Group theoretical construction of planar noncommutative phase spaces

@article{Ngendakumana2013GroupTC,
  title={Group theoretical construction of planar noncommutative phase spaces},
  author={Ancille Ngendakumana and Joachim Nzotungicimpaye and Leonard Todjihounde},
  journal={Journal of Mathematical Physics},
  year={2013},
  volume={55},
  pages={013508}
}
Noncommutative phase spaces are generated and classified in the framework of centrally extended anisotropic planar kinematical Lie groups as well as in the framework of noncentrally abelian extended planar absolute time Lie groups. Through these constructions the coordinates of the phase spaces do not commute due to the presence of naturally introduced fields giving rise to minimal couplings. By symplectic realizations methods, physical interpretations of generators coming from the obtained… 
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References

SHOWING 1-10 OF 47 REFERENCES

Noncommutative Phase Spaces by Coadjoint Orbits Method

We introduce noncommutative phase spaces by minimal couplings (usual one, dual one and their mixing). We then realize some of them as coadjoint orbits of the aniso- tropic Newton{Hooke groups in two-…

PARA-GALILEAN VERSUS GALILEAN NONCOMMUTATIVE PHASE SPACES

This paper deals with the construction of noncommutative phase spaces as coadjoint orbits of noncentral extensions of Galilei and Para-Galilei groups in two-dimensional space. The noncommutativity is…

Galilei-Newton law by symplectic realizations

We classify all the coadjoint orbits of the central extension Ĝ of the one spatial dimensional Galilei group G. Taking them in support of the generic symplectic realizations of the Galilei group G,…

Nonrelativistic conformal groups and their dynamical realizations

Nonrelativistic conformal groups, indexed by l=N/2, are analyzed. Under the assumption that the "mass" parametrizing the central extension is nonvanishing the coadjoint orbits are classified and…

Exotic Galilean symmetry in the non-commutative plane and the Hall effect

Quantum mechanics in the non-commutative plane is shown to admit the 'exotic' symmetry of the doubly centrally extended Galilei group. When coupled to a planar magnetic field whose strength is the…

Clifford algebras and possible kinematics.

We review Bacry and Levy-Leblond's work on possible kinematics as applied to 2-dimensional spacetimes, as well as the nine types of 2-dimensional Cayley-Klein geo- metries, illustrating how the…

Integrability criterion for abelian extensions of Lie groups

We establish a criterion for when an abelian extension of infinite-dimensional Lie algebras integrates to a corresponding Lie group extension , where is connected, simply connected and for some…

The Non-commutative Landau Problem

Abstract The Landau problem is discussed in two similar but still different non-commutative frameworks. The “standard” one, where the coupling to the gauge field is achieved using Poisson brackets,…

Classical mechanics in non-commutative phase space *

In this paper the laws of motion of classical particles have been investigated in a non-commutative phase space. The corresponding non-commutative relations contain not only spatial non-commutativity…

Conformal mechanics in Newton-Hooke spacetime