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.
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 8
nov 2009