Dirac particle in a magnetic field: Symmetries and their breaking by monopole singularities

Abstract

Some rules governing motion of a charged particle obeying the Dirac equation are assembled, including exact helicity conservation for scattering on an arbitrary finite magnetic field configuration. The singularity at the location of a magnetic monopole invalidates the derivation of the rules mentioned, leaving the Dirac Hamiltonian $H$ undefined for the lowest angular momentum state of the electron in the field of the pole. Specifying the behavior of $H$ under the discrete $P$, $T$, and $C$ symmetries determines it almost uniquely. One result is that $H$ may possess a bound state of zero energy, contrary to assertions in early papers on the subject. Zero-energy bound states which violate the superselection rule for electric charge are also studied, including one which is the point limit of a solution for a fermion multiplet interacting with a finite-energy soliton monopole. Implications of such a bound state for second quantization have been considered previously by others and are further analyzed here. The suggestion that monopoles may possess half-integral fermion number is shown to be unwarranted by present evidence.