Partially classical limit of the Nelson model

TL;DR

This paper investigates the Nelson model's behavior when the number of Bose excitations becomes infinite and the coupling constant diminishes, revealing a transition from quantum to classical dynamics for the Bose field and its influence on particles.

Contribution

It provides a rigorous analysis of the partially classical limit of the Nelson model, connecting quantum field behavior to classical wave equations and particle dynamics.

Findings

Bose field converges to a classical wave or Klein-Gordon solution.

Quantum fluctuations satisfy wave or Klein-Gordon equations.

Particle evolution is governed by a potential from the classical field.

Abstract

We consider the Nelson model which describes a quantum system of nonrelativistic identical particles coupled to a possibly massless scalar Bose field through a Yukawa type interaction. We study the limiting behaviour of that model in a situation where the number of Bose excitations becomes infinite while the coupling constant tends to zero in a suitable sense. In that limit the appropriately rescaled Bose field converges to a classical solution of the free wave or Klein-Gordon equation depending on whether the mass of the field is zero or not, the quantum fluctuations around that solution satisfy the wave or Klein-Gordon equation and the evolution of the nonrelativistic particles is governed by a quantum dynamics with an external potential given by the previous classical solution.