Snell's law starting from QED?

Question

Can one "interpret" Snell's law in terms of QED and the photon picture? How would one justifiy this interpretation with some degree of mathematical rigour? At the end I would like to have a direct path from QED to Snell's law as an approximation which is mathematically exact to some degree and gives a deeper physical insight (i.e. from a microscopic = qft perspective) to Snell's law.

Answer 1

Sure. Start with QED, obtain Maxwell's equations, do the paraxial approximation and finally use Fermat's principle.

Answer 2

This appears to be explained in detail in Feynman's "QED the strange theory of light and matter" in Chapter 2, page 39 to 45, of the 2006 edition, in more or less plain English.

Answer 3

In regard to the single photon aspect of the question, I speculate that the explanation is similar to the Mossbauer effect, ie the photon is absorbed and re-emitted by the entire mirror/crystal rather than a single atom. If you insist on thinking of a single photon being absorbed and re-emitted by a single atom, you are going to have to invoke the regularity of the mirror or crystal and use constructive and destructive interference, as mentioned in other answers.

You can see the necessity of a coherent many body solution from the fact that a single atom does not refract or reflect, it scatters, and similarly for a rough mirror.