Adiabatic Klein-Gordon dynamics for the renormalized Nelson model.

Abstract

We study the renormalized Nelson model in a semiclassical regime where the field becomes classical while the particle remains quantum. The degree of classicality is measured by a small parameter $\epsilon \ll 1$ . In this scaling the particle evolves on microscopic times, whereas the field exhibits nontrivial dynamics only on the macroscopic scale $t=O\left({\epsilon }^{-2}\right)$ . The natural semiclassical model is the coupled Schrödinger-Klein-Gordon (SKG) system, which encodes the time-scale separation through an explicit $\epsilon$ -dependence. Based on this scale separation in SKG, we apply the adiabatic principle to derive a new PDE for the classical field, the $\epsilon$ -free adiabatic Klein-Gordon (aKG) equation, where the field is driven by the instantaneous ground state of the particle. Our main result is a norm approximation of the Nelson dynamics by the aKG solution corrected by a quasi-free fluctuation dynamics around the classical field, generated by a renormalized Bogoliubov-Nelson Hamiltonian. As a corollary, we obtain convergence of the reduced one-body densities for both subsystems, where the fluctuation correction vanishes, thereby justifying aKG as a semiclassical Born-Oppenheimer type approximation of the renormalized Nelson model.