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 S (a circle). Since S 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 (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. To be published in Progress of Theoretical Physics. e-mail address : [email protected]