Derivation of the Maxwell-Schr\"odinger Equations from the Pauli-Fierz Hamiltonian

TL;DR

This paper derives the Maxwell-Schr"odinger equations as an effective description of a large quantum system of charged particles coupled to a quantized electromagnetic field, demonstrating condensation phenomena in the mean-field limit.

Contribution

It extends the 'method of counting' to analyze the dynamics of charged particle condensates interacting with radiation, deriving effective classical field equations.

Findings

Charged particles and photons exhibit condensation over time.

The Maxwell-Schr"odinger system accurately describes the large N limit dynamics.

The method of counting is successfully extended to this setting.

Abstract

We consider the spinless Pauli-Fierz Hamiltonian which describes a quantum system of non-relativistic identical particles coupled to the quantized electromagnetic field. We study the time evolution in a mean-field limit where the number $N$ of charged particles gets large while the coupling to the radiation field is rescaled by $1/\sqrt{N}$. At time zero we assume that almost all charged particles are in the same one-body state (a Bose-Einstein condensate) and we assume also the photons to be close to a coherent state. We show that at later times and in the limit $N \rightarrow \infty$ the charged particles as well as the photons exhibit condensation, with the time evolution approximately described by the Maxwell-Schr\"odinger system, which models the coupling of a non-relativistic particle to the classical electromagnetic field. Our result is obtained by an extension of the "method of counting", introduced by Pickl, to condensates of charged particles in interaction with their radiation field.