Ab initio derivation of the quantum Dirac equation by conformal differential geometry: the "Affine Quantum Mechanics'"

TL;DR

This paper derives the Dirac equation from conformal differential geometry, proposing 'Affine Quantum Mechanics' as a new framework that links quantum features to geometric curvature, with potential applications in unifying fundamental forces.

Contribution

It provides an ab initio derivation of the Dirac equation using conformal geometry, introducing a novel geometric approach called 'Affine Quantum Mechanics' that extends to general relativity.

Findings

Quantum features emerge from conformal curvature and Weyl's pre-potential.

The derivation is exact and closed-form, connecting geometry with quantum wave functions.

The framework can be extended to arbitrary space-time metrics, including those in General Relativity.

Abstract

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.