Spectra of fermions in monopole fields—Exactly soluble models

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Abstract
We investigate the structure of the energy spectrum of an isospin-1/2 Dirac particle in the field of the SU(2) magnetic monopole of 't Hooft and Polyakov. We show that aside from the zero-energy mode, which is always present, there are at most a finite number of bound states. To clarify the interaction of the fermion with the various components of the monopole field, we consider two different extrapolations of the background field to limiting forms. The corresponding Dirac equations turn out to be exactly soluble. In the first limiting model, only the Higgs field is retained, and the Dirac equation is found to be equivalent to the nonrelativistic Coulomb problem. The second model is just the point monopole, and our problem is equivalent to a doublet of massive Dirac particles interacting with an Abelian magnetic monopole. This classical problem admits a simple treatment in the context of non-Abelian gauge theories; we present its solution in this formulation; we point out the hitherto unnoticed fact that the Hamiltonian is not self-adjoint on the customary domain of nonsingular wave functions and we study its self-adjoint extensions and bound states.

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