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Register a new userAb initio derivation of the quantum Dirac equation by conformal differential geometry: the Affine Quantum Mechanics
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A rigorous textit{ab initio} derivation of the (square of) Diracs equation for a single particle with spin is presented. The general Hamilton-Jacobi equation for the particle expressed in terms of a background Weyls 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 Diracs 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 Weyls 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.
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