Non-Commutative Theories in Physics  

In General > s.a. [non-commutative geometry]; lagrangian systems; quantum spacetime; symmetry breaking.
* 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; Finite-dimensional systems and quantum mechanics are simpler than field theory and provide useful toy models.
* 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.
* 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-in [rev]; Schucker ht/01-in, ht/01-in [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-a0906-in [and quantum gravity]; Banerjee et al FP(09) [overview].
@ Relativistic particles: Deriglazov ht/02, PLB(03)ht/02; Malik IJMPA(07)ht/05 [in electromagnetic field, symmetries]; Wohlgenannt ht/06-in [intro]; Balachandran et al JHEP(07)ht [discrete time and energy non-conservation]; Joseph PRD(09)-a0811; > s.a. quantum particles.
@ Spinors: Grimstrup et al EPJC(02)ht/01; Paschke & Sitarz LMP(06) [Dirac operators on non-commutative T2].
@ Spectral action: Chamseddine & Connes CMP(97)ht/96 [proposal]; Besnard JGP(07) [canonical quantization, example]; > s.a. non-commutative geometry.
@ Hamiltonian formalism: Kalau JGP(96); Hawkins CMP(97)gq/96; Lukierski ht/98-in [deformed phase space]; Gomis et al PRD(01)ht/00 [and path integral]; Giunashvili mp/02 [and phase space]; Malik MPLA(03) [and Lagrangian]; Li et al MPLA(05)ht/04 [oscillator with non-commutative phase space]; Cortese & García PLA(06) [eom and symplectic form]; Cuesta et al ht/06 [gauge theories with non-commutative phase space]; Cortese & García IJGMP(07)ht [variational formulation]; Li & Dulat a0708 [non-commutative phase space and spacetime symmetries].
@ Related topics: Gruppuso JPA(05) [classical mechanics]; Kochan Sigma(08)ht/06 [classical Lagrangian mechanics]; Pinzul & Stern NPB(08)-a0705 [gauging the star product].

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 gq/98, gq/99; Rovelli PRL(99)gq; Wess mp/99-in [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-a0709, PRD(08)-a0810; Giri a0802; Jing & Lin PLA(08) [new representation]; Scholtz et al JPA(09)-a0812 [general formalism and applications].
@ Quantum mechanics in non-commutative spacetime: Balachandran et al JHEP(04)ht [Moyal plane], JHEP(04)ht [cylinder]; Vaquera-Araujo & Lucio mp/05 [plane]; Calmet & Selvaggi PRD(06)ht; Wachter qp/07, qp/07, qp/07; Noui PRD(08)-a0807.
@ Different C: Morariu & Polychronakos NPB(01)ht [on torus]; Deriglazov PLB(02)ht [on S2].
@ Related topics: Dulat & Li a0802 [Landau problem]; Bemfica & Girotti PRD(09)-a0906 [as a gauge theory]; Bastos et al a0907 [Wigner measures]; Dias et al a0911 [deformation quantization].
> Related topics: see coherent states; crystals [particle in quasicrystal]; dissipation; scattering; time in quantum theory.

Other Theories and Topics > s.a. Burgers Equation; non-commutative field theories; fluids; random processes; velocity.
@ Astrophysics / cosmology: Tamaki et al PRD(02)gq/01 [-rays and uhe cosmic rays]; Romero & Vergara MPLA(03)ht [solar system]; Fukuma et al NPB(04) [cmb spectrum]; Pimentel & Mora GRG(05) [quantum cosmology]; Klammer & Steinacker PRL(09)-a0903; Bertolami & Zarro a0908 [stars and astrophysical objects]; > s.a. inflationary phenomenology, non-commutative gravity.
@ Integrable systems: Hamanaka & Toda PLA(03); Giachetta et al PLA(07)qp/06 [quantization].
@ Harmonic oscillators: Hatzinikitas & Smyrnakis JMP(02); Parmeggiani CMP(08) [spectrum, semiclassical].
@ Other systems: Chaichian et al EPJC(04)ht/02 [H atom].
@ Other phenomenology: Acatrinei MPLA(05)ht/03 [experimental effect]; Kauffman qp/03 [discrete physics]; Calmet EPJC(05)hp/04 [bounds]; Balachandran et al PRD(06)ht/05 [waves]; Colatto et al PRD(06)ht/05 [spin effects]; Abel et al JHEP(06) [vacuum birefringence]; Zhang et al PRD(08)-a0901 [electron momentum spectroscopy].
> Related topics: see modified lorentz symmetry, symplectic structures [non-commutative ]; uncertainty.

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