Mean-field Dynamics for the Nelson Model with Fermions

TL;DR

This paper studies the mean-field and semiclassical limits of a fermionic Nelson model, proving convergence of many-body states to a product of a Slater determinant and a coherent state evolving under coupled equations.

Contribution

It establishes the convergence of fermionic many-body dynamics to a fermionic Schroedinger-Klein-Gordon system in the mean-field and semiclassical limits.

Findings

Convergence of reduced density matrices to a tensor product state.

Derivation of fermionic Schroedinger-Klein-Gordon equations.

Validation of mean-field approximation for fermions in the Nelson model.

Abstract

We consider the Nelson model with ultraviolet cutoff, which describes the interaction between non-relativistic particles and 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. In this case, the limit is known to be also a semiclassical limit. We prove convergence in terms of reduced density matrices of the many-body state to a tensor product of a Slater determinant with semiclassical structure and a coherent state, which evolve according to a fermionic version of the Schroedinger-Klein-Gordon equations.