# On Spin II

## TL;DR

This paper explores the mathematical foundations of photon fields and spinorial degrees of freedom using complex, quaternionic, and projective geometric frameworks, aiming to unify physical and geometric concepts.

## Contribution

It introduces a novel approach to incorporate spin into photon field models via quaternionic representations and geometric constructions based on Lie's transfer principle.

## Key findings

- Mapping lines to spins using quaternionic representations.
- Relating group structures SU(4) and SL(2,H) to physical spin models.
- Connecting geometric and algebraic frameworks in photon and spin theories.

## Abstract

Having previously identified the photon field with a (special) linear Complex, we give a brief account on identifications and reasoning so far. Then, in order to include spinorial degrees of freedom into the Lagrangean description, we discuss the mapping of lines to spins based on an old transfer principle by Lie. This introduces quaternionic reps and relates to our original group-based approach by SU(4) and SU*(4)~SL(2,H), respectively. Finally, we discuss some related geometrical aspects in terms of (spatial) projective geometry which point to a projective construction scheme and algebraic geometry.