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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).

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    $\begingroup$ Classically, you don't need to get the same frequency back when scattering off matter. Think of the Doppler shift if scattering off a moving particle. Or suppose a laser bounces off a stationary mirror, and is received by a moving detector. $\endgroup$ – mmesser314 Feb 28 '16 at 20:38
  • $\begingroup$ This doesn't work; if your emitter and detector have no relative motion, there's no frequency shift, and that's the setup for Compton. We know Compton scattering has no easy classical explanation because it was one of the main pieces of evidence for quantum mechanics when it was discovered. $\endgroup$ – knzhou Feb 28 '16 at 20:43
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    $\begingroup$ But the particle recoils. Working classically, at the beginning of the interaction there is no frequency shift, but at the end there is. Take the average if you like. This just illustrates that classically you might expect a frequency shift. But on a small scale, you really need QM. $\endgroup$ – mmesser314 Feb 28 '16 at 20:47
  • $\begingroup$ Not sure of anything, but as of my short thinking, the light accelerates the electron, this exchanging energy. This results in a change of momentum on side of the electron, and a change of wavelength on side of light. $\endgroup$ – Gyro Gearloose Feb 28 '16 at 20:52
  • $\begingroup$ @mmesser314 I still have no idea what you mean. The Doppler effect only depends on the velocities of the source (the light source in the experiment) and the receiver (the light detector). Classically, the recoil of the electron has no effect whatsoever. $\endgroup$ – knzhou Feb 28 '16 at 20:56
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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.

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

By stripping some wave energy off one wave and adding it to another.

Classically, when light scatters off matter, the frequency of the light must stay the same.

Wave mechanics are classical. Classically, when we combine two waves, the result is a higher-frequency wave. Classically, we might partially combine two waves of frequency f, to yield two waves one with a frequency greater than f, the other with a frequency lower than f.

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.

So the continuity argument is of limited application.

In the 1920's, this result was paradoxical, and was considered to have no classical explanation.

Can you provide a reference for that please? Only de Broglie proposed the wave nature of matter in 1924.

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!

The photon is a wave, and so is the electron.

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

Because it fails for waves.

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?

It's true, but it hasn't been demonstrated because there's no accepted electron model. Papers such as Is the electron a photon with toroidal topology? struggle to get into high-impact journals and receive very little publicity.

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).

I don't think you can reconcile this. There is no magic. The wave nature of the electron is scientific fact, along with pair production, the Einstein-de Haas effect, magnetic moment, and annihilation. Everybody knows that in atomic orbitals electrons "exist as standing waves". Whatever your preference for the wave nature of the electron, it's crystal clear that that wave isn't propagating linearly at c.

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