Classical field theory limit of many-body quantum Gibbs states in 2D and 3D

TL;DR

This paper rigorously derives nonlinear Gibbs measures from many-body quantum systems in 2D and 3D, linking quantum Gibbs states to classical field theories near Bose-Einstein condensation.

Contribution

It establishes the convergence of quantum Gibbs states to classical nonlinear Gibbs measures in higher dimensions, including renormalization procedures.

Findings

Quantum Gibbs states converge to classical Gibbs measures.

The classical measure requires Wick renormalization due to singularities.

A new entropy estimate and quantum variance control method were developed.

Abstract

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 Schr{\"o}dinger-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.