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 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.
Highlights
We shall analyze the perturbative construction of a Bose–Einstein condensate for a relativistic charged scalar field theory at finite temperature
Bose–Einstein condensation is discussed in the realm of nonrelativistic quantum theories. (See, e.g., [58] and references therein.) Bose–Einstein condensation in the noninteracting case is the phenomenon that below a certain critical temperature the ground state becomes macroscopically populated
If the charge density is above this threshold, the chemical potential has to satisfy μ2 = m2; the states corresponding to pure phases have nonvanishing one-point functions which are related by the gauge symmetry. (See [10] for the concept of chemical potential in an algebraic formulation.)
Summary
We shall analyze the perturbative construction of a Bose–Einstein condensate for a relativistic charged scalar field theory at finite temperature. We shall analyze various possible equilibrium states at finite temperature for a complex scalar quantum field with mass m and chemical potential μ both for free and self-interacting theories. The traditional construction of nonzero temperature equilibrium states (KMS states) for perturbatively defined interacting quantum field theories suffers, even in the massive case, from spurious infrared divergences at higher-loop order (see [2,64]) It was recently shown how these divergences can be circumvented [30,53]. In the case of a massless scalar field, this problem could be circumvented by taking into account that the interaction produces at finite temperature a thermal mass If this term is included in the free theory, the correlations of the unperturbed state decay sufficiently fast [24]. We shall see that the symmetry is effectively broken in the background theory while it is recovered in the exact theory where it is spontaneously broken
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