Relativistic Remnants of Non-Relativistic Electrons

TL;DR

This paper explores how non-relativistic limits of electrons obeying the Dirac equation still exhibit light-cone singularities, indicating charge and current densities propagate at light speed despite electron mass.

Contribution

It provides a detailed analysis of the relativistic remnants in non-relativistic limits, revealing persistent light-cone singularities in electron charge and current densities.

Findings

Charge and current densities travel at the speed of light.

Light-cone singularity persists in non-relativistic limit.

Solutions involve derivatives of relativistic invariant functions.

Abstract

Electrons obeying the Dirac equation are investigated under the non-relativistic $c \mapsto \infty$ limit. General solutions are given by derivatives of the relativistic invariant functions whose forms are different in the time- and the space-like region, yielding the delta function of $(ct)^2 - x^2$. This light-cone singularity does survive to show that the charge and the current density of electrons travel with the speed of light in spite of their massiveness.