Doing some theory research on Superconductivity, I stumbled on a few of the monsters hidden in the closet of QFT. Among which is the fact that BEC (Bose Einstein condensation) has not been proven to happen (not even to bosons), and that there is a no go theorem forbidding spontaneous breaking of the phase symmetry for non degenerate ground states... Everything seems very messy. Superconductivity is often discussed as the BEC of cooper pairs or spontaneous breaking of the U(1) symmetry. Now I have a few questions: 1) We know BEC of fermions has been done in the lab in that case the fermions pair to form bosons. Are there any suggestions what a creation operator of a boson consisting of fermions could look like. Whenever I try to stuff more then one electron in one state the fermionic operators cancel out. 2) Since such a creation operator doesn't seem to be a simple product of a number of raising operators, is the underlying Fock space even a good ground to build these bosonic operators on. 3) I read somewhere in relation to Bloch waves, and symmetry groups, that a symmetry of a Hamiltonian forces the same symmetry upon its Eigenstates as a consequence of Schur's lemma (which I couldn't see). When I look at the cheap explanation of spontaneous symmetry breaking with the Mexican hat potential, the ground state breaks the Symmetry of the Hamiltonian. How does it cheat its way around group theory? Sorry that the questions are probably full of bad implicit assumptions. I'd be glad if someone could elucidate this even just a bit, or give me some pointers to helpful literature.