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In an earlier source theory investigation it was shown that the vacuum polarization of a massive gauge field is finite, provided that the conservation of current is imposed everywhere, including the interior of the sources. It is shown in the present paper that radiative corrections to a vertex function (two-particle production by an external source) are also finite if the same requirement about current conservation is imposed. In other words one-loop corrections turn out to be finite in both calculations. The cancellations leading to convergence may be understood in terms of Ward identities. The three form factors that appear are not only shown to be finite but are also explicity found.

Parallel transfer is generalized to allow for the existence of a local gauge group. In spaces with torsion such a group may be introduced in a natural way and physically interpreted as a generalization of the electromagnetic gauge group. It is possible to construct simple theories of this type which correctly represent the isospin symmetries of the known fields. According to the program here proposed the local connection of microspace is restricted by the symmetries of the observed fields; the dynamics is then determined by the curvature of space, just as in the known macroscopic limit. In the models here considered the discrete groups are not discussed and all conservation laws are exact.

The massive gauge field, either free or coupled to a massless fermion field, is shown to be the Lagrangian equivalent of a composite massless field containing pseudoscalar as well as vector components. These fields are minimally coupled to each other while the pseudoscalar part has no direct fermion interaction (vector dominance). In general the total boson field consists of both vector and pseudoscalar components, but there exists a privileged gauge in which the pseudoscalar components are transformed away. The vector field in this special gauge is formally identical to the pseudoscalar current in the general gauge; therefore the field algebra in the privileged gauge is identical to the current algebra in the general gauge. That is, vector fields go into pseudoscalar currents under a local gauge transformation. In the formalism the pseudoscalar field plays the role of a coordinate system on the group space while the vector field correlates the group space at distinct space-time points. The nonlinear Lagrangian may be expressed in terms of geometric invariants of the group space and is manifestly invariant under general transformation of the pseudoscalar field.

General covariance implies that the physical continuum is characterized by its connection as well as by its local structure; and therefore by the symmetries of the connection as well as of the local Lorentz and gauge groups. The connection of the local Lorentz group is as usual associated with the gravitational field; the corresponding connection of the gauge group is associated with the universal weak and electromagnetic couplings. The strong couplings have the complete symmetry of the local gauge group. The weak couplings have the lesser symmetry of the connection, which in turn corresponds to a subgroup of the local gauge group. There are four weak, charged vector bosons, associated in pairs with the right and left-handed connections of the gauge group; the only neutral vector is the photon.