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Algebraic approach to Bose-Einstein condensation in relativistic quantum field theory: spontaneous symmetry breaking and the Goldstone theorem. (English) Zbl 1461.81070

Summary: 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 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 K. Fredenhagen and F. Lindner [Commun. Math. Phys. 332, No. 3, 895–932 (2014; Zbl 1305.82012)] and N. Drago et al. [ibid. 18, No. 3, 807–868 (2017; Zbl 1362.81064)]. 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.

MSC:

81T10 Model quantum field theories
81V73 Bosonic systems in quantum theory
82B26 Phase transitions (general) in equilibrium statistical mechanics
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
82B30 Statistical thermodynamics
81R40 Symmetry breaking in quantum theory
46L60 Applications of selfadjoint operator algebras to physics
46L10 General theory of von Neumann algebras
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