**Quantum Dynamics from Pure Spinor Geometry**

**Abstract:**

P.A.M. Dirac wrote in 1937 that from the available data in cosmology
and atomic physics, he could induce the existence of "a deep connection
in Nature between cosmology and atomic theory". In this paper we try to
show how this connection indeed may be found in the elementary part of
pure spinor geometry (two component spinors) which naturally contains
the two systems of equations of motion for massless systems: the Weyl's
and the consequent Maxwell's ones. Then the implied conformally
compactified space-time (Robertson-Walker Universe) and the dual
momentum space offer the possibility of computing some of the data (like
the elementary time h/Mc^{2}) on which Dirac based his conjecture.

For higher component pure spinors the associated Clifford algebras
geometry notoriously explain several properties of elementary particles.
But specially suggest that quantum dynamics may be formulated and
rationally understood, rather than in space-time, in momentum space
where pure spinors define Poincaré invariant spheres where quantum
problems may be formulated and solved; we will illustrate it for the H-atom
stationary states problem, dealt by V. Fock, where the pure spinor
sphere (S^{3}) explains the relativistic form of the formula for the energy
levels, and suggests further possibilities which are outlined and discussed.