On a Microscopic Representation of Space-Time VII -- On Spin

TL;DR

This paper explores a geometric framework for representing spin using line and complex geometry, linking classical projective geometry with quantum spin representations and discussing related symmetry groups.

Contribution

It introduces a natural geometric description of spin via lines and complexes, connecting classical geometry with quantum spin and symmetry group structures.

Findings

Identifies SU(2) spin in terms of projective geometry.

Constructs a Lagrangian using line/complex invariants.

Associates SU(4) and related real forms with geometric structures.

Abstract

We recall some basic aspects of line and line Complex representations, of symplectic symmetry emerging in bilinear point transformations as well as of Lie transfer of lines to spheres. Here, we identify SU(2) spin in terms of (classical) projective geometry and obtain spinorial representations from lines, i.e.~we find a natural non-local geometrical description associated to spin. We discuss the construction of a Lagrangean in terms of line/Complex invariants. We discuss the edges of the fundamental tetrahedron which allows to associate the most real form SU(4) with its various related real forms covering SO($n$,$m$), $n+m=6$.