  We prove that Gibbs measures of nonlinear Schr\"odinger 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
many-body quantum 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 $d =2,3$, the Gibbs measures are supported
on singular distributions, and a renormalization of the chemical potential is
necessary. On the many-body quantum side, the need for renormalization is
manifested by a rapid growth of the number of particles. We relate the original
many-body quantum problem to a renormalized version obtained by solving a
counterterm problem. Our proof is based on ideas from field theory, using a
perturbative expansion in the interaction, organized by using a diagrammatic
representation, and on Borel resummation of the resulting series.
