In quantum field theory, how can Compton scattering change the frequency of light?

Classically, when light scatters off matter, the frequency of the light must stay the same. This follows directly from a continuity argument: if you put in $f$ field oscillations per second, you'd better get $f$ oscillations per second out, because you can just follow each peak through. However, we observe a frequency shift in Compton scattering. In the 1920's, this result was paradoxical, and was considered to have no classical explanation.

In quantum mechanics, the frequency shift is explained by treating light as a particle, the photon. However, in quantum field theory, which also produces the correct result for Compton scattering, light is again treated as a field!

• Why does the continuity argument described above for classical fields fail for quantum fields?


• In quantum field theory, Compton scattering is tree-level, and tree-level behavior is equivalent to classical field theory. Therefore, there should be a classical explanation for Compton scattering, i.e. Compton scattering is not a quantum effect. Is this true, and has this been demonstrated?

Note: I am not asking for a quantum mechanical explanation of the Compton effect. I've already seen this plenty of times. My question is how to reconcile the argument that Compton has no classical explanation (in the first paragraph) with my heuristic argument that Compton does have a classical explanation (the last bullet point).

Answer

The "continuity argument" fails for quantum fields because quantum fields are operator-valued distribution that do not take definite "values".

Compton scattering has a classical equivalent, but not in the way you are thinking. In the non-relativistic regime, we get back Thompson scattering where the frequency of the electromagnetic wave doesn't change. The classical picture is an electromagnetic wave scattering off a point particle.

The relativistic Compton scattering from QFT corresponds to a classical high intensity regime, where the electric field of the infalling wave is strong enough to accelerate the electron such that it essentially Doppler shifts the outgoing, scattered wave, cf. "Limits on the Applicability of Classical Electromagnetic Fields as Inferred from the Radiation Reaction" by K. T. McDonald. This effect is variously known as "radiation-pressure recoil", "radiation reaction", "radiation damping" and other names. Classically, this Compton scattering-like effect vanishes when going to low intensities.

The more general observation to make here is that the "classical limit" of a quantum field theory may not be the classical theory we naively expect. Indeed, the presence of the fermionic electron field in the QFT alone, which is absent from classical electrodynamics, should show that the heuristic "$\hbar\to 0$ argument" that shows that generically tree-level computations for an action $S[\phi_1,\dots, \phi_n]$ correspond to classical field theory computations for the same action does not directly imply that tree-level QED computations correspond to CED computation.