ASHTEKAR VARIABLES AND UNIFICATION OF GRAVITATIONAL AND ELECTROMAGNETIC-INTERACTIONS

It is shown that Ashtekar new canonically conjugate variables (based on SU(2) valued connection one-forms A(i), potentials for the (anti) self dual Weyl regime) recently extracted from the space-time geometry, enable one to view the gravitational field as a (self-interacting) Yang-Mills field, thus setting premises to a unified field theory. We prove that these connections can be pulled back to a principal fibre bundle P over space-time (M, g(ij)), with SU(2) gauge group. A metric h(g, A, k) (where k is a gauge invariant metric on the fibre) can be introduced on P, whose Ricci scalar curvature provides an action density R(g, A, k), sum of the Ricci scalar curvature of (M, g) of the curvature of the fibre, and of the self action density of the gauge field A. Extrema of the action integral occur provided g satisfies Einstein's equation and A satisfies the Yang-Mills equation. Field equations thus arise from a single variational principle and display General Relativity as a gauge theory in the (anti) self-dual regime. It is shown that geodesics of (P, h) project down to paths of charged particles in (M, g) which are accelerated by the Yang-Mills field strength and where the charge (fibre component of the geodesic in P) is of purely gravitational origin. These results bring physical significance to the geometry of (P, h). We then present an interaction which could govern the large scale structure of the universe. This interaction is the result of an SU(2) x U(1) unification of gravitational and electromagnetic degrees of freedom, and is mediated by a field of the Yang-Mills type. This Yang-Mills field is shown to carry a charge which encompasses both gravitational and Maxwellian aspects and which could play a role in the understanding of long range waves.