On ergodic states, spontaneous symmetry breaking and the Bogoliubov quasi-averages

It is shown that Bogoliubov quasi-averages select the pure or ergodic states in the ergodic decomposition of the thermal (Gibbs) state. Our examples include quantum spin systems and many-body boson systems. As a consequence, we elucidate the problem of equivalence between Bose-Einstein condensation and the quasi-average spontaneous symmetry breaking (SSB) discussed for continuous boson systems. The multi-mode extended van den Berg-Lewis-Pul\'{e} condensation of type III demonstrates that the only physically reliable quantities are those that defined by Bogoliubov quasi-averages.