Wigner measures approach to the classical limit of the Nelson model: Convergence of dynamics and ground state energy

Zied Ammari (1) , Marco Falconi (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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-Schrödinger equation. Also, we show that the ground state energy level of $N$ nucleons, when $N$ 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.

Mots clés: Classicla limit; Wigner measures; Nelson model; Klein-Gordon-Schrödinger equation; Ground state energy

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Zied Ammari, Marco Falconi. Wigner measures approach to the classical limit of the Nelson model: Convergence of dynamics and ground state energy. Journal of Statistical Physics, 2014, 157 (2), pp.330-362. ⟨10.1007/s10955-014-1079-7⟩. ⟨hal-00958725⟩