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

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.