Wave-like description of Compton scattering and photoelectric effect I have found in the wikipedia page for QFT the following statement:

... Although the photoelectric effect and Compton scattering strongly suggest the existence of the photon, it is now understood that they can be explained without invoking a quantum electromagnetic field ...

But there are no references. Can somebody provide a reference or guideline to do this? I am looking for a proper description with mathematical form.
 A: As pointed in the comment by @CuriousOne, there is indeed a paper providing such explanation ("The photoelectric effect without photons" by Lamb and Scully).
It was however later found to be faulty. A very good review and references on this can be found in these answers to a different question posted on this website, but apparently there is no consensus on the validity of the explanation.
A: I've been collecting wave theory explanations of QM phenomena for years. The photo-electric effect is the easiest one. The much-talked about frequency dependence is an obvious consequence of the Schroedinger equation. The Compton effect is different: here, the coupling between the e-m field and the electron states is not controlled by the shared frequency of the two systems, but by the shared wavelength. Schroedinger published a very simple paper in 1927 pointing this out, but he was howled down by the Copenhagen school. I thought I would win the Nobel Prize when I figured this out ten years ago, so I was disappointed to find my explanation was the same as Schroedingers. 
Schroedinger was however baffled by the flecks of silver on a photographic plate from the light of a distant star. I have come up with an original explanation of this called "Quantum Siphoning". You have to start be realizing the photographic plate is not exactly a passive source in the sense that it needs to absorb a full "quantum" of energy to record a transition. The loophole that nobody else figured out is that at the very tiny concentrations of metallic silver needed to create a developable photographic image, the thermodynamics actually favors the transition. 
All of these arguments can be found on my blog, and they are mostly linked from this recent article, "There Are No Pea-Shooters For Photons."
