Mean-field Dynamics for the Nelson Model with Fermions

Nikolai Leopold, Sören Petrat

Published 2018-07-18Version 1

The Nelson model (with ultraviolet cutoff) describes a quantum system of non-relativistic particles coupled to a positive or zero mass quantized scalar field. We take the non-relativistic particles to obey Fermi statistics and discuss the time evolution in a mean-field limit of many fermions which is coupled to a semiclassical limit. At time zero, we assume that the bosons of the radiation field are close to a coherent state and that the state of the fermions is close to a Slater determinant with a certain semiclassical structure. We prove that the many-body state approximately retains its Slater determinant and semiclassical structure at later times and that its time evolution can be approximated by the fermionic Schroedinger-Klein-Gordon equations. We prove the convergence for reduced densities with explicit rates and for all semiclassical times.