Dirac's magnetic monopole and the Kontsevich star product

Understand

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.