## A soft-photon theorem for the Maxwell-Lorentz system.(English)Zbl 1431.78003

Summary: For the coupled system of classical Maxwell-Lorentz equations, we show that $$\mathfrak{F}(\hat{x}, t) = \lim_{| x | \rightarrow \infty} | x |^2 F(x, t)$$ and $$\mathcal{F}(\hat{k}, t) = \lim_{| k | \rightarrow 0} | k | \hat{F}(k, t)$$, where $$F$$ is the Faraday tensor, $$\hat{F}$$ is its Fourier transform in space, and $$\hat{x} := \frac{x}{| x |}$$, is independent of $$t$$. We combine this observation with the scattering theory for the Maxwell-Lorentz system due to A. Komech and H. Spohn [Commun. Partial Differ. Equations 25, No. 3–4, 559–584 (2000; Zbl 0970.35149)], which gives the asymptotic decoupling of $$F$$ into the scattered radiation $$F_{sc, \pm }$$ and the soliton field $$F_{v_{\pm \infty}}$$ depending on the asymptotic velocity $$v_{ \pm \infty }$$ of the electron at large positive (+), respectively, negative $$(-)$$ times. This gives a soft-photon theorem of the form $$\mathcal{F}_{\text{sc}, +}(\hat{k}) - \mathcal{F}_{\text{sc}, -}(\hat{k}) = -(\mathcal{F}_{v_{+ \infty}}(\hat{k}) - \mathcal{F}_{v_{- \infty}}(\hat{k}))$$, 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.

### MSC:

 78A25 Electromagnetic theory (general) 78A40 Waves and radiation in optics and electromagnetic theory 78A35 Motion of charged particles 78A45 Diffraction, scattering 81V10 Electromagnetic interaction; quantum electrodynamics 35Q60 PDEs in connection with optics and electromagnetic theory 35B40 Asymptotic behavior of solutions to PDEs 78M35 Asymptotic analysis in optics and electromagnetic theory 78A60 Lasers, masers, optical bistability, nonlinear optics

Zbl 0970.35149
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