On a Microscopic Representation of Space-Time IV

TL;DR

This paper explores a geometric and algebraic framework linking space-time, Lie groups, and quantum field theories, emphasizing the role of complex geometry and projective structures in unifying physical descriptions.

Contribution

It introduces a novel approach connecting complex projective geometry, Lie group chains, and quantum field representations to deepen the understanding of space-time structure.

Findings

Relation of Dirac algebra to SU* (4) and real/quaternionic Lie groups

Use of projective geometry and transfer principles in physical theories

Unification of electromagnetism, relativity, and quantum field concepts

Abstract

We summarize some previous work on SU(4) describing hadron representations and transformations as well as its noncompact 'counterpart' SU$*$(4) being the complex embedding of SL(2,$\mathbb{H}$). So after having related the 16-dim Dirac algebra to SU$*$(4), on the one hand we have access to real, complex and quaternionic Lie group chains and their respective algebras, on the other hand it is of course possible to relate physical descriptions to the respective representations. With emphasis on the common maximal compact subgroup USp(4), we are led to projective geometry of real 3-space and various transfer principles which we use to extend previous work on the rank 3-algebras above. On real spaces, such considerations are governed by the groups SO($n$,$m$) with $n+m=6$. The central thread, however, focuses here on line and Complex geometry which finds its well-known counterparts in descriptions of electromagnetism and special relativity as well as - using transfer principles - in Dirac, gauge and quantum field theory. We discuss a simple picture where Complexe of second grade play the major and dominant r\^{o}le to unify (real) projective geometry, complex representation theory and line/Complex representations in order to proceed to dynamics.