Non-Abelian monopoles break color. I. Classical mechanics

Abstract

Monopoles which are sources of non-Abelian magnetic flux are predicted by many models of grand unification. It has been argued elsewhere that a generic transformation of the "unbroken" symmetry group H cannot be globally implemented on such monopoles for reasons of topology. In this paper, we show that similar topological obstructions are encountered in the mechanics of a test particle in the field of these monopoles and that the transformations of H cannot all be globally implemented as canonical transformations. For the SU(5) model, if H is SU(3)_C×U(1)_em, a consequence is that color multiplets are not globally defined, while if H is SU(3)_C×SU(2)_WS×U(1)_Y, the same is the case for both color and electroweak multiplets. There are, however, several subgroups K_T, K_T',... of H which can be globally implemented, with the transformation laws of the observables differing from group to group in a novel way. For H=SU(3)_C×U(1)_em, a choice for K_T is SU(2)C×U(1)em, while for H=SU(3)_C×SU(2)_WS×U(1)_Y, a choice is SU(2)_C×U(1)×U(1)×U(1). The paper also develops the differential geometry of monopoles in a form convenient for computations.