Chern-simons Mechanics In Noncommutative Phase Space

The concept of spatial noncommutativity was introduced morethan half a century. It receives much attention in the past decadeowing to the development of string theory. Recent researches findthat in order to maintain Bose-Einstein statistics in noncommutativespace，we should put both the coordinates and momenta to benoncommutative. In this thesis, we apply this latest discovery tostudy the Chern-Simons mechanics in noncommutative phase space.Our model describes a charged particle confined by a harmonicpotential in noncommutative phase space (both positions andmomenta are noncommutative) interacting with a perpendicularuniform magnetic field. We show that there are two specificparameters in this model. They will lead to two reduced modelswhen they tend to certain values. Interestingly, we find that thesolutions to the classical equations of motion of the full theory canreduce smoothly to the reduced models’. However, the energyspectrum of the full theory will be infinite when these two parameters tend to certain values. In order to get meaningful resultswe propose schemes for regularization. We resort Dirac theory toprove our regularization schemes are reasonable.The organization of this paper is as follows: We shall introducethe background of our researches in chapter1. And then in chapter2,we shall review the classical and quantum aspects of Chern-Simonsmechanics in commutative space respectively. Chapter3is the mainpart of our thesis. In this chapter, we shall study the Chern-Simonsmechanics in noncommutative phase space and its reduced theoriesin great details. The degeneracy of energy spectrum and theproperties of angular momentum are also studied. The last chapter isdevoted to conclusions and further discussions.