Electron is a charge, what means concrete configuration of electric field, growing to infinity in the center (if it would be a point particle). It is also a magnetic dipole moment - concrete configuration of magnetic field growing to infinity. And a mass - concrete configuration of gravitational field, again growing to infinity in the center. ... So particle is a sophisticated localized configuration of fields - is it also something more? If there would be such configuration identical to of some particle, would it be already this particle, or there still would be something missing? Stable, localized configurations of fields are generally called solitons - are particles solitons? (there are quantization methods for solitons and topological solitons like fluxons undergo quantum phenomenas like interference)
A particle is an excitation of a quantum field corresponding to that particle (so an electron is an excitation of an "electron field"). This has nothing to do with the electromagnetic field (or all other fields the electron interacts with), and it cannot be understood in a classical way.
So analogously positron is just an excitation of "positron field" ... and hundreds/thousands of other particles have their own fields? Sure we can say something like this for density: of electrons, of positrons etc. ... but fundamental fields correspond rather to interactions: EM, weak, strong, gravitational. In analogy to density, we can introduce e.g. your "electron field", but it is an effective description counting them (including quantum correlations) - what electron creation operator does, is not only just "adding single electron to electron field", but it also would drastically change electric field of the whole Universe ... and so Gauss law forbids creating a single electron - it has to be done e.g. as electron-positron pair. Particle physicists live in the perturbative approximation, often forgetting that it is only an approximation - in fact e.g. particle is not a plane wave, what would mean that this single particle is spread over the whole Universe. Still using this approximation, we get Feynman ensembles among possible "classical" scenarios: e.g. electron meets positron, producing two 511keV photons. For each of such scenario there are also real fields involved, e.g. first there is strong electric field of this electron and positron, attracting them together and finally their mass/energy is released in this EM field - as its excitations (usually carrying angular momentum): photons. So looking at such single scenario in Feynman ensemble, is particle just its own configuration of fields, or is also something more? Sure it would be e.g. a Dirac delta in density function for this particle (in space or momentum representation), but it is only an effective description - is particle objectively something more than just "the sum of its fields"?
There are just ~50 fields or so, corresponding to the various particle types. What does that mean? The electric charge of the electron is by far not as fundamental for the electron as you think. It has an electric charge, this means it couples to photons. So what? It has a weak charge, it couples to W and Z bosons. Neutrinos do not have an electric charge, and you still cannot produce them without producing (or annihilating) some other particle in the process. They do not. Physics can be described with those fields alone, everything else is philosophy.
Fundamentally there is no distinction between the electron (and other particles) fields (note the plural) and the fields associated with the interactions EM, weak, strong, and gravity (with the possible but unlike exception of gravity). All these fields are associated with quanta which are interpreted as particles. The photon is as much a particle as the electron is and the electron fields are as much a field as the photon field is (AKA the electromagnetic field).
While I would distinguish fundamental fields (of interactions) from the effective ones (counting different types of particles), I agree here - particles are just localized excitations of some fields (so formally they are solitons). I know that in quantum description you don't like the "localized" word as you usually work in the momentum space approximation, but in reality e.g. protons are usually localized for example in the centers of concrete atoms. You are talking about perturbative approximation, in which you imagine interaction between two charged objects as neverending stream of photons. It is a useful picture for this approximation, but the reality is very different - photon has e.g. spin 1, so both interacting particles would have to change spin every time ... this spin corresponds to very real "mechanical" angular momentum - so both interacting particles would have to change angular momentum every time (and also energy and momentum) ... Photons are sophisticated EM waves - e.g. electron changes angular momentum, so like behind marine propeller, Noether theorem says there have to be produced a wave carrying this angular momentum - where, beside description, you need some additional fields here??? ... and their number is not conserved in nonlinear optics ... and you can have much more complex EM configurations, like topologically knotted ones: http://physicsworld.com/cws/article/news/2013/oct/16/physicists-tie-light-into-knots - how do they look in your "photon field"? From the other side, electron is just the lightest charge ... knowing electric field around, if Gauss law says that there is negative charge inside, you can be almost certain that there is electron inside (or anti-proton).
You're not listening because you think you know something that you really don't. A photon is a quanta of electromagnetic field and an electron is a quanta of the electron field. The number of electrons is not a conserved quantity. Se for instance the beta-decay where an electron is produced while a neutron converts into a proton. Electric charge is conserved, that's true but the number of electrons isn't.
One of your confusions - localized or not, elementary particles are not solitons. Solitons are solutions to certain nonlinear wave equations. Localized elementary particles are linear superpositions of plane waves.
Indeed, the number of electrons is not conserved ... what is ultimately conserved is charge: guarded by electric field around due to Gauss law - and so often more appropriate than "electron field" is just "charge density rho" - something basic while considering electromagnetic field and definitely not fundamental. Sure you can introduce additional fields, like density of electrons or similarly your "electron fields", but it is only for an effective description. The difference between density scalar field and quantum fields of particles is that the latter additionally contains information about their correlations/superpositions. Like with photons - electron changes e.g. orbital angular momentum, so Noether law says there have to be produced wave carrying angular momentum ... where do you need some additional fundamental fields here? So you are saying that e.g. proton is just "a linear superposition of plane waves"? Superposition of plane waves can have practically any shape, which can further evolve, disperse due to velocity differences ... while proton always looks nearly the same and is well localized - there are required nonlinearities holding it together. Even in perturbative approximation you will get nowhere without nonlinearities (interactions) - usually approximated by the lowest term there.
