We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions d=1,2,3. The quantum many-body thermal states that we consider are the grand canonical ensemble for d=1 and an appropriate modification of the grand canonical ensemble for d=2,3. In dimensions d=2,3, these Gibbs measures are supported on singular distributions, and a renormalization of the interaction by Wick ordering is necessary. On the quantum many-body side, the need for renormalization is manifested by a rapid growth of the number of particles. We also relate the original quantum many-body problem to its renormalized version, by analysing the associated counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized using a diagrammatic representation, and on Borel resummation of the resulting series.
