Diracian theory of fields — Spinogravitation

Abstract

The paper involves two essential parts. First, using the modern spinor approach, we prove that Maxwell and Yang-Mills fields, can be described by means of tensor products of some Dirac fields. An analogous method, using some properties of the complete lifts of Yano and Ishihara, from the space time manifold to its tangent bundle allows us to consider the Einstein equations (exterior case) as a consequence of Dirac equations for some spin 1 and massless spinor field. There is a natural association with gravitational field and some electromagnetic field. Our methods use two classical isomorphisms, essentially: the Kähler-Atiyah isomorphism between a Clifford algebra and an exterior algebra, and the isomorphism of the tensor product of a spinor space and its dual with the Clifford algebra. The application of the K-A isomorphism to spinors needs a symmetry breaking by means of some subgroup of the pseudo-orthogonal group, called “spinoriality group” defined with many details in [6b]. Note also in the second part the role of the harmonic coordinates. The reader has to know essential results about differential geometry and spinor bundles. The complete references are in [6b].