Bose-Einstein Condensation and Spontaneous Symmetry Breaking

Abstract

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 Bogoliubov's $c$-number substitution for the ${\bf k}={\bf 0}$ mode operator $a_{\bf 0}$.Comment: Contribution to the proceedings of the 21st Max Born Symposium, Wroclaw, Poland, June 26--28, 2006. (References added. To be published in Reports on Mathematical Physics.