Wigner Measures Approach to the Classical Limit of the Nelson Model: Convergence of Dynamics and Ground State Energy

We consider the classical limit of the Nelson model, a system of stable nucleons interacting with a meson field. We prove convergence of the quantum dynamics towards the evolution of the coupled Klein-Gordon-Schrodinger equation. Also, we show that the ground state energy level of nucleons, when is large and the meson field approaches its classical value, is given by the infimum of the classical energy functional at a fixed density of particles. Our study relies on a recently elaborated approach for mean field theory and uses Wigner measures.