Fermions and bosons in a unified framework. II. Interacting models

In this paper we construct some fully interacting field theories. The first model has a colored, curved Weinberg-Salam-type action. It is formed by taking the Hilbert product of the (generalized) curvature of a (given) superalgebra with an auxiliary (generalized) curvature. Note that pieces of simple superalgebras are gauged; the effective superalgebra of gauge fields is not simple. The auxiliary curvature was needed to obtain the linear pieces of the action, and it thus appears to be somewhat ad hoc. In contrast we show how to construct an action using only the curvature of a local superalgebra without the auxiliary curvature (it is therefore quadratic). Nonetheless, linear terms arise as crossterms between pieces of the curvature. In fact, since we have chosen to use a special-unitary flavor algebra and four-component spinors, we discover we have already specified a unique simple supergroup whose other Bose gauge fields are in U(2,2), the Lie algebra formed by all the Dirac matrices. These fields gauge the spin structure of the fermions. Color causes certain complications discussed in the paper. The tensor piece of the U(2,2) curvature consists of the usual curvature plus a term identifiable as the old auxiliary tensor. Thus both linear and quadratic terms for the space-time curvature arise when the full curvature is squared. The field associated with the identity generator is electromagnetism; with the vector, torsion; with the tensor, curvature and auxiliary terms. We call the fields associated with the axial generators axial torsion and axial electromagnetism. When the fields which couple to Dirac spinors are assumed proportional to their scalar counterparts, an experimental value for a conserved axial electromagnetic coupling is ${10}^{-3\mathrm{}}e$. We present a qualitative argument for the renormalizability of this action, since it is almost that of a standard Yang-Mills gauge theory, based on preservation of recoordinatization invariance by the quantization procedure.