Magnetic-monopole interactions

We study the long-range interactions between an arbitrary number of finite-energy monopoles in non-Abelian gauge theories with spontaneous symmetry breakdown. The Higgs fields may belong to either the adjoint representation or to an arbitrary representation of the gauge group $G$ with residual symmetry group U(1). We use the conservation properties of the stress-energy tensor to calculate the instantaneous force on monopoles starting from given initial field configurations. We show that when the Higgs fields belong to the adjoint representation of $G$, the stress-energy tensor vanishes everywhere in the Prasad-Sommerfield limit leading to a "no-interaction" result for a system of monopoles (or antimonopoles, but not both). When the monopoles are widely separated, one may picture each of them as consisting of a "core" outside which (the exterior region) the Yang-Mills gauge potentials obey free-field Yang-Mills equations. We find exact solutions in the exterior region (exterior solutions) and show how they determine the desired long-range interactions through the stress-energy tensor. One is led to a very simple physical interpretation of the interactions as consisting of Coulomb-type attractive or repulsive forces due to magnetic charges and Newtonian or Yukawa-type attractive forces due to Higgs fields. We show how these forces differ when we have massive and massless Higgs fields. In the massless case, the Coulomb and Newtonian forces do not have the same strength in general. From this result and also the mass spectrum, we find that the conjectured symmetry between gauge particles and finite-energy monopoles is limited to the case when the Higgs fields belong to the adjoint representation.