Notes on non-commutative Chern–Simons quantum mechanics

Abstract

We investigate the classical and quantum aspects of non-commutative topological (Chern–Simons) mechanics. We introduce the magnetic field by the minimal substitution in a way which preserves the original symplectic structures. We find that the classical aspect, say, the solutions to the equations of motion, converges to the reduced theory which is obtained by turning off the mass term smoothly. However, the quantum aspect, i.e., the spectra and the angular momenta, does not have such continuous limits. The spectra will become divergent when the mass term is turned off. A scheme is proposed to regularize the spectra so as to get a finite result. In order to verify our regularization scheme, we resort to Dirac theory. We find that there are two constraints during the reduction from the full theory to the reduced one which alter the symplectic structures. The eigenvalues of angular momenta also have no continuous limits, and this situation is similar to the one which has been studied some years ago. The possibility of taking an additional limit is also discussed.