Fermion-monopole system reexamined

The problem of a Dirac particle in a fixed Abelian monopole field is reexamined. We find that the boundary conditions adopted by Kazama, Yang, and Goldhaber can be generalized. When the generalization is made, we are led to the existence of $\theta$ vacua. For massless fermions, chiral symmetry is spontaneously broken. A change in $\theta$ is equivalent to a chiral rotation, and the physics is independent of $\theta$. For massive fermions, $\mathrm{CP}$ invariance is broken except for $\theta =0, \pi$. The vacuum charge for a unit pole obeys the Witten formula $Q=-\frac{e\theta }{2\pi }$ and the monopole becomes a dyon. The results for the two cases are related by an analog of Levinson's theorem. We also note the connection of our results with fractional quantum numbers on solitons, and with the $\eta$ invariant of Atiyah, Patodi, and Singer.