Are electrostatic and magnetostatic fields quantised? Most things in fundamental physics come quantised. There is even a hunt for the hypothesised quantum of gravity, the graviton. Magnetic fields are quantised under certain specific circumstances, such as for superconductors, but this appears to be a consequence of the circumstance rather than the nature of the field itself. In the wider context, This question received no satisfactory explanation as to how quantum field theory might account for quantisation of these (nominally) static fields. Is there any such theory of quantisation for the electrostatic and magnetostatic fields?
 A: I can only guess what you mean by a quantized field here.
If you mean, that the field is quantized in its strenght, then there is no such thing. Quantized theories allow for continuus energies. Photons for example can have any energy. The strength of a specific interaction is not restricted to come is specific quanta of strength.
If you mean, how a classical field can be described in a quantized manner, then this answer describes the facts fairly well. The answer here just describes how one would come from the first order (so called tree level) feynman diagram to an approximation for the coulomb potential. There is some approximations nescesarry, because feynman diagrams usually describe particle-particle interactions. Thus we use scattering theory and have to "fix" one of the momenta to get an approximately static field.
If you just want an explanation for the idea of quatization of fields:
The quantization comes from the fact that we have quantized ammounts of charge. The electrons are excitations of the QED field. QFT does not mean, that the classical field is quantized. It rather links the objects creating the field to excitations of a more fundamental field. Thus you will not see actual quantization of the electrostatic field. Only of the field creating the electrostatic field. Simply because it is not the quantized field QED is about.
