A technique for the analysis of the invariance identities of classical gauge field theory by means of functional equations.

Abstract

In order to obtain the equations of motion for a particle in a classical gauge field, a variational principle is considered. The theory is general in that the structural group is an arbitrary r-dimensional Lie group and the base space is an arbitrary n-dimensional psuedo-Riemannian manifold. An n + r dimensional principal fiber bundle is constructed in order to introduce the usual gauge potentials and field strengths. In addition, a set of r quantities (called "coupling parameters") which transform as the components of an adjoint type (0,1) object and also depend upon the parameter of the particle's trajectory are constructed. The gauge potentials and coupling parameters are evaluated on the identity section of the principle bundle, and the Lagrangian is assumed to be a C³ scalar function of these and of the components of the metric tensor and tangent vector on the base space. The Lagrangian is not gauge-invariant, but it is stipulated that when the arguments of the Euler-Lagrange vector (evaluated on the identity section) are replaced by their counterparts (which may be evaluated on an arbitrary section) the resulting vector must be gauge-invariant. A novel application of methods from the theory of functional equations is applied together with standard techniques inherent in the theory of differential equations to show that the arguments of the Lagrangian must occur together in certain prescribed combinations. The invariance postulates uniquely determine the Lagrangian in terms of its arguments other than the coupling parameters and r functions of the coupling parameters. The Lagrangian is shown to separate into a free-field term and an interaction term, and the functions of the coupling parameters are found to be the components of an adjoint type (0,1) quantity whose adjoint absolute derivative vanishes. This agrees with the equations of certain approaches to the Yang-Mills theory for isotopic spin particles.¹ Standard initial conditions are shown to determine a unique (local) solution to the derived equations of motion. ftn¹ The equations have the same formal structure as systems obtained in the classical limit of quantum mechanical results found by Wong (1), pp. 691-693.