Studies On Classical And Quantum Aspects Of Noncommutative Chern-Simons Mechanics

Chern-Simons mechanics is first studied by R. Jackiw, et.al. It describes a planer charged particle confined by a harmonic oscillator potential in the background of a perpendicular magnetic field. Previous studies show that this model can be generalized to a noncommutative plane. In this thesis, we shall present a new kind of noncommutative generalization. The difference between ours and previous studies is that the symplectic structures are intact when the magnetic field is coupled.We solve the classical equations of motion directly in the noncommutative plane. In order to get the spectra exactly, we map the noncommutative plane to a conventional one by means of Bopp shift. There is a reduced model which is obtained by turning off the mass term. We also study this reduced model from both classical and quantum aspects. We find that the solutions to the classical equations of motion for this reduced model can also be gained by turning off the mass term from the solutions to the classical equations of motion of the original model. It shows that the classical aspect has a continuous limit. However, we find that the quantum aspect does not have the same continuous limit. The spectra of the original model will be divergent when the mass term is turned off. We propose a reasonable scheme to regularize the spectra in the process of turning off the mass term. We employ the Dirac theory to verify that our regularization is correct.The contents of this thesis is as follows:In the next chapter, we shall review the noncommutative theories and Chern-Simons mechanics briefly and then in chapterⅡ, we shall review the work of R. Jackiw which introduce the Chern-Simons quantum mechanics firstly. In chapterⅢ, we shall review two noncommutative generalizations of Chern-Simons mechanics which introduce the magnetic field by means of modifying the Poisson brackets and minimal coupling respectively. ChaperⅣis a kind of noncommutative generalization which keep the original symplectic structures intact. We shall study this type of noncommutative Chern-Simons mechanics in details in this chaper. Further remarks and discussions will be given at the last chapter.