On a Microscopic Representation of Space-Time III

TL;DR

This paper explores a geometric and algebraic framework connecting space-time representation, Lie algebras, and line geometry, aiming to relate mathematical structures to physical concepts like electromagnetism and relativity.

Contribution

It extends previous work on algebraic structures and their geometric interpretations, linking Lie algebras to physical space-time concepts through line geometry.

Findings

Identification of su*(4) as a complex embedding of sl(2,ℍ)

Relation of algebraic structures to projective and line geometry of ℝ³

Application to physical concepts like electromagnetism and relativity

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.