The Abdus Salam International Centre for Theoretical Physics,
Strada Costiera, 11 34014 Trieste, ITALY

Internal symmetry from division algebra in pure spinor geometry

The E'. Cartan's equations defining "simple" spinors (renamed "pure" by C. Chevalley) are interpreted as (quantum) equations of motion for fermion multiplets in momentum spaces. The Cartan's conjecture on the non elementary nature of euclidean geometry is adopted; it conceives euclidean vectors as sums (or integrals) of null vectors bilinearly constructed in terms of pure or simple spinors. Consequentely those momentum spaces, constructed with pure spinors, result lorentzian and compact, isomorphic to invariant mass spheres imbedded in each other.

The equations found are most of those traditionally adopted ad hoc by theoretical physics in order to represent the observed phaenomenology of elementary particles. In particular it is shown how, the known internal symmetry groups, might derive from the 3 complex division algebras correlated with the associated Clifford algebras. Precisely complex numbers generate U(1), at the origin of charges of fermions, which steadly appear in charged-neutral pairs of fermions, or of fermion multiplets; quaternions generate SU(2) isospin and SU(2)L of the electroweak model; octonions generate SU(3) both of flavour and of color. They also explain some of the elementary particle properties such as the 3 lepton-hadron families and the 3 colors, both in number equal to the 3 immaginary units of quaternions (or Pauli matrices). The possible role of pure spinors, such as those correlated  the constraint relations, are presented and discussed.

The adopted Cartan's conjecture  allows a striking parallelism between geometry and physics, in so far, while notoriously classical mechanics of macroscopic bodies is well represented with euclidean geometry in space-time (example: celestial mechanics), neither macroscopic bodies nor, according to Cartan, euclidean geometry are elementary and then the mechanics of the "elementary  constituents" of matter: the fermions, has to be represented with pure spinors: the "elementary constituents" of euclidean geometry, and what is obtained is wave - or quantum - mechanics (in first quantization): the "elementary constituent" of classical mechanics, as it should, and, in this frame, the euclidean concept of point-event has to be abandoned; it could be sobstituted by a continuous sum, or integral, of null vectors (bilinear in pure spinors) which happens to be, as it is shown, precisely a string. In this way this approach is not at all orthogonal to the now prevailing one of strings and  superstrings, as it could have appeared at first sight. Some further consequences are drawn from this parallelism.