  The Cartan's equations definig simple spinors (renamed pure by C. Chevalley)
are interpreted as equations of motion in momentum spaces, in a constructive
approach in which at each step the dimesions of spinor space are doubled while
those momentum space increased by two. The construction is possible only in the
frame of geometry of simple or pure spinors, which imposes contraint equations
on spinors with more than four components, and the momentum spaces result
compact, isomorphic toinvariant-mass-spheres imbedded in each other, since the
signatures appear to be unambiguously defined and result steadily lorentzian;
up to dimension ten with Clifford algebra Cl(1,9), where the construction
naturally ends. The equations of motion met in the construction are most of
those traditionally postulated ad hoc for multicomponent fermions. The 3
division algebras: complex numbers, quaternions and octonions appear to be
strictly correlated with this spinor geometry, from which they appear to
gradually emerge in the construction, where they play a basic role for the
physical interpretation. In fact they seem then to be at the origin of
electroweak and strong charges, of the 3 families and of the groups of the
standard model. In this approach there seems to be no need of higher
dimensional (>4) space-time, here generated merely by Poincare translations,
and dimensional reduction from Cl(1,9) to Cl(1,3) is equivalent to decoupling
of the equations of motion.

High Energy Physics - Theory