Gibbs measures for nonlinear Schrodinger equations and many-body quantum mechanics

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The nonlinear Schrödinger equation (NLS) is a nonlinear PDE that arises in the dynamics of many-body quantum systems. An instance of this correspondence can be seen in the phenomenon of Bose-Einstein condensation. The solution of the NLS corresponds to the Bose-Einstein condensate. This in general gives us a correspondence between a nonlinear PDE and a quantum problem. The latter is linear, albeit non-commutative. It is posed on the bosonic Fock space, in which the number of particles is not fixed.The NLS possesses a Hamiltonian structure, that allows us to (at least formally) define a Gibbs measure, which is invariant under the flow. The construction of such a measure dates from the constructive quantum field theory in the 1970s (the work of Nelson, Glimm-Jaffe, Simon), and later work of Lebowitz-Rose-Speer and McKean-Vaninsky. Its invariance was first rigorously shown in the pioneering work of Bourgain in the 1990s. Today, Gibbs measures are used as a fundamental tool in the study of probabilistic low-regularity well-posedness theory. This is due to the fact that Gibbs measures are typically supported on low-regularity Sobolev spaces.The main goal of my proposal is to understand how Gibbs measures arise in the correspondence between the NLS and many-body quantum theory. In the quantum problem, one works with quantum Gibbs states. These are equilibrium states on Fock space corresponding to the many-body Hamiltonian at a fixed (positive) temperature. By using the (classical) Gibbs measure, one can similarly construct the classical Gibbs states. The correspondence that we want to verify is the convergence of correlation functions of the quantum Gibbs state to those of the classical Gibbs state in an appropriately defined mean-field limit.When working in higher dimensions, one should take special care to eliminate the divergences that arise in the problem. This is done by applying the procedure of Wick-ordering. This procedure is well-known in the classical theory and it has a clear quantum analogue.Earlier results on this problem were obtained by Lewin-Nam-Rougerie, by the author in collaboration with Fröhlich-Knowles-Schlein, and by the author himself. The methods used to study the problem came from analysis, but also from probability, and statistical mechanics. There is still a substantial gap with what is known in this problem and what is known in the classical theory. Namely, in the classical theory it is possible to construct Gibbs measures for the NLS with very singular interaction potentials. A major challenge in the quantum problem is the lack of commutativity. In this proposal, I aim to tackle this problem. The techniques come from different aspects of analysis, probability, and statistical mechanics. One goal would be to understand connections between techniques from the analysis of the NLS (which are primarily based on harmonic analysis) and the methods of quantum field theory.