That's where you going wrong. The electron fields are the fundamental description while the particle is obtained as a localized quanta of the field. The same thing is true about the electromagnetic fields. The fields are the fundamental description. There is no meaningful way in which one field is more or less fundamental than the other.
The photon is described by a field A_{μ} which is a 4-vector. It gives the photon's polarization. For light waves the polarization is transverse, and the photon's angular momentum is ±1 along the direction of propagation. The photons associated with the Coulomb interaction have a timelike polarization. That is, they are described by the fourth component A_{0}. The angular momentum for this polarization is zero.
Indeed we need a definition of "more fundamental" for this discussion - let us think about it. Generally thermodynamical description is seen effective (effect) instead of fundamental (reason) - so having a classical particle in a box, you could give its trajectory or average it over time and give just density as averaged trajectory. Clearly the trajectory is more fundamental here than the density - as you can infer density from trajectory, but cannot do the other way. So I think the definition of "A is more fundamental than B" can be: you can infer B from A in unique way, but you cannot do it the opposite way (?) - it is a partial order. So let us now look at electromagnetic field and electron field. Assume there is only a single electron in the Universe - EM field says not only where this electron is, but can also tell a lot about its history ... what electron field can tell here?
No. no non-"linearities" are required to keep a photon together. If the photon is in a vacuum there will be no dispersion and if it is in a medium it will be the non-linearity of the medium response to the photon that creates the dispersion to begin with, so it is kind of nonsensical to say the non-linearity keep it together. They actually pull the wave packet apart. Note also that doesn't mean the photon is torn apart. It only means that its location becomes more uncertain.
The electron field can tell you that electrons can be created or destroyed so your assumption of a universe with just a single electron in it is physically inconsistent. That's been known to be the case for about 80 years now since the advent of Quantum Field Theory.
Ok, so what is happening when a single optical photon goes through a prism? If you don't have something holding it together, shouldn't different components of its Fourier transform choose different angles - dispersing it? Does EM field forbids it? The Gauss law only guards total charge, does it forbid e.g. spontaneous pair creation? Sure, there is something missing in pure EM field: that charges are quantized - and quantum description only assumes that. EM field can be easily extended to make that Gauss law counts topological charge (still behaving accordingly to Maxwell's equations), which just have to be integer as in the nature - see e.g. Faber's model I discuss here: https://www.physicsforums.com/showthread.php?t=710042
Yes a photon wave packet disperses when it goes through a prism even if it is a single photon wave packet - that is the position of the photon becomes more uncertain over time, and yet the photon doesn't dissolve over time and when detected it is still observed as a localized particle. But the place where that photon is found becomes more uncertain since it can be found anywhere inside of the wave packet which is becoming broader as it disperses. I'm not sure I understand where you're going with that. How does that link with your previous statements?
My point here is that, in contrary to your linear superposition, in reality optical photon does not disperse, but its energy remains localized to finally e.g. excite a single atom with the whole energy ... that you need something more than linear superposition to really understand optical photons. You were claiming that quantum field theory is somehow better because it allows for various number of particles, so I have replied that EM field also allows for it. Especially when we extend it to enforce charge quantization as topological charge - then you get soliton particle models in which you also naturally have various number of particles. For example here is a nice animation of soliton-antisoliton annihilation in 1D: http://en.wikipedia.org/wiki/Topological_defect#Images . Run it backward and you get pair creation. And to describe solitons you also can perform quantization procedure, getting Feynman ensemble among possible scenarios - exactly like in QFT, but with particles obtaining internal structure - being solitons.
And my point is that you're mistaken to believe they don't disperse. They do disperse. You're correct that the photon's energy will be absorbed by a localized atom but that does not mean the photon didn't disperse. All it means is that the photon is quantized and must be absorbed all at once. The atom is much smaller then the wavelength of the light absorbed. Try creating a soliton with a length much more localized than its own wavelength. let me know if you're successful. Clearly the photon was NOT localized prior to being absorbed. You're trying to understand a quantum phenomenon using a classical model. No surprise you're getting all tangled up with misconceptions and non-sense.
If you also see e.g. electron or proton as just "linear superposition of waves", any dependence on momentum while a scattering, would also analogously disperse this particle ... so you are claiming that single particles can disperse ... So how does e.g. electric field look around such dispersed elementary (indivisible!) charge?
Well, classical electrodynamics don't have photons at all so I'm still have no idea what do you mean by variable number of particles here. Unless you quantize the theory - that is unless you use QFT, there are no photons, just waves. Do not confuse a localized wave such as a soliton with a quanta such as a photon. They are not remotely related to each other.