Topology and Quantum States: the Electron-Monopole system

TL;DR

This paper explores the classical and quantum dynamics of the electron-monopole system, proposing a novel Hilbert space formulation using exterior differential forms to handle topologically non-trivial configuration spaces, and demonstrating its advantages.

Contribution

It introduces a new Hilbert space framework based on exterior differential forms for quantum systems on non-trivial topologies, and shows how to realize all irreducible representations for SU(2).

Findings

Hilbert spaces of exterior differential forms can model quantum states on topologically complex spaces

All unitary irreducible representations of SU(2) can be obtained within this framework

Scalar Dirac type operators can be constructed following Kähler's approach

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.