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.

我们构造了一个状态，该状态描述了玻色–爱因斯坦在有限温度下具有\（| \ varphi | ^ 4 \）相互作用的相对论性复标量场的凝聚态。我们从经典冷凝物的线性化理论开始，并通过扰动理论构造相互作用场。使用热质量的概念，可以通过Fredenhagen和Lindner（Commun Math Phys 332：895，2014）和Drago等人开发的方法构造有限温度下的平衡态。（Ann HenriPoincaré18：807，2017）。在这里，摄动一致的原则起着至关重要的作用。通过线性化理论打破了U的事实，解决了与Goldstone定理的明显矛盾。（1）对称性；因此，该定理仅适用于全数列，而不适用于有限阶的截断，因此无红外散度。