Notes on Spinors and Polyforms II: Quaternions and Octonions

TL;DR

This paper explores the generalization of Pauli matrices using quaternions and octonions, relating these models to creation/annihilation operators and detailing Weyl spinors of Spin(4,4).

Contribution

It explicitly connects quaternionic and octonionic models of Clifford algebras to creation/annihilation operator frameworks and describes Weyl spinors of Spin(4,4).

Findings

Explicit models for quaternionic and octonionic generalizations of Pauli matrices.

Detailed description of Weyl spinors of Spin(4,4).

Connections established between algebraic models and operator constructions.

Abstract

Pauli matrices are 2x2 tracefree matrices with a real diagonal and complex (complex-conjugate) off-diagonal entries. They generate the Clifford algebra Cl(3). They can be generalised by replacing the off-diagonal complex number by one taking values in either quaternions or octonions (or their split versions). These quaternionic and octonionic generalisations generate well-known models of Cl(5) and Cl(9) respectively. The main aim of the paper is to explicitly relate these models to the models arising via the creation/annihilation operator construction. We describe in details the models related to quaternions and octonions, as well as to the split quaternions and octonions. In particular, we record the description of the possible types of Weyl spinors of Spin(4,4), which does not seem to have appeared in the literature.