We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schrödinger-type classical field theory, in terms of partition functions and reduced density matrices. The Gibbs measure thus describes the behavior of the infinite Bose gas at criticality, that is, close to the phase transition to a Bose–Einstein condensate. The Gibbs measure is concentrated on singular distributions and has to be appropriately renormalized, while the quantum system is well defined without any renormalization. By tuning a single real parameter (the chemical potential), we obtain a counter-term for the diverging repulsive interactions which provides the desired Wick renormalization of the limit classical theory. The proof relies on a new estimate on the entropy relative to quasi-free states and a novel method to control quantum variances.

中文翻译：

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2D 和 3D 中多体量子吉布斯态的经典场论极限

我们从热平衡中的多体量子系统开始，在两个和三个空间维度中提供了非线性吉布斯测量的严格推导。更准确地说，我们证明了大型玻色子量子系统的大正则吉布斯态在分配函数和约简密度矩阵方面收敛于非线性薛定谔型经典场论的吉布斯测度。因此，吉布斯测量描述了无限玻色气体在临界状态下的行为，即接近玻色-爱因斯坦凝聚的相变。吉布斯测度集中在奇异分布上，必须适当地重新归一化，而量子系统是明确定义的，没有任何重新归一化。通过调整单个真实参数（化学势），我们获得了发散排斥相互作用的反项，它提供了极限经典理论所需的 Wick 重整化。该证明依赖于相对于准自由态的熵的新估计和控制量子方差的新方法。