Adiabatic Klein–Gordon dynamics for the renormalized Nelson model

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 ε << 1. In this scaling the particle evolves on microscopic times, whereas the field exhibits nontrivial dynamics only on the macroscopic scale t = O(ε^−2). The natural semiclassical model is the coupled Schrödinger–Klein–Gordon (SKG) system, which encodes the time-scale separation through an explicit ε-dependence. Based on this scale separation in SKG, we apply the adiabatic principle to derive a new PDE for the classical field, the ε-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 aKGsolution 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.