Group Theoretical Construction of Planar Noncommutative Systems

TL;DR

This thesis constructs and classifies planar noncommutative phase spaces using group theory, revealing new structures and couplings, and linking them to deformations of Poisson brackets.

Contribution

It introduces a systematic group-theoretical framework for noncommutative phase spaces, including new couplings and their physical interpretations, extending previous classifications.

Findings

Noncommutative structures arise from central and noncentral extensions.

New minimal couplings involving position, momentum, and dual potentials are introduced.

Noncommutative parameters are constant fields from group extensions.

Abstract

In this thesis, we construct and classify planar noncommutative phase spaces by the coadjoint orbit method on the anisotropic and absolute time kinematical groups. We show that noncommutative symplectic structures can be generated in the framework of centrally extended anisotropic kinematical groups as well as in the framework of noncentrally abelian extended absolute time kinematical groups. However, noncommutative phase spaces realized with noncentral abelian extensions of the kinematical groups are algebraically more general than those constructed on their central extensions. As the coadjoint orbit construction has not been carried through some of these planar kinematical groups before, physical interpretations of new generators of those extended structures are given. Furthermore, in all the cases discussed here, the noncommutativity is measured by naturally introduced fields, each corresponding to a minimal coupling. This approach allows to not only construct directly a dynamical system when of course the symmetry group is known but also permits to eliminate the non minimal couplings in that system. Hence, we show also that the planar noncommutative phase spaces arise naturally by introducing minimal coupling. We introduce here new kinds of couplings. A coupling of position with a dual potential and a mixing model (that is minimal coupling of the momentum with a magnetic potential and of position with a dual potential). Finally we show that this group theoretical discussion can be recovered by a linear deformation of the Poisson bracket. The reason why linear deformation of Poisson bracket is required here is that the noncommutative parameters (which are fields) are constant (they are coming from central and noncentral abelian extensions of kinematical groups).