A Lagrangian consisting of an Abelian Chern-Simons term and N identical point particle sources is known to lead to fractional statistics for the sources. In this paper, we investigate the non-Abelian generalization of this system with special emphasis on source statistics. All solutions for the Yang-Mills potential in the presence of identical or nonidentical sources are found. For two or more sources, they fall in many gauge inequivalent classes whereas in the Abelian problem, there is only one such class. An effective Lagrangian for N sources is found for each of these solutions. The quantum mechanics and statistics of the sources are sensitive to the potential leading to the effective Lagrangian. There is for instance, a class of solutions for identical sources which are not invariant under exchange of sources. For these solutions, the identity of the sources obliges us to consider such a potential and all its exchange transforms at the same time, and to introduce a Hilbert space of states which is the direct sum of the Hilbert spaces associated with each of these potentials. There are also exchange invariant potentials for identical sources. For SU(3) and N = 3, all exchange invariant potentials are shown to lead to statistics defined by S3 representations. The nature of statistics for SU (M) for higher M as also the creation of intrinsic spin by self interaction are briefly considered.
Lorentz-Marcinkiewicz property of direct sum of operators
