Bose-Einstein Condensation and Spontaneous Symmetry Breaking

After recalling briefly the connection between spontaneous symmetry breaking and off-diagonal long-range order for models of magnets a general proof of spontaneous breaking of gauge symmetry as a consequence of Bose-Einstein condensation is presented. The proof is based on a rigorous validation of Bogoliubovʹs c-number substitution for the k = 0 mode operator α0.