Existence of a Unique Local Solution to the Many-body Maxwell-Schr\"odinger Initial Value Problem

TL;DR

This paper proves the local existence and uniqueness of solutions for a coupled quantum-classical many-body system involving particles and electromagnetic fields, advancing understanding of quantum electrodynamics models.

Contribution

It establishes the first rigorous proof of local well-posedness for the many-body Maxwell-Schrödinger system with a contraction mapping approach.

Findings

Existence of a unique local solution is proven.

The solution is constructed via contraction mapping.

The model couples quantum particles with a classical electromagnetic field.

Abstract

We study the many-body problem of charged particles interacting with their self-generated electromagnetic field. We model the dynamics of the particles by the many-body Maxwell-Schr\"odinger system, where the particles are treated quantum mechanically and the electromagnetic field is a classical quantity. We prove the existence of a unique local in time solution to this nonlinear initial value problem using a contraction mapping argument.