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).