Amended Proof of Pitaevskii-Stringari's Theorem

TL;DR

This paper provides a rigorous, approximation-free proof of Pitaevskii-Stringari's theorem, arguing against BEC as a phase transition via SSB in 1D systems at zero temperature, using quantum field theory and thermodynamic limits.

Contribution

It offers a mathematically rigorous proof avoiding Bogoliubov approximation and clarifies the non-existence of BEC as a phase transition in 1D systems at zero temperature.

Findings

No Bose-Einstein Condensation as a phase transition in 1D at zero temperature.

The proof is based on quantum field theory without Bogoliubov approximation.

Thermodynamic limit is crucial for the argument.

Abstract

An amended proof of Pitaevskii-Stringari's theorem is given in mathematically coherent manner without resorting to the Bogoliubov approximation. This approach is based on the orthodox quantum field theory which rigorously maintains the canonical commutation relations. Moreover, we make a sound argument by taking the thermodynamic limit which the authors of the original papers did not refer to. We conclude that there is no Bose Einstein Condensation (BEC) as a phase transition based on the Spontaneous Symmetry Breaking (SSB) of the global $U(1)$ gauge symmetry in flat one dimensional geometry even absolute zero temperature.