Geometric momentum and angular momentum for charge-monopole system

TL;DR

This paper explores the geometric momentum and angular momentum in a charge-monopole system, revealing an $so(3,1)$ algebra structure and implications for phase shifts and quantization.

Contribution

It introduces the $so(3,1)$ algebra of geometric and orbital angular momentum for a charge-monopole system and links it to observable quantum effects.

Findings

The geometric momentum satisfies the $so(3,1)$ algebra with angular momentum.

The algebra implies the Aharonov-Bohm phase shift.

Charge and flux quantization are discussed in this framework.

Abstract

For a charge-monopole pair, though the definition of the orbital angular momentum is different from the usual one, and the transverse part of the momentum that includes the vector potential as an additive term turns out to be the so-called geometric momentum that is under intensive study recently. For the charge is constrained on the spherical surface with monopole at the origin, the commutation relations between all components of geometric momentum and the orbital angular momentum satisfy the $so(3,1)$ algebra. With construction of the geometrically infinitesimal displacement operator based on the geometric momentum, the $so(3,1)$ algebra implies the Aharonov-Bohm phase shift. The related problems such as charge and flux quantization are also addressed.