From Pure Spinor Geometry to Quantum Mechanics

Abstract

It is shown how pure spinors might play a fundamental role in building up a mathematical basis for quantum mechanics. First they are the elementary constituents of strings, in spaces with lorentzian signature, where they replace the concept of point‐event, when dealing with quantum physics. Therefore the corresponding quantum field theory results fundamentally non local. Furthermore, if the Cartan’s equations defining pure spinors are interpreted as equations of motion for fermions, or fermion multiplets, several complex phenomena in elementary particle physics find plain explanations, which derive from the corresponding Clifford’s algebras and the correlated division algebras. In particular the U(1) at the origin of charges appear evident in most equations for fermion doublets. This could explain the frequency in nature of charged‐neutral doublets. The geometry of pure spinors contemplate the existence of compact manifolds in momentum space (spheres) where problems of quantum dynamics might be mathematically formulated. One of these may be identified with that S3 in which V. Fock (1935) formulated and solved, with great geometrical evidence, the quantum problem of the H‐atom stationary states, setting also in evidence its SO(4) symmetry: a mathematical way to quantum physics which deserves to be further explored.