Fundamental algebra for quantum mechanics on <i>S</i> <i>D</i> and gauge potentials

Abstract

Quantum mechanics on SD is formulated. The algebra of (D+1)-dimensional Euclidean group is set up as fundamental commutation relation, which plays the role of the ordinary canonical algebra on the flat space. Compared to the standard procedure of using Dirac’s formalism for quantization of the constrained system, our method leads to more general results. Monopolelike structure appears inside the SD, which becomes visible when the space is cut by 3-dimensional hyperplane. Gauge potential for this structure is obtained. We also show, that no higher dimensional singularity can appear inside the SD.