Tetrahedral structure of idempotents of the Clifford algebra C_3,1

Abstract

In a series of papers, we have developed the principles of a variant of standard gauge theories which we call’ spin Gauge Theories’, based upon Clifford algebra, and have progressed towards the formulation of a unified theory of particle and gravitational interactions. In the first paper [1], we presented a theory of the electroweak interactions of one family of leptons and quarks, introducing the concept of the ‘Frame Field’. We conjectured that the frame field was intimately involved with gravitation, and in the second paper [2], we used spin gauge theory principles and the frame field to derive the Einstein-Hilbert gravitational Lagrangian as a particle interaction; we found that the theory automatically added a’ spin gravity’ term, quadratic in the curvature, modifying the Einstein theory at short distances. In the third paper [3], we showed that the electroweak and gravitational interactions could be combined into a unified theory. The only observed particle interaction which was not included in the theory was the strong interaction; the standard’ strong’ interaction theory is generated by the SU(3) colour symmetry group of quarks. We have outlined [4] a spin gauge theory, using the Clifford algebra C_4,7, which unifies the electroweak, gravitational and SU(3) interactions. In this and the accompanying paper [5], we describe in some detail how the colour group is introduced; we also report on a modification of the standard theory which is suggested to us by the structure of our unified model. We have not yet completed our work on this modification.