A second order differential equation for the relativistic description of electrons and photons

TL;DR

This paper introduces a novel relativistic wave equation for electrons and photons based on a reformulated Klein-Gordon equation using space-time algebra, unifying quantum electrodynamics descriptions.

Contribution

It presents a new second-order differential equation framework for electrons and photons that incorporates spin effects and derives Maxwell's equations from this wave equation.

Findings

Wave equation is equivalent to the quadratic Dirac equation.

Maxwell's equations are derivable from the photon wave equation.

The approach unifies electron and photon descriptions in a relativistic quantum framework.

Abstract

A new relativistic description of quantum electrodynamics is presented. Guideline of the theory is the Klein-Gordon equation, which is reformulated to consider spin effects. This is achieved by a representation of relativistic vectors with a space-time algebra made up of Pauli matrices and hyperbolic numbers. The algebra is used to construct the differential operator of the electron as well as the photon wave equation. The properties of free electron and photon states related to this wave equation are investigated. Interactions are introduced as usual with the minimal substitution of the momentum operators. It can be shown that the new wave equation is equivalent to the quadratic form of the Dirac equation. Furthermore, the Maxwell equations can be derived from the corresponding wave equation for photons.