A soft-photon theorem for the Maxwell-Lorentz system

For the coupled system of classical Maxwell-Lorentz equations, we show that $F(x̂,t)=lim|x|→∞|x|2F(x,t)$ and $F(k̂,t)=lim|k|→0|k|F^(k,t)$⁠, where F is the Faraday tensor, $F̂$ is its Fourier transform in space, and $x̂≔x|x|$⁠, is independent of t. We combine this observation with the scattering theory for the Maxwell-Lorentz system due to Komech and Spohn, which gives the asymptotic decoupling of F into the scattered radiation F_sc,± and the soliton field $Fv±∞$ depending on the asymptotic velocity v_±∞ of the electron at large positive (+), respectively, negative (−) times. This gives a soft-photon theorem of the form $Fsc,+(k̂)−Fsc,−(k̂)=−(Fv+∞(k̂)−Fv−∞(k̂))$⁠, and analogously for $F$⁠, which links the low-frequency part of the scattered radiation to the change of the electron’s velocity. Implications for the infrared problem in QED are discussed in the Conclusions.