Theories with an additive gauge group

This paper investigates the structure of theories that obtain from variational statements that are invariant under the action of a gauge group. Partial differential equations are obtained for the gauge group and for the infinitesimal generators of the gauge group. These are used to derive partial differential equations whose solutions give Lagrangian functions that result in action functionals that are strongly invariant under the gauge group. Properties of the Euler equations for such theories are analyzed, where it is shown that it is always possible to add a gauge condition to such theories when the data is of Neumann type. The results are illustrated by theories for the interaction of a vector 4-potential with a finite number of matter fields. The current and the charge-current potentials are shown to be determined from knowledge of the Lagrangian function and of the infinitesimal generators of the gauge transformations of the matter fields.