Let's look at a general multi-particle Hamiltonian $H_0$ (e.g. many non-interacting particles). Now we add a background electric field. The new Hamiltonian is then $H=H_0+ \Delta H$ with $\Delta H=-J \cdot A$, where $J$ is the quantum operator associated to the electric current and $A$ is the magnetic vector potential (gauge: $A_t=0)$.
My question is: Where does this expression come from?