U 2 ■SU 2 ×U 1 INVARIANCE OF PRINCIPAL BUNDLES FOR MAGNETIC MONOPOLES AND ITS PHYSICAL IMPLICATIONS

Abstract

In this paper it is shown that the principal U1-bundles for magnetic monopoles—Hopf bundles S3/ZD (D being any integer) possess U2(?)SU2×U1 invariance which proserves the bundle structure.Starting from this invariance, we show that among the generators of U2(?)SU2×U1 transformations on the principal bundle, those of SU2 lead naturally to the angular momentum operators (including the extra term -Zeg r/r) for spatial rotation around the magnetic monopole, and that of U1 gives the charge operator.Furthermore, we deduce the spherical harmonics around the monopole which can be expressed in a form analogous to the angular momentum of a symmetric top and its wave functions, Hence, the principal bundle for a magnetic monopolc and its Us(?)SU2×U1 invariance not only characterize the spatial rotation around the monopole, but also give a description unifying the angular momentum operator and charge operator.Becaasc the subgroup SU2 is a double covering of the usual rotation group SO3 and possesses spinor representations, the results of this paper imply that the existence of magnetic monopoles may provide a reasonable explanation, from the global group-theoretical point of view, for the existence of particles with half-integer spin.