CIRCLE-MINUS-STRUCTURES IN QUANTUM-THEORY IN VIEW OF GEOMETRIC-QUANTIZATION

Because the various topological effects (theta-structures) such as the Aharonov-Bohm effect, the anyon system, and non-Abelian statistics are pure quantum effects and should emerge naturally in a quantization procedure, we systematically discuss a general quantization scheme in a geometric formalism where wave-functions are smooth sections of some vector bundles over configuration space. Following ideas of L. Schulman, M. Laidlaw, J. S. Dowker, and others, we choose those vector bundles to be the associated bundles of the universal covering space of configuration space. The theta-structures are shown to result from the fact that various vector bundles can be built over the universal covering space, which are labeled by the nonequivalent irreducible unitary representations of the fundamental group of configuration space. A flat connection description of theta-structures is also possible, owing to Milnor's theory.