Clifford Valued Differential Forms, Algebraic Spinor Fields, Gravitation, Electromagnetism and "Unified" Theories

Abstract

In this paper we show how to describe the general theory of a linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the algebra of Clifford bivectors is isomorphic to the Lie algebra of Sl(2,C). In that way the pullback of the linear connection under a trivialization of the bundle is represented by a Clifford valued 1-form. That observation makes it possible to realize Einstein's gravitational theory can be formulated in a way which is similar to a Sl(2,C) gauge theory. Some aspects of such approach is discussed. Also, the theory of the covariant spinor derivative of spinor fields is introduced in a novel way, allowing for a physical interpretation of some rules postulated for that covariant spinor derivative in the standard theory of these objects. We use our methods to investigate some polemical issues in gravitational theories and in particular we scrutinize a supposedly "unified" field theory of gravitation and electromagnetism proposed by M. Sachs and recently used in a series of papers. Our results show that Sachs did not attain his objective and that recent papers based on that theory are ill conceived and completely invalid both as Mathematics and Physics.