Quantum mechanics on nontrivial fiber bundles associated with a monopole

Scattering wave functions and Green's functions are found in the global space of the principal fiber bundle corresponding to the Dirac monopole. Hidden symmetries of the Dirac charge – monopole system are found, and also transformations that connect states relating to different topological charges $n\in\mathbb Z$. We show that the concept of spatial reflection does not exist when the physical states are defined on the bundle that is usually associated with the Dirac charge – monopole system. In other words, there does not exist an operator that lifts spatial reflection to the global space of such a bundle. A well-defined operator of dyon permutation is constructed on the two-dyon bundle. Its action on local sections can therefore be correctly defined. It is shown that the symmetric wave function defined on this bundle cannot be transformed into an antisymmetric wave function by means of a gauge transformation, in contrast to the well-known assertion first made in connection with the problem of dyon spin.