Dirac and Maxwell systems in split octonions

TL;DR

This paper explores the representation of Dirac and Maxwell systems within split octonions, establishing connections with triality, group invariants, and analyticity conditions in a (4+4)-dimensional space.

Contribution

It provides explicit split octonionic representations of SO(4,4) and Spin(4,4), and constructs generalized Lagrangians for Dirac and Maxwell systems using these algebraic structures.

Findings

Explicit split octonionic representations of SO(4,4) and Spin(4,4)

Construction of generalized Dirac and Maxwell Lagrangians

Connection between equations and split octonionic analyticity

Abstract

The known equivalence of 8-dimensional chiral spinors and vectors, also referred to as triality, is discussed for (4+4)-space. Split octonionic representation of SO(4,4) and Spin(4,4) groups and the trilinear invariant form are explicitly written and compared with Clifford algebraic matrix representation. It is noted that the complete algebra of split octonionic basis units can be recovered from the Moufang and Malcev relations for the three vector-like elements. Lagrangians on split octonionic fields that generalize Dirac and Maxwell systems are constructed using group invariant forms. It is shown that corresponding equations are related to split octonionic analyticity conditions.