A Note on Polarization Vectors in Quantum Electrodynamics

TL;DR

This paper addresses the mathematical treatment of photon polarization vectors in quantum electrodynamics, proposing a method to eliminate their discontinuities and improve the theory's mathematical properties.

Contribution

It introduces a novel approach to remove the problematic polarization vectors in QED, enhancing the mathematical consistency of the theory.

Findings

Elimination of discontinuous polarization vectors

Improved decay properties of the Fourier transforms

Enhanced mathematical consistency in QED

Abstract

A photon of momentum k can have only two polarization states, not three. Equivalently, one can say that the magnetic vector potential A must be divergence free in the Coulomb gauge. These facts are normally taken into account in QED by introducing two polarization vectors epsilon_\lambda(k) with lambda in {1,2}, which are orthogonal to the wave-vector k. These vectors must be very discontinuous functions of k and, consequently, their Fourier transforms have bad decay properties. Since these vectors have no physical significance there must be a way to eliminate them and their bad decay properties from the theory. We propose such a way here.