Ehrenfest theorem in relativistic quantum theory

TL;DR

This paper proves the Ehrenfest theorem within relativistic quantum theory for charged particles, using conservation laws and avoiding issues with the relativistic position operator, thus extending classical results to the relativistic domain.

Contribution

It introduces a universal method to derive the Ehrenfest theorem in relativistic quantum theory without relying on the explicit form of the wave equation.

Findings

Ehrenfest theorem holds in relativistic quantum theory for charged particles.

The derivation relies on conservation laws rather than specific wave equations.

The approach circumvents problems with the relativistic position operator.

Abstract

Ehrenfest theorem is proven in relativistic quantum theory of charged particles, moving under the influence of an external electromagnetic field. In order to extend the classic Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton's second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the conserved electric four-current built from the particle wave function. The derivation is based on the conservation laws for the energy and momentum.