In this article, we postulate SO(3, 1) as a local symmetry of any relativistic theory. This is equivalent to assuming the existence of a gauge field associated with this noncompact group. This SO(3, 1) gauge field is the spinorial alhnity which usually appears when we deal with weighting spinors, which, as is well known, cannot be coupled to the metric tensor field. Furthermore, according to the integral approach to gauge fields proposed by Yang, it is also recognized that in order to obtain models of gravity we have to introduce ordinary affinities as the gauge field associated with GL(4) (the local symmetry determined by the parallel transport). Thus if we assume both GL(4) and SO(3, 1) as local independent symmetries we are led to analyze the dynamical gauge system constituted by the Einstein field interacting with the SO(3, 1) Weyl-Yang gauge field. We think this system is a possible model of strong gravity. Once we give the first-order action for this Einstein-Weyl-Yang system we study whether the SO(3, 1) gauge field could have a tetrad associated with it. It is also shown that both fields propagate along a unique characteristic cone. Algebraic and differential constraints are solved when the system evolves along a null coordinate. The unconstrained expression for the action of the system is found working in the Bondi gauge. That allows us to exhibit an explicit expression of the dynamical generator of the system. Its signature turns out to be nondefinite, due to the nondefinite contribution of the Weyl-Yang field, which has the typical spinorial behavior. A conjecture is made that such an unpleasant feature could be overcome in the quantized version of this model.
1. Introduction
Jn Yang's integral approach [l] to gauge theories, there is assigned a fundamental meaning to the phase factor associated with each group, independently of whether it is compact or not.
It has been shown in [l] that when it is assumed that the underlying space-time variety is Riemannian and that the only relevant group is GL(4), one obtains a thirdorder model of gravity which, in the absence of sources, contains all the solutions to Einstein's field equations in the vacuum.
However, it can be suggested that GL(4) strictly corresponds to the local symmetry generated by the translations. Therefore it can be argued that in order to reach a more relativistic gauge theory there must be introduced as a local gauge symmetry the other basic element of relativistic physics: SO(3, 1) [2], the Lorentz group.
Other authors [3] have also supported this point of view, which leads to more or less generalized gauge models carrying a geometrical version of the fundamental fields.