The general classical solution of the superparticle

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 × 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.

Division algebras and supersymmetry II

“…Following Schray [30], we call this identity the "3-ψ's rule". This identity is an example of a "Fierz identity" -roughly, an identity that allows one to reorder multilinear expressions made of spinors.…”

Gravity as the square of Yang–Mills?

“…This is itself tied to the earlier observation of Kugo and Townsend [80] that the existence of minimal super Yang-Mills mulitplets in only D = 3, 4, 6, 10 is related to the uniqueness of R, C, H, O. This was followed-up by a number of authors [105][106][107][108][109], sharpening the correspondence between supersymmetry and division algebras. The final case of D = 10, A = O was developed most carefully in [98], where the link between supersymmetry and the alternativity of O was emphasised.…”

Octonionic representations of Clifford algebras and triality

“…We expect that a similar, fully octonionic treatment of supersymmetrical theories will make their symmetries more transparent. In fact, we have successfully applied the methods of this article to the CBS-superparticle [29]. We hope to be able to find a parallel treatment of the Green-Schwarz superstring.…”

“…The choice of an idempotent will turn out to be equivalent to the choice of a basis of the carrier space of the representation, which may be understood as a change of the multiplication rule of the octonions. An immediate application of the ideas presented here can be found in [29].…”