Classical limit of the Nelson model with cut off

TL;DR

This paper investigates the classical limit of the Nelson model with a cutoff, demonstrating convergence of quantum observables to classical solutions and revealing a surprising quantum residue in transition amplitudes.

Contribution

It provides a rigorous analysis of the classical limit for the Nelson model with cutoff, including quantum fluctuations and transition amplitude behavior.

Findings

Quantum observables converge to classical equations

Quantum fluctuations evolve around classical solutions

Normal ordered products yield an average of classical solutions

Abstract

In this paper we analyze the classical limit of the Nelson model with cut off, when both non-relativistic and relativistic particles number goes to infinity. We prove convergence of quantum observables to the solutions of classical equations, and find the evolution of quantum fluctuations around the classical solution. Furthermore we analyze the convergence of transition amplitudes of normal ordered products of creation and annihilation operators between different types of initial states. In particular the limit of normal ordered products between states with a fixed number of both relativistic and non-relativistic particles yields an unexpected quantum residue: instead of the product of classical solutions we obtain an average of the product of solutions corresponding to varying initial conditions.