A Canonical Field Theory with Spinors

Abstract

In this paper, we have constructed a general field theory in covariant form which incorporates the usual covariant generalization of Dirac matrices. The field equations are derived from a lagrangian that is a second-order differential covariant (a scalar density of weight 1) constructed in the covariant spinor formalism by the same method that in riemannian geometry leads to the curvature tensor. It is possible to show that, in spite of the apparently greater wealth of geometrical elements, this theory is completely equivalent to the general theory of relativity. The field equations satisfy the usual differential identities and, in addition, "spin" identities'; there are four "strong" conservation laws which can be used to obtain equations of motion for singularities. Since we do not known at present whether the equivalence with the theory of relativity may not be lost in the process of quantization, we consider eventual quantization desirable and have, in this paper, converted the theory into the canonical form.