Topology and Quantum States: the Electron-Monopole system

Abstract

This paper starts by describing the dynamics of the electron-monopole system at both classical and quantum level by a suitable reduction procedure. This suggests, in order to realise the space of states for quantum systems which are classically described on topologically non trivial configuration spaces, to consider Hilbert spaces of exterior differential forms. Among the advantages of this formulation, we present in the case of the group ${\rm SU}(2)$, how it is possible to obtain all unitary irreducible representations on such a Hilbert space, and how it is possible to write scalar Dirac type operators, following an idea by K\"ahler.