The general classical solution of the superparticle

TL;DR

This paper develops a comprehensive octonionic formalism for vectors and spinors in 9+1 dimensions, providing a complete classical solution for the CBS superparticle's equations of motion using Jordan matrices.

Contribution

It introduces an octonionic representation of Clifford algebra and constructs a superspace variable with Jordan matrices, solving the superparticle equations to all orders.

Findings

Complete classical solution of superparticle equations

Octonionic formalism for 9+1D vectors and spinors

Representation of supersymmetry transformations with Jordan products

Abstract

The theory of vectors and spinors in 9+1 dimensional spacetime is introduced in a completely octonionic formalism based on an octonionic representation of the Clifford algebra $\Cl(9,1)$. The general solution of the classical equations of motion of the CBS superparticle is given to all orders of the Grassmann hierarchy. A spinor and a vector are combined into a $3 \times 3$ Grassmann, octonionic, Jordan matrix in order to construct a superspace variable to describe the superparticle. The combined Lorentz and supersymmetry transformations of the fermionic and bosonic variables are expressed in terms of Jordan products.