# Dirac’s magnetic monopole and the Kontsevich star product

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 ##IMG## [http://ej.iop.org/images/1751-81...] {$\hbar$}