Infrared problem vs gauge choice: scattering of classical Dirac field

TL;DR

This paper investigates the scattering of classical Dirac fields in external electromagnetic backgrounds, establishing gauge conditions and asymptotic states that simplify the analysis of scattering processes.

Contribution

It demonstrates the existence of gauges and asymptotic states for Dirac fields in scattering settings, and introduces a special evolution picture simplifying the scattering analysis.

Findings

Existence of gauges with asymptotic free states

Asymptotic completeness of wave operators

Well-defined limits of free evolution at infinity

Abstract

We consider the Dirac equation for the classical spinor field placed in an external, time-dependent electromagnetic field of the form typical for scattering settings: $F=F^\mathrm{ret}+F^\mathrm{in}=F^\mathrm{adv}+F^\mathrm{out}$, where the current producing $F^{\mathrm{ret}/\mathrm{adv}}$ has past and future asymptotes homogeneous of degree $-3$, and the free fields $F^{\mathrm{in}/\mathrm{out}}$ are radiation fields produced by currents with similar asymptotic behavior. We show the existence of the electromagnetic gauges in which the particle has 'in' and 'out' asymptotic states approaching free field states, with no long-time corrections of the free dynamics. Using a special Cauchy foliation of the spacetime we show in this context the existence and asymptotic completeness of the wave operators. Moreover, we define a special 'evolution picture' in which the free evolution operator has well-defined limits for $t\to\pm\infty$, thus the scattering wave operators do not need the free evolution counteraction.