Prom pure spinor geometry to quantum physics: A mathematical way

摘要

In the search of a mathematical basis for quantum mechanics, in order to render it self-consistent and rationally understandable, we find that the best approach is to adopt E. Cartan's way for discovering spinors; that is to start from 3-dimensional null vectors and then show how they may be represented by two-dimensional spinors We have now only to go along this path, however in the opposite direction; with these spinors (which are pure) we construct bilinearly null vectors and we find that they naturally generate Minkowski momentum space, where the simplest Cartan equations defining pure spinors determine all equations of motion for massless systems: both the quantum (Weyl's) and the classical (Maxwell's) ones These equations are conformal covariant, from which we get, in the conformally compactified phase space, a self-dual torus, from which we may rigorously derive the atomic time units like At = h/Mc2, which P.. A M Dirac introduced in 1938, as a sign of a "deep connection in Nature between cosmology and atomic physics". We have then the possibility of a new, purely mathematical, determination of h: the Planck constant, and thus the possible mathematical starting point for the representation of quantum mechanics A fundamental property of pure spinors is that of generating bilineaily null vectors in momentum spaces which, if Lorentzian, defines Poincare-invariant spheres where quantum-dynamical problems should be formulated and solved This was anticipated in 1935 by V Fock, who formulated on the sphere S3, the one-point compactification of ordinary 3-dimensional momentum space, the non-relativistic integral equation for the H-atom stationary states He solved it and pointed out the great geometrical visibility of quantization, since the discrete values of the H-atom energy levels En are simply due to the, obviously discrete, eigenvibrations of S3 (in contrast to the cumbersome historical approach). We show that the H-atom sphere S3 must be one of those Poincare-invariant spheres generated by pure spinors, and in fact En = ~%-Ip-,ra = 1,2,3, , of obvious relativistic form. In general pure spinors are also at the origin of the algebraic explanation of the internal symmetry of the "standard model" in elementary-particle theory. They are also at the mathematical origin of strings which substitute, in quantum wave mechanics, the concept of point event.