I couldn't help but noticed that the main physical arguments are the same as they can be found in practically all the books on many body published in English. The finite temperature case is very well understood, at least in the thermodynamic equilibrium case, but the $T = 0$ case is still very problematic. Indeed, starting with Abrikosov et al., and going through Pines, Nozieres, Pitaevskii, Fetter, etc., one starts with the adiabatic switching-on of interparticle interaction and with the celebrated Gell-Mann & Low theorem (GMLT). This theorem is correct, but it doesn't guarantee that one obtains the interacting ground-state (g.s.) state vector in this way. Indeed, the theorem ONLY shows that starting from the free g.s. one obtains an eigenvector of the full (i.e., with interaction) Hamiltonian, but it doesn't show that the corresponding energy eigenvalue is the minimum one, so that this interacting eigenvector to REALLY represent the interacting g.s. Can one prove that the eigenvalue is a minimum, or one just has to POSTULATE this? Any ideas and references to the literature would be greatly appreciated.
Concerning systems with symmetry breaking, again at $T = 0$, such as BEC and BCS one introduces the corresponding condensates by hand. In the BEC case it's not that bad since if one follows the GMLT construction starting with the free g.s., there is a condensate to begin with for the free g.s. (and this can be rigorously proven from a math point of view) for a free system. However, one still has to show that free g.s. is REALLY mapped onto the interacting g.s. as I've already asked. But in the BCS case there is no condensate for the free system. One really needs an interaction to form a condensate. One way is, of course, to introduce symmetry breaking terms into the free system but in the end, after the thermodynamic limit is taken, one has to take the limit of their strength (coupling constant) to zero. How can one justify the GMLT construction in this case? How can one prove that the energy is minimum?
I would be very grateful if you could give me your insight and possible answers to my questions above.