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

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