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 p osition must be bounded. Consideration on the uncertainty of position demands delicate treatment. Recently Ohnuki and Kitakado have formulat ed quantum mechanics on S(D) (a D-dimensional sphere). Armed with thei r formulation, we examine this subject. We also consider parity and fi nd a phenomenon similar to the spontaneous symmetry breaking. We discu ss problems which we encounter when we attempt to formulate quantum me chanics on a general manifold.