Some typical delusions in the theory of Bose-Einstein condensation

TL;DR

This paper clarifies common misconceptions in Bose-Einstein condensation theory, emphasizing the importance of symmetry breaking, ensemble equivalence, and correcting misunderstandings about approximations and fluctuations.

Contribution

It elucidates fundamental points in Bose-Einstein condensation theory, correcting misconceptions and clarifying the roles of symmetry, fluctuations, and approximations.

Findings

Global gauge symmetry breaking is necessary and sufficient for BEC.

No grand canonical catastrophe occurs in stable systems.

Symmetry-broken averages are essential for correct analysis.

Abstract

Despite the long history of the theory of Bose-Einstein condensation, there exist till nowadays some slippery points that are often misunderstood and result in confusion. The report touches some of these points, explaining the following: Global gauge symmetry breaking is the necessary and sufficient condition for the existence of Bose-Einstein condensate. There is no any ``grand canonical catastrophe". The stability of the ideal Bose gas depends on the spatial dimensionality and the shape of a trap. Symmetry-broken averages cannot be neglected. The so-called ``Popov approximation", ascribed to Popov, suggesting to neglect anomalous averages, is neither an approximation nor has anything to do with Popov. There are no thermodynamically anomalous fluctuations in stable equilibrium systems. Representative statistical ensembles are equivalent.