Notes on Spinors and Polyforms II: Quaternions and Octonions

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.