GAUGE FIELD, PARITY AND UNCERTAINTY RELATION OF QUANTUM-MECHANICS ON S(1)

We consider the uncertainty relation between position and momentum of a particle on S1 (a circle). Since S1 is compact, the uncertainty of position must be bounded. Consideration on the uncertainty of position demands delicate treatment. Recently Ohnuki and Kitakado have formulated quantum mechanics on S(D) (a D-dimensional sphere). Armed with their formulation, we examine this subject. We also consider parity and find a phenomenon similar to the spontaneous symmetry breaking. We discuss problems which we encounter when we attempt to formulate quantum mechanics on a general manifold.