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

Abstract

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.

Concentration of cylindrical Wigner measures

“…Motivation. This technical note is motivated by some problems we encountered in the study of the semiclassical properties of bosonic quantum systems [1,11]. Let us discuss briefly a concrete case.…”

Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit

“…We conclude this preliminary discussion by recalling a result about the convergence of states in the Fock space Γ sym pHq originally proven in [AN08, AN09, AN11, AN15a] (see also [AF14,AF16] for further applications). The detailed version of the result is stated in Sect.…”

“…ωz α 2 H , α 2 ď 2, as well as the convergence of the corresponding evaluation of dΓpωq is proved in [AN15a, Lemma 3.13]. The convergence of the expectation of the field operator also follows along the same guidelines, an interested reader might consult, e.g., [AF14].…”

On the dynamics of the mean-field polaron in the high-frequency limit

“…We remark that the Nelson model, which is similar to the Fröhlich model, in a classical limit leads to the Schrödinger-Klein-Gordon system [AF14]. The designation of (1)-(2) as mean-field polaron in the title of our paper is adopted from [BNAS00], where these equation, apparently, have been studied for the first time.…”