  A rigorous \textit{ab initio} derivation of the (square of) Dirac's equation
for a single particle with spin is presented. The general Hamilton-Jacobi
equation for the particle expressed in terms of a background Weyl's conformal
geometry is found to be linearized, exactly and in closed form, by an
\textit{ansatz} solution that can be straightforwardly interpreted as the
"quantum wave function" $\psi_4$ of the 4-spinor Dirac's equation. In
particular, all quantum features of the model arise from a subtle interplay
between the conformal curvature of the configuration space acting as a
potential and Weyl's "pre-potential" closely related to $\psi_4$, which acts on
the particle trajectory. The theory, carried out here by assuming a Minkowsky
metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the
one adopted in the context of General Relativity. 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.
