# A soft-photon theorem for the Maxwell-Lorentz system

August 07, 2019

For the coupled system of classical Maxwell-Lorentz equations we show that
the quantities \begin{equation*}
\mathfrak{F}(\hat x, t)=\lim_{|x|\to \infty} |x|^2 F(x,t), \quad
\mathcal{F}(\hat k, t)=\lim_{|k|\to 0} |k| \widehat{F}(k,t), \end{equation*}
where $F$ is the Faraday tensor, $\hat{F}$ its Fourier transform in space and
$\hat{x}:=\frac{x}{|x|}$, are 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 $F_{\mathrm{sc},\pm}$ and the soliton field $F_{v_{\pm\infty}}$
depending on the asymptotic velocity $v_{\pm\infty}$ of the electron at large
positive (+), resp. negative (-) times. This gives a soft-photon theorem of the
form \begin{equation*} \mathcal{F}_{\text{sc},+}(\hat{k}) -
\mathcal{F}_{\text{sc},-}(\hat{k})= -(
\mathcal{F}_{v_{+\infty}}(\hat{k})-\mathcal{F}_{v_{-\infty}}(\hat{k})),
\end{equation*} and analogously for $\mathfrak{F}$, 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.

Keywords:

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