Analytical mechanics of a relativistic particle in a positional potential

TL;DR

This paper introduces a new Hamiltonian formulation for a relativistic particle under positional forces, leading to modified Dirac and Klein-Gordon equations with specific energy spectrum corrections.

Contribution

It presents a reparametrization-invariant action for relativistic particles in positional potentials, enabling a consistent Hamiltonian approach and revealing corrections to standard quantum equations.

Findings

Hamiltonian formulation invariant under time reparametrization

Modified Dirac and Klein-Gordon equations with energy corrections

Potential applications in phenomenological models

Abstract

We propose a form for the action of a relativistic particle subject to a positional force that is invariant under time reparametrization and therefore allows for a consistent Hamiltonian formulation of the dynamics. This approach can be useful in the study of phenomenological models. Also the Dirac and Klein-Gordon equation differ from the standard formulation, with corrections of order (E-m)/m in the energy spectra.