How are photons of the electrostatic and magnetostatic fields different from electromagnetic radiation? I understand that a photon of electromagnetic energy (light or radio) has both wave and particle characteristics and is best modelled using quantum field theory. I also understand that static and electric fields are encompassed by the same theory. I have read variously that the photons of these static fields are virtual, or that they are polarized (or an analogous property) differently from electromagnetic photons.
Without going deeply into the maths, can anybody explain the relationship between the three types of field or photon?
 A: My understanding is that the question is very hard. Photons as interaction messengers appear only in quantum field theory. In quantum mechanic they are not present, they are replaced by a potential (in Schrödinger equation). Description of "electrostatic situation" in quantum field theory is difficult because the most understood description we have is based on perturbation theory. There you have incoming states (particles from infinite past and infinite distance) and outgoing states (which exist in infinite future and infinite distance) and transition from infinite past to infinite future is described by "corrections" to initial states so as to form final states (one says: "S-matrix evolution"). The frame of perturbation theory certainly does not fit "electrostatic" problems. Such problems contain spatially distributed charge (so not free single non-interacting particles) which exist "always", from time minus infinity to time plus infinity. I am not aware of how such interaction should be interpreted in terms of photons. As far as my knowledge goes one uses to describe mesons (or other particles) in strong interactions by ladder gluon diagrams. Maybe such description might (under certain assumptions) be valid in quantum electrodynamics. In that case one could imagine electrostatic situation as constantly interchanging virtual photons. Yes, virtual, they do not "really" exist (what "really" exists are external legs of Feynman diagrams or asymptotic incoming and outgoing states if you want).
A: 
I understand that a photon of electromagnetic energy (light or radio) has both wave and particle characteristics and is best modeled using quantum field theory.

Light, radio, and lower frequency are best modeled using classical electromagnetic theory, where there are self propagating  sinusoidal  (hence wavess)varying electric and magnetic fields the energy given by the Poynting vector.
Higher frequencies as X-rays and Gamma rays are treated at the quantum mechanical level, as photons of energy$=hν$.
In general though, all electromagnetic waves can be modeled by field theory to be built up by the confuence of an innumerable number of photons ( elementary particles)

I also understand that static and electric fields are encompassed by the same theory. I have read variously that the photons of these static fields are virtual, or that they are polarized (or an analogous property) differently from electromagnetic photons.

There is a mathematical way to model static electric and static magnetic fields using virtual photons, as discussed here .

Static force fields are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces arising from the interactions between two bodies mediated by virtual particles, particles that exist for only a short time determined by the uncertainty principle. The virtual particles, also known as force carriers, are bosons, with different bosons associated with each force

Thus virtual photons can be used to model mathematically the static electric and magnetic fields. BUT keep in mind that the term "virtual" particle means that it is under an integration and has a changing four vector within the limits of integration for the given boundary condition problem. Virtual photons have the quantum numbers of a photon but not its mass ( which should be zero for a real photon). Virtual particles are always exchanged within a Feynman integral.
So there are the classical static electric and static magnetic fields, there are classical electromagnetic waves and there is the quantum mechanical elementary particle called a photon. A mathematical field theoretical  method exists ( for example here ) to derive the classical from the quantum, is the simplest summation.
