Infinite-dimensional representations of the rotation group and Dirac's monopole problem

TL;DR

This paper explores infinite-dimensional representations of the rotation group to analyze the Dirac monopole problem, proposing a generalized quantization rule that allows arbitrary magnetic charges and introduces a new quantum number called topological spin.

Contribution

It introduces a framework using infinite-dimensional representations in indefinite metric spaces to generalize the Dirac quantization condition and define the topological spin.

Findings

Arbitrary magnetic charges are permitted within this framework.

The Dirac quantization condition is replaced by a generalized rule.

A new quantum number, topological spin, is introduced.

Abstract

Within the context of infinite-dimensional representations of the rotation group the Dirac monopole problem is studied in details. Irreducible infinite-dimensional representations, being realized in the indefinite metric Hilbert space, are given by linear unbounded operators in infinite-dimensional topological spaces, supplied with a weak topology and associated weak convergence. We argue that an arbitrary magnetic charge is allowed, and the Dirac quantization condition can be replaced by a generalized quantization rule yielding a new quantum number, the so-called topological spin, which is related to the weight of the Dirac string.