Scalar QED is supposed to be a good effective description of superconductors. That being said, a better microscopic description should be that of fermion QED, since at short distances we know that the good degrees of freedom are electrons and not Cooper pairs.
Is it possible to derive the macroscopic theory (scalar QED) from the microscopic one (fermion QED) explicitly?
My guess is that it should be possible to show that fermionic QED at large chemical potential flows to bosonic QED. At zero potential QED is of course weakly coupled so, if the claim is true at all, it should be provably true. But I'm not sure if large chemical potential makes the theory strongly coupled (in which case, the claim would depend on the choice of UV completion, which sounds wrong). I'm not even sure if a chemical potential is the right setting -- under which conditions is fermionic QED supposed to become a superconductor? (Is there even a good notion of chemical potential in fermionic QED? Fermion number is not conserved, so the answer should be "no".)
