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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.

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This is a general answer to this series of questions that have to be broken into individual entries.

Maxwell equations are a unification of electromagnetic , +electric, + magnetic observations which had been distilled into laws when the study of charges became serious.

Here are the four laws that Maxwell unified into one mathematical model for electricity, magnetism and light to start with.

Now physics models use mathematics as a tool by adding extra axioms to pick from the solutions of the mathematical equations those that describe and predict observations. So the four laws in their various mathematical formulations, are axiomatic constraints so that the solutions of the equations are predictive.

Since the time of Maxwell physics obseravations showed in the data the need for a quantum mechanical underlying mathematical model for all physics observations, and classical electromagnetic fields, electric, magnetic, and electromagnetic, emerge from the underlying quantum level.

The beauty of Maxwell's equations is that they are the same in the quantum and in the classical frame. In the classical frame they describe the electromagnetic spectrum from infrared to gamma rays, and used as operators operating on a wave function they describe the behavior of photons, the quanta of electromagnetic waves.

Photons are not waves in four dimensional space. The wave function of the photon $Ψ$ is the one that has sinusoidal solutions in four dimensional spacetime. The photon itself has a probability of measurement given by $Ψ^*Ψ$ which gives interference patterns typical of waves. Probability waves as interpreted by the axioms of quantum mechanics. See these observations of single photons, points on the screen, and their accumulation.

singlphdblsl

. Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

So for your questions to make sense within the physics models that are currently accepted to model reality, you have to clear up the quantum from the macroscopic classical level.

Photons are elementary point particles with zero mass that carry energy/momentum and spin. No measurable classical fields. Only in a quantum mechanical superposition of zillions of photons the classical electric and magnetic fields can be measured/observed.

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The answer below is only about your fourth question(s).

My understanding is that if one photon is given off in a complete vacuum, then it would propagate spherically (due to the symmetry).

It’s all about photons. Whenever one thinks about EM radiation, at the end the conclusions should be consistent with the radiation of quantized particles. EM radiation is the macroscopic effect of the emission of photons. Without energy changes of disturbed subatomic particles no EM radiation occurs.

A photon, emitted from an electron, moves straight forwards, or more precise along its geodesic path (influenced only from the geometry of the space with its gravitational sources). And, every photon is a indivisible unit from its emission until its absorption. The first is self- evident, wether observing a laser beam or even a lighthouse beam.

For EM radiation it is another thing; for short distances. If the emitter allows a symmetrical emission of photons, the EM radiation is spherical symmetrical, like in some approximation for stars or for electric bulbs (excluding the lamp socket). And in the case of the bulb and AC power one indeed get a swelling radiation.

An EM wave one get from the periodical and somehow synchronized acceleration of electrons on the surface of an antenna rod. BTW, this waves are not spherically symmetric because of the antenna rod.

In all of the described cases in a big enough distance from the source a sensitive detector will detect the arrival of single photons, be this the light from the star, the bulb or the radio wave.

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.

You name the reason why the concept of a spherical wave dislocation of a propagating photon is not helpful. It would be nice to get an example of an phenomenon, where such a concept is helpful at all.

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?

Good point to show the misconception.

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.

Applause.

Again, it would be nice to get an example of an phenomenon, where such a concept of spherical distribution is helpful at all. The responses, if any, will occur around the double slit experiment. I refer to my answer to the question Is photon something that already exists before measurement?

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