Algebraic approach to Bose-Einstein Condensation in relativistic Quantum Field Theory. Spontaneous symmetry breaking and the Goldstone Theorem

TL;DR

This paper develops an algebraic framework for describing Bose-Einstein condensates at finite temperature in relativistic quantum field theory, addressing symmetry breaking and Goldstone's theorem.

Contribution

It introduces a method to construct finite-temperature condensate states using perturbation theory and resolves conflicts with Goldstone's theorem.

Findings

Finite-temperature condensate states constructed for relativistic scalar fields.

Linearised theory breaks U(1) symmetry, affecting Goldstone's theorem.

Perturbative approach avoids infrared divergences in symmetry breaking.

Abstract

We construct states describing Bose Einstein condensates at finite temperature for a relativistic massive complex scalar field with $|\varphi|^4$-interaction. We start with the linearised theory over a classical condensate and construct interacting fields by perturbation theory. Using the concept of thermal masses, equilibrium states at finite temperature can be constructed by the methods developed in arXiv:1306.6519 and arXiv:1502.02705. Here, the principle of perturbative agreement plays a crucial role. The apparent conflict with Goldstone's Theorem is resolved by the fact that the linearised theory breaks the $U(1)$ symmetry, hence the theorem applies only to the full series but not to the truncations at finite order which therefore can be free of infrared divergences.