The O(3,1) symmetry problem of the charge–monopole interaction

Abstract

The question of whether there exists a smooth, time-independent conserved vector observable in the classical mechanics of the charge–monopole interaction that spans an O(3,1) Lie algebra together with the angular momentum or not is examined. It turns out that any candidate for such dynamical symmetry algebra has hard singularities at that part of the phase space that corresponds to charge–monopole collisions. In the course of the investigation we use the transformation of Boulware et al. [D. G. Boulware, L. S. Brown, R. N. Cahn, S. D. Ellis, and C. Lee, Phys. Rev. D 14, 2708 (1976)] relating the charge–monopole system to a point mass moving in an inverse square potential. This transformation is shown to be a complete isomorphism between the scattering parts of the related Hamiltonian systems; its global behavior is described in terms of an U(1) principal fiber bundle of nontrivial topology. Several remarks on the symmetries of various monopole problems are made, e.g., the most general O(4) symmetry algebra is given for a special form of the charge–dyon interaction.