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 $$|\varphi |^4$$ | φ | 4 -interaction. We start with the linearized 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 Fredenhagen and Lindner (Commun Math Phys 332:895, 2014) and Drago et al. (Ann Henri Poincaré 18:807, 2017). Here, the principle of perturbative agreement plays a crucial role. The apparent conflict with Goldstone’s theorem is resolved by the fact that the linearized 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.