From quantum electrodynamics to a geometric gauge theory of classical electromagnetism

TL;DR

This paper introduces a new classical field theory formulation of electromagnetism that is gauge invariant and approximates quantum electrodynamics in the classical limit, using a geometric approach to relate the four-current to the phase velocity of the field.

Contribution

It presents a novel gauge-invariant classical electromagnetism theory derived from a geometric approach that approximates QED in the classical limit, including gravitational mass.

Findings

The formulation reproduces classical electromagnetism for charged continua.

The four-current aligns with the extremal phase velocity of the field.

The theory links rest density to the squared modulus of the classical field.

Abstract

A relativistic version of the correspondence principle, a limit in which classical electrodynamics may be derived from QED, has never been clear, especially when including gravitational mass. Here we introduce a novel classical field theory formulation of electromagnetism, and then show that it approximates QED in the limit of a quantum state which corresponds to a classical charged continua. Our formulation of electromagnetism features a Lagrangian which is gauge invariant, includes a classical complex field from which a divergenceless four-current may be derived, and reproduces all aspects of the classical theory of charged massive continua without any quantum effects. Taking a geometric approach, we identify the four-current as being in the direction of extremal phase velocity of the classical field; the field equations of motion determine this phase velocity as being equal to the mass, which makes the rest density proportional to the squared modulus of the field.