Almost radial gauge

TL;DR

This paper introduces an almost radial gauge for electromagnetic potentials, which vanishes rapidly in timelike directions, aiming to address the infrared problem in quantum electrodynamics by extending the free electromagnetic field algebra.

Contribution

It constructs a new gauge potential with specific decay properties and extends the algebra of smeared operators to include more general smearing functions, facilitating infrared analysis in QED.

Findings

Potential vanishes rapidly in timelike directions.

Extended algebra includes vector fields with typical scattered current behavior.

Framework applicable to infrared problems in QED.

Abstract

An almost radial gauge $A^\mathrm{ar}$ of the electromagnetic potential is constructed for which $x\cdot A^\mathrm{ar}(x)$ vanishes arbitrarily fast in timelike directions. This potential is in the class introduced by Dirac with the purpose of forming gauge-invariant quantities in quantum electrodynamics. In quantum case the construction of smeared operators $A^\mathrm{ar}(K)$ is enabled by a natural extension of the free electromagnetic field algebra introduced earlier (represented in a Hilbert space). The space of possible smearing functions $K$ includes vector fields with the asymptotic spacetime behavior typical for scattered currents (the conservation condition in the whole spacetime need not be assumed). This construction is motivated by a possible application to the infrared problem in QED.