Dirac's magnetic monopole and the Kontsevich star product

TL;DR

This paper explores different quantization methods for a charged particle in a magnetic monopole field, demonstrating their equivalence and connecting the magnetic Weyl product with Kontsevich's deformation quantization formula.

Contribution

It introduces invariant geometric quantization schemes for monopole systems and links the magnetic Weyl star product to Kontsevich's deformation quantization, providing explicit formulas and asymptotic expansions.

Findings

Both quantization approaches induce the same star product.

The magnetic Weyl product matches Kontsevich's formula for Poisson structures.

Explicit integral kernels and asymptotic expansions are derived.

Abstract

We examine relationships between various quantization schemes for an electrically charged particle in the field of a magnetic monopole. Quantization maps are defined in invariant geometrical terms, appropriate to the case of nontrivial topology, and are constructed for two operator representations. In the first setting, the quantum operators act on the Hilbert space of sections of a nontrivial complex line bundle associated with the Hopf bundle, whereas the second approach uses instead a quaternionic Hilbert module of sections of a trivial quaternionic line bundle. We show that these two quantizations are naturally related by a bundle morphism and, as a consequence, induce the same phase-space star product. We obtain explicit expressions for the integral kernels of star-products corresponding to various operator orderings and calculate their asymptotic expansions up to the third order in the Planck constant $\hbar$. We also show that the differential form of the magnetic Weyl product corresponding to the symmetric ordering agrees completely with the Kontsevich formula for deformation quantization of Poisson structures and can be represented by Kontsevich's graphs.