# Bogoliubov quasi-averages: spontaneous symmetry breaking and algebra of fluctuations

## TL;DR

This paper advocates the use of Bogoliubov quasi-averages to analyze phase transitions, spontaneous symmetry breaking, and quantum fluctuations in Bose-Einstein condensation and related quantum systems.

## Contribution

It demonstrates the effectiveness of Bogoliubov quasi-averages in studying SSB, BEC, and quantum fluctuations, providing solutions to existing problems and clarifying their physical relevance.

## Key findings

- Quasi-averages reliably define physical quantities in BEC.
- Solution to the Lieb, Seiringer, Yngvason problem in BEC.
- Quasi-averages effectively describe algebra of quantum fluctuations.

## Abstract

The paper advocates the Bogoliubov method of quasi-averages for quantum systems. First, we elucidate its applications to study the phase transitions with Spontaneous Symmetry Breaking (SSB). To this aim we consider example of Bose-Einstein condensation (BEC) in continuous systems. Our analysis of different type of generalised condensations demonstrates that the only physically reliable quantities are those that defined by Bogoliubov quasi-averages. In this connection we also give a solution of the problem posed by Lieb, Seiringer and Yngvason in [SY07]. Second, using the scaled Bogoliubov method of quasi-averages and taking the structural quantum phase transition as a basic example, we scrutinise a relation between SSB and the critical quantum fluctuations. Our analysis shows that again the quasi-averages give an adequate tool for description of the algebra of critical quantum fluctuation operators in the both commutative and noncommutative cases.