On a Microscopic Representation of Space-Time VIII—On Relativity

Abstract

Based on the previous work on the Dirac algebra and su*(4) Lie algebra generators, using Lie transfer we have associated spin to line and Complex reps. Here, we discuss the construction of a Lagrangian in terms of invariant theory using lines or linear Complex reps like F^μν, its dual F^αβ, or even quadratic terms like e.g. F^μνF_μν, or F^aμνF^a_μν with respect to regular linear Complexe. In this context, we sketch briefly the more general framework of quadratic Complexe and show how special relativistic coordinate transformations can be obtained from (invariances with respect to) line transformations. This comprises the action of the Dirac algebra on 4×2 “spinors”, real as well as complex. We discuss a classical picture to relate photons to linear line Complexe so that special relativity emerges naturally from a special line (or line Complex) invariance, and compare to Minkowski’s fundamental paper on special relativity. Finally, we give a brief outlook on how to generalize this approach to general relativity using advanced projective and (line) Complex geometry related to P⁵ and the Plücker–Klein quadric as well as transfer principles.