Mean-field limits of particles in interaction with quantized radiation fields

TL;DR

This paper introduces a new method combining counting and coherent states to analyze mean-field limits of quantum particles interacting with quantized radiation fields, providing explicit error bounds and applications to models like Schrödinger-Klein-Gordon.

Contribution

It develops a novel technique for mean-field analysis of quantum systems with radiation fields, improving upon existing methods and enabling explicit error estimates.

Findings

Derived the Schrödinger-Klein-Gordon system from the Nelson model.

Provided explicit convergence rates for the reduced density matrix.

Applicable to complex models like the Pauli-Fierz Hamiltonian.

Abstract

We report on a simple strategy to treat mean-field limits of quantum mechanical systems in which a large number of particles weakly couple to a second-quantized radiation field. Extending the method of counting, introduced in [Lett. Math. Phys. 97, 151-164], with ideas inspired by [http://www.mathematik.uni-muenchen.de/%7Ebohmmech/theses/Matulevicius_Vytautas_MA.pdf] and [J. Math. Phys. 54(1), 012303] leads to a technique that can be seen as a combination of the method of counting and the coherent state approach. It is similar to the coherent state approach but might be slightly better suited to systems in which a fixed number of particles couple to radiation. The strategy is effective and provides explicit error bounds. As an instructional example we derive the Schr\"odinger-Klein-Gordon system of equations from the Nelson model with ultraviolet cutoff. Furthermore, we derive explicit bounds on the rate of convergence of the one-particle reduced density matrix of the non-relativistic particles in Sobolev norm. More complicated models like the Pauli-Fierz Hamiltonian can be treated in a similar manner [arXiv:1609.01545].