Derivation of Dirac's equation from conformal differential geometry

A rigorous ab initio derivation of the (square of) Dirac’s equation for a single particle with spin is presented. The theory is carried out in the framework of an approach where it is assumed that quantum phenomena originate from the interplay between the motion of a relativistic spherical top and the non trivial background geometry of its configuration space. We require full conformal invariance in each step of the theory, which is achieved by replacing the mass of the top withWeyl’s curvature. The curvature acts on the particle as a scalar potential and the particle, in turn, acts back on curvature modifying Weyl’s pre-potential. The mechanism is similar to the one at the basis of the general relativity, with the difference that curvature is originated here by the affine connections of space rather than by the metric tensor, which can be prescribed at will. The theory is intrinsically nonlinear, but it is linearized, exactly and in closed form, by an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” y4 of the 4-spinor Dirac’s equation. This novel theoretical scenario, referred to as “Affine Quantum Mechanics”, appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation.