Weyl correspondence for a charged particle in the field of a magnetic monopole

We construct a generalized Weyl correspondence for an electrically charged particle in the field of the Dirac magnetic monopole. Our starting points are a global Lagrangian description of this system as a constrained system with $U(1)$ gauge symmetry given in terms of the fiber bundle theory and a reduction of the presymplectic structure arising on the constraint surface. In contrast to the recently proposed quantization scheme based on using a quaternionic Hilbert module, the quantum operators corresponding to classical observables in our construction act in the complex Hilbert space of $U(1)$-equivariant functions introduced by Greub and Petry. These functions are defined on the total space of a fiber bundle that is topologically equivalent to the Hopf fibration.