A variational formulation for relativistic mechanics based on Riemannian geometry and its application to the quantum mechanics context

TL;DR

This paper introduces a variational approach rooted in Riemannian geometry for relativistic mechanics, leading to the derivation of the Klein-Gordon equation within a quantum mechanics framework.

Contribution

It presents a novel variational formulation for relativistic mechanics based on Riemannian geometry, connecting classical and quantum descriptions.

Findings

Derived the Klein-Gordon equation as an approximation of the variational formulation.

Established a geometric foundation for relativistic quantum mechanics.

Unified classical and quantum mechanics perspectives through geometric methods.

Abstract

This article develops a variational formulation of relativistic nature applicable to the quantum mechanics context. The main results are obtained through basic concepts on Riemannian geometry. Standards definitions such as vector fields and connection have a fundamental role in the main action establishment. In the last section, as a result of an approximation for the main formulation, we obtain the relativistic Klein-Gordon equation.