Geometric tri-product of the spin domain and Clifford algebras

TL;DR

This paper explores the relationship between the triple product in the spin domain and Clifford algebra's geometric product, developing a geometric framework for spin particles and Lorentz group representations.

Contribution

It introduces a novel geometric tri-product in the spin domain and demonstrates its connection to Clifford algebras, enabling new models of spin particles and Lorentz group actions.

Findings

Established the relation between the spin domain's triple product and Clifford algebra's geometric product.

Developed a geometric spectral theorem for the spin domain.

Constructed spin 1 and spin 1/2 Lorentz group representations within this framework.

Abstract

We show that the triple product defined by the spin domain (Bounded Symmetric Domain of type 4 in Cartan's classification) is closely related to the geometric product in Clifford algebras. We present the properties of this tri-product and compare it with the geometric product. The spin domain can be used to construct a model in which spin 1 and spin1/2 particles coexist. Using the geometric tri-product, we develop the geometry of this domain. We present a geometric spectral theorem for this domain and obtain both spin 1 and spin 1/2 representations of the Lorentz group on this domain.