Stability and Instability of Relativistic Electrons in Classical Electro magnetic Fields

TL;DR

This paper investigates the stability of relativistic electrons in classical electromagnetic fields, demonstrating that proper projection onto positive energy states ensures matter stability even with magnetic fields involved.

Contribution

It shows that combining relativistic dynamics with magnetic fields can maintain matter stability if the positive energy states are correctly defined, resolving previous instability issues.

Findings

Proper projection onto positive energy states guarantees stability.

Including magnetic fields in the positive energy definition is crucial.

Incorrect definitions always lead to instability.

Abstract

The stability of matter composed of electrons and static nuclei is investigated for a relativistic dynamics for the electrons given by a suitably projected Dirac operator and with Coulomb interactions. In addition there is an arbitrary classical magnetic field of finite energy. Despite the previously known facts that ordinary nonrelativistic matter with magnetic fields, or relativistic matter without magnetic fields is already unstable when the fine structure constant, is too large it is noteworthy that the combination of the two is still stable provided the projection onto the positive energy states of the Dirac operator, which defines the electron, is chosen properly. A good choice is to include the magnetic field in the definition. A bad choice, which always leads to instability, is the usual one in which the positive energy states are defined by the free Dirac operator. Both assertions are proved here.