Star-product in the presence of a monopole

TL;DR

This paper develops a new star-product for quantum particles in a magnetic monopole field, using quaternionic Hilbert spaces, which is associative only for quantized monopole charges, aligning with Dirac's quantization.

Contribution

It introduces a self-dual quantization scheme with a quaternionic Hilbert space to define a star-product in monopole backgrounds, ensuring associativity for quantized charges.

Findings

The star-product reproduces the classical Poisson structure at first order in hbar.

Associativity of the product holds only for quantized monopole charges.

The method extends the deformation quantization framework to monopole systems.

Abstract

We present a deformed star-product for a particle in the presence of a magnetic monopole. The product is obtained within a self-dual quantization-dequantization scheme, with the correspondence between classical observables and operators defined with the help of a quaternionic Hilbert space, following work by Emch and Jadczyk. The resulting product is well defined for a large class of complex functions and reproduces (at first order in hbar) the Poisson structure of the particle in the monopole field. The product is associative only for quantized monopole charges, thus incorporating Dirac's quantization requirement.