How quantum field explains magnetic forces? Imagine a charged matter particle is oscillating and the energy and momentum are transferred from matter field (electron field) to gauge field (electromagnetic field), now I am going to add a new matter particle with columb charge close to it. So how does the energy and momentum transfers bring both charges toward each other? Is it possible to apply Fleming right hand rule in quantum field?
 A: Quantum mechanics does not deal with forces so much as energy and momentum. Force is the rate of change of momentum with time or energy with position. If a couple of electrons, thought of for now as point particles, are placed in a photon field then the energy in the photon field will depend on the distance between the electrons. This is computed by solving the appropriate differential equations or working out the eigen values of the operators. This is turn gives the energy as a function of the distance of separation. In principle, if the electron is now seen also to be a field, the computation is similar, but rather more difficult. Related to this is the use of the Feynmann diagrams to expand the problem into a series solution, and then to add up the terms in the series to approximate the values.
But, also, in Quantum Mechanics, the case is considered using the scalar and vector potentials, rather than the electric and magnetic fields. So, the left hand rule, in and of itself does not apply in quite that sense.
To start looking at this, you could look at the scalar and vector potential theory and also the theory of poynting flux.
