2011/07/15 by Enrico Santamato, Santamato, Enrico, Francesco De Martini +1 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #FOS: Physical sciences #Noncommutative and Quantum Gravity Theories #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Relativity and Gravitational Theory #quant-ph
paper · pdf · doi:10.48550/arxiv.1107.3168
14 pages
arxiv created 2011/07/15 · openalex publication_date 2011/07/15 · arxiv updated 2011/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A rigorous 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 ansatz solution that can be straightforwardly interpreted as the "quantum wave function" ψ4 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 ψ4, 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.