Composite bundles in Clifford algebras. Gravitation theory. Part I

TL;DR

This paper explores the structure of Clifford algebra bundles in gravitation theory, focusing on their relation to tangent bundles, spinor subbundles, and the use of composite bundles to classify different spinor fields.

Contribution

It introduces a novel approach using composite bundles to characterize various Clifford algebra bundles and spinor fields in the context of gravitation theory.

Findings

Clifford algebra bundles can contain spinor subbundles and relate to tangent bundles.

Different Clifford algebra bundles are not necessarily isomorphic.

The technique of composite bundles helps describe diverse spinor fields under general covariant transformations.

Abstract

Based on a fact that complex Clifford algebras of even dimension are isomorphic to the matrix ones, we consider bundles in Clifford algebras whose structure group is a general linear group acting on a Clifford algebra by left multiplications, but not a group of its automorphisms. It is essential that such a Clifford algebra bundle contains spinor subbundles, and that it can be associated to a tangent bundle over a smooth manifold. This is just the case of gravitation theory. However, different these bundles need not be isomorphic. To characterize all of them, we follow the technique of composite bundles. In gravitation theory, this technique enables us to describe different types of spinor fields in the presence of general linear connections and under general covariant transformations.