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We prove that Gibbs measures of nonlinear Schr\"odinger equations arise ashigh-temperature limits of thermal states in many-body quantum mechanics. Ourresults hold for defocusing interactions in dimensions $d =1,2,3$. Themany-body quantum thermal states that we consider are the grand canonicalensemble for $d = 1$ and an appropriate modification of the grand canonicalensemble for $d =2,3$. In dimensions $d =2,3$, the Gibbs measures are supportedon singular distributions, and a renormalization of the chemical potential isnecessary. On the many-body quantum side, the need for renormalization ismanifested by a rapid growth of the number of particles. We relate the originalmany-body quantum problem to a renormalized version obtained by solving acounterterm problem. Our proof is based on ideas from field theory, using aperturbative expansion in the interaction, organized by using a diagrammaticrepresentation, and on Borel resummation of the resulting series.

Gibbs measures of nonlinear Schr\"odinger equations as limits of many-body quantum states in dimensions $d \leq 3$