1108 [3D space, and rotational symmetry]; Kupriyanov JMP(13)-a1204 [with non-commutative coordinates]; Alavi & Abbaspour JPA(14); Nakamura a1402 [canonical structure]; Gouba IJMPA(16)-a1603 [four formulations: canonical, path-integral, Weyl-Wigner and systematic]; Santos et al a1802 [non-Hermitian, phase-space formalism].
@ Different configuration space: Morariu & Polychronakos NPB(01)ht [on a torus]; Deriglazov PLB(02)ht [on S2]; > s.a. quantum theory on generalized spaces.
@ Specific systems: Castello-Branco & Martins JMP(10)-a0803 [free-falling particle in a gravitational field]; Gáliková & Prešnajder JMP(13)-a1112, a1302 [Coulomb problem]; > s.a. Landau Model.
@ Related topics: Bemfica & Girotti PRD(09)-a0906 [as a gauge theory]; Bastos et al CMP(10)-a0907 [Wigner measures]; Dias et al JMP(10)-a0911 [deformation quantization]; Rohwer et al JPA(10)-a1004 [spatial extent and structure]; Salminen & Tureanu PRD(11)-a1101 [non-commutative time in quantum field theory]; Ruuge & Van Oystaeyen CMP(11) ["non-commutative Planck constant"]; Martina IJGMP(12) [using the Lagrange-Souriau 2-form formalism]; Benatti & Gouba JMP(13)-a1302, OSID(15)-a1409 [classical limit on a non-commutative configuration space]; Santos a1906 [thermal diffusion of Gaussian states].
> Related topics: see coherent states; composite systems; crystals [particle in quasicrystal]; dissipation; scattering; time in quantum theory.

Other Theories and Phenomenology > s.a. lattice field theory; non-commutative field theory; non-commutative gravity [including strings]; fluids.
@ Astrophysics: Tamaki et al PRD(02)gq/01 [γ-rays and uhe cosmic rays]; Romero & Vergara MPLA(03)ht [solar system]; Fukuma et al NPB(04) [cmb spectrum]; Klammer & Steinacker PRL(09)-a0903; Bertolami & Zarro PRD(10)-a0908, JPCS(10)-a1002 [stars and astrophysical objects]; Nelson et al PRL(10)-a1005 [pulsar timing]; Haghighat & Stern a1008 [and pulsar redshifts]; Rahaman et al GRG(12)-a1011 [galaxy rotation curves]; Lambiase et al JCAP(13)-a1302 [constraints from Gravity Probe B data]; Harikumar & Zuhair a1707; > s.a. inflationary phenomenology; non-commutative cosmology.
@ Integrable systems: Hamanaka & Toda PLA(03); Giachetta et al PLA(07)qp/06 [quantization].
@ Harmonic oscillators: Hatzinikitas & Smyrnakis JMP(02); Streklas PhyA(07); Parmeggiani CMP(08) [spectrum, semiclassical]; Ghosh & Roy PLB(12)-a1110 ["stringy" coherent states]; Vega JMP(14)-a1304.
@ Spinors: Grimstrup et al EPJC(02)ht/01; Paschke & Sitarz LMP(06) [Dirac operators on non-commutative T2]; Gomes et al PRD(10); Burić et al CQG(15)-a1502 [in curved non-commutative space]; Nguyen & Schenkel a2004 [non-commutative hypersurface Dirac operators].
@ H atom: Chaichian et al EPJC(04)ht/02; Moumni et al ARP-a1106; Kupriyanov JPA(13)-a1209 [curved space]; Zaim & Delenda JPCS(13)-a1212 [relativistic]; Ghoderao & Ramadevi a1912.
@ Wave equations: Balachandran et al PRD(06)ht/05; Majid CMP(12) [wave operators]; Schroers & Wilhelm Sigma(14)-a1402 [2+1].
@ Other phenomenology: Acatrinei MPLA(05)ht/03 [experimental effect]; Kauffman qp/03 [discrete physics]; Calmet EPJC(05)hp/04 [bounds]; Colatto et al PRD(06)ht/05 [spin effects]; Zhang et al PRD(08)-a0901 [electron momentum spectroscopy]; Singh et al FP(10) [muon decay, non-symmetric gravity]; Saitou et al PTEP(14)-a1408 [and hydrodynamics of granular materials]; Liang a1702 [proposal based on the Aharonov-Bohm effect]; Lizzi a1805-proc [particle physics]; Gnatenko & Tkachuk a1808; Yu & Ma a1810 [fermion generations]; Karimabadi et al a1812 [gravity tests]; Gnatenko & Tkachuk a1905 [upper bound on momentum non-commutativity parameter from Mercury perihelion precession]; Harikumar et al a1908 [Casimir effect]; Ghegal a2010 [Bhabha scattering]; > s.a. non-commutative spacetime and gauge theories; stochastic processes.
> Related topics: see brownian motion; Burgers Equation; modified lorentz symmetry; gravitomagnetism [effective non-commutative structure]; interactions; pauli exclusion principle; random walk; time; Topological Insulators; uncertainty relations; velocity.

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