Lagrangian and Hamiltonian descriptions of Yang-Mills particles

Abstract

A new Lagrangian $L$ is proposed for the description of a particle with a non-Abelian charge in interaction with a Yang-Mills field. The canonical quantization of $L$ is discussed. At the quantum level $L$ leads to both irreducible and reducible multiplets of the particle depending upon which of the parameters in $L$ are regarded as dynamical. The case which leads to the irreducible multiplet is the minimal non-Abelian generalization of the usual Lagrangian for a charged point particle in an electromagnetic field. Some of the Lagrangians proposed before for such systems are either special cases of ours or can be obtained from ours by simple modifications. Our formulation bears some resemblance to Dirac's theory of magnetic monopoles in the following respects: (1) Quantization is possible only if the values of certain parameters in $L$ are restricted to a certain discrete set, this is analogous to the Dirac quantization condition; (2) in certain cases, $L$ depends on external (nondynamical) directions in the internal-symmetry space. This is analogous to the dependence of the magnetic-monopole Lagrangian on the direction of the Dirac string.