On a Microscopic Representation of Space-Time III

Abstract

Using the Dirac (Clifford) algebra $\gamma^{\mu}$ as initial stage of our discussion, we summarize and extend previous work with respect to the isomorphic 15dimensional Lie algebra su$*$(4) as complex embedding of sl(2,$\mathbb{H}$), the relation to the compact group SU(4) as well as associated subgroups and group chains. The main subject, however, is to relate these technical procedures to the geometrical (and physical) background which we see in projective and especially in line geometry of $\mathbb{R}^{3}$. This line geometrical description, however, leads to applications and identifications of line complexes and the discussion of technicalities versus identifications of classical line geometrical concepts, Dirac's 'square root of $p^{2}$', the discussion of dynamics and the association of physical concepts like electromagnetism and relativity. We outline a generalizable framework and concept, and we close with a short summary and outlook.