QUANTIZATION AND SUPERSELECTION SECTORS II: DIRAC MONOPOLE AND AHARONOV-BOHM EFFECT

Abstract

The quantization procedure of the preceding paper is applied to study two generic topological quantum effects, viz. the charge quantization induced by (abelian) magnetic monopoles, and the Aharonov-Bohm effect. Prior to these applications, a general procedure is given for reducing unitary representations of a locally compact G which are induced by nontrivial unitary representations of H⊂G. This involves the use of spherical trace functions, and is useful in the determination of the eigenfunctions of the Hamiltonian of the particle in a given superselection sector. Such Hamiltonians, implementing the time-evolution on the given abstract C*-algebra, are explicitly constructed and analyzed. The relevant quantum effects are found to be a consequence of the representation theory of the appropriate algebras of observables. In this way a group- and operator-theoretic elucidation of the mathematical structure of the given systems is attempted. This paper may be read independently of its predecessor.