My questions are related to the question asked at Are EM radiation and EM waves the same thing?. My background is in math (my Ph.D. thesis was in geometric analysis), and I have only taken basic physics courses as an undergraduate. Does quantum field theory provide a complete mathematical answer to all of my following questions? If so, are there any good references which could be understood by someone with a strong math but weak physics background? There seems to be a large overlap between terms in quantum electrodynamics and 4-manifold topology, but I do not understand the physical significance.
As I understand it, Maxwells equations are a classical mechanics interpretation of some local properties of charges. In the absence of any actual charges, EM fields (resulting from some charge now removed from the system) propagate according the the linear wave equation (which is computed from Maxwell's equations with these constraints.)
1) Are all solutions of the linear wave equation physical? Are there any boundary or regularity assumptions?
2) How does quantization affect which solutions are physical? Is this related to why we can assume we can take the Fourier transform with (spacial) periodic boundary conditions to obtain a discrete set of (spacial) frequencies? (as opposed to taking the FT on Euclidean space.) I want to emphasize the question: why is a spacial/temporal frequency almost always associated with EM radiation? Why do many sources seem to suggest that all EM radiation is a sinusoidal wave in space and time?
3) What is the exact distinction between the "near" and "far" fields as described on wikipedia? I am bothered by the fact that EMR appears to exist only in the absence of charge, but there had to be some some charge to create the EMR in the first place. Does this require quantum field theory to fully explain?
My next two questions (4 and 4b below) seem to be related to a non-physical system due to the answer of Is a single photon emitted as a spherical EM wavefront?, but my questions boil down to how can I understand wave (non-local) - particle (local) duality when it seems to me that the quantized photon must have an instantaneous interaction (all energy transfered as one chunk) but it would take time for the corresponding effect to "travel" to the whole wave.
4) My understanding is that if one photon is given off in a complete vacuum, then it would propagate spherically (due to the symmetry). When this photon interacts with a charge (such as an electron), the whole photon would be "absorbed" by the charge, changing its energy. This interaction happens locally, but must affect the whole electric field propagating spherically. Does this interaction happen "instantaneously", or is there a speed of propagation of the effect on the EM field? If it is not instantaneous, how did the photon get absorbed without half a photon being absorbed at some time in between?
4b) If one photon of EMR was propagating in such a spherically symmetric manner and ran into two (symmetrical) electrons exactly symmetrically on opposite side the origin of the EMR, where would the photon interact? Would no interaction occur at all? I guess I am very confused by the (local, quantized) interaction of the photon, the local interaction of the EM field, the global behavior of the EM field, and the finite speed of propagation of information due to relativity.