Relativistic spin 1/2 particle in the total space of the principal $U(1)$ monopole bundle

The quantum dynamics of a relativistic spin 1/2 particle moving in the total space $P$ of the principal $U(1)$ bundle associated with the Dirac monopole is described. It is completely characterized in terms of an additional $U(1)$ bundle $P^*$ that is intimately assoicated with the scattering problem. A representation in which the dynamics takes its simplest form is obtained. The lift of the Lorentz group into the total space is constructed.