Simultaneous geometrization of classical and quantum mechanics of spinless particles

Abstract

It is shown that classical and quantum equations of motion of a relativistic spinless particle (the Lorentz and Klein-Gordon equations) allow for a geometrization on the same manifold$\begin{array}{*{20}c} * \\ V \\ \end{array}$ ₄. A classical particle on$\begin{array}{*{20}c} * \\ V \\ \end{array}$ ₄ is described as a free particle ($\mathop \Delta \limits^*$p^μ=0), while the quantum particle, as a free wave ($\begin{array}{*{20}c} * \\ \nabla \\ \end{array}$s=0).