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I understand why the classical Thomson scattering computation should not match the QED one, since a classical field is composed of many photons, while Compton scattering involves a single photon. This paper did the relativistic classical computation, and indeed it does not match.

Therefore, the fact that they both match at low frequencies is somewhat puzzling to me. How can we argue for that?

One possible explanation I can think of is that one cannot exactly determine the number of photons at low energies. If I have a detector of photons, and I send a beam of photons with very low energy, Heisenberg uncertainty principle gives me large time uncertainty which forbids from pinpointing exactly when I detected a photon, therefore I can't say for sure how many photons the beam has. Hence, in the low energy limit it does not matter how many photons I have, one or many, the result should be the same, which is Thomson scattering. However, I tried computing the QED tree-level process with 2 incoming identical photons + 1 incoming electron and 1 outgoing photon + 1 outgoing electron , and this process obviously occurs at $O(e^3)$ so there's no possible way it can match the $O(e^2)$ computation.

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  • $\begingroup$ I think I do not agree with your explanation of "why the classical Thomson scattering computation should not match the QED one" (the first sentence). I think this is because the electromagnetic field is not quantized in Thomson scattering. Klein and Nishina got their formula simply by nothing else but quantizing the EM field. What you supposed in the third paragraph is actually infrared divergence whose energy level is much lower than Tomson scattering. Also, there should not be any amplitude of the two-in-one-out diagram as this violated the conservation of energy. $\endgroup$
    – Youran
    May 29, 2021 at 13:45
  • $\begingroup$ Regarding first comment, what Klein-Nishina did was to compute scattering of an electron with a photon. How is this classical? A classical EM field is made up of many photons, not one. About second comment, I meant with electrons involved, as in usual Compton scattering. I edited it for clarity. I don't think momentum conservation forbids having 2 photons + 1 electron $\to$ 1 photon + 1 electron. $\endgroup$ May 29, 2021 at 13:58
  • $\begingroup$ We believed that all tree-level diagrams are classical effects. So I call the Klein-Nishina equation a classical effect. The pain of Compton scattering cannot be cured by phone interference, but by field quantization. Sorry, the conservation of energy does not forbid a two-in-one-out diagram. It is the conservation of energy and charge forbids it. This is called Furry's theorem. $\endgroup$
    – Youran
    May 29, 2021 at 14:05

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The limit you have to take to go from the Klein-Nishina computation to the Thomson case is the non-relativistic limit.

The KN computation "is "quantum" in the sense that you are treating the photon as a point-like object having a well defined momentum and having an hard scattering with an electron with an arbitrarily large momentum transfer.

In the Thomson case the computation is "classical" in the sense that you are considering an incoming EM wave with a given frequency. The Thomson computation is however also "non-relativistic", since the frequency of the outgoing wave is computed using non-relativistic equations of motion. This is basically neglecting any electron recoil and forces the outgoing frequency to be the same as the ingoing frequency. This approximation is good when $\hbar \omega \ll m_e c^2$. This is indeed the limit in which the Klein-Nishina and Thomson computation match.

If you take the formula for the ougtoing frequency of the photon

$$\omega' = \frac{\omega}{1+\frac{\omega}{m}(1-\cos\theta)}$$

you see that if $\omega \ll m$ then $\omega \sim \omega'$. Taking the same limit in the Klein-Nishina formula gives back the Thomson cross section.

Additional complications might arise, and this could be what it is been computed in the paper you linked (not sure though), when the classical amplitude of the electric field is large (for example for strong lasers). In that case you might have additional photons in your scattering process and you obtain a non-linear Compton scattering, but that's a whole other thing and it's unrelated to both Thomson and KN computations.

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