Bose-Einstein Condensation and Spontaneous Symmetry Breaking

TL;DR

This paper provides a rigorous proof linking Bose-Einstein condensation to spontaneous gauge symmetry breaking, emphasizing the role of off-diagonal long-range order and validating Bogoliubov's substitution.

Contribution

It offers a general, rigorous proof that Bose-Einstein condensation necessarily leads to spontaneous gauge symmetry breaking, clarifying the underlying mathematical structure.

Findings

Proof of spontaneous gauge symmetry breaking from Bose-Einstein condensation

Validation of Bogoliubov's c-number substitution for the zero mode

Connection between off-diagonal long-range order and 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 $a_{\bf 0}$.