Special symmetries for the Utiyama Lagrangian with external Yang-Mills fields.

Abstract

This paper demonstrates how conditions for the existence of symmetries for particles interacting with external background gauge fields arise easily from the standard symmetry equations. The general phenomenon is described by the Lagrangian ${L}_{\mathrm{UA}}$ obtained from Utiyama's original Lagrangian ${L}_{U}$ by regarding the gauge fields A as a given set of fields. An infinitesimal transformation X+Y, consisting of certain combined infinitesimal coordinate and local gauge transformations, will be an infinitesimal symmetry of ${L}_{\mathrm{UA}}$ precisely when the Lie derivative (${\mathrm{L}}_{\mathrm{UA}}$) yields a Lagrangian with identically vanishing Euler-Lagrange equations. This Lie derivative is explicitly computed and is shown to contain the expressions derived recently by several papers in the literature, as well as a new expression with possible physical importance.