Mean Field Dynamics of Boson Stars

TL;DR

This paper proves that the dynamics of a large system of relativistic bosons with Coulomb interactions can be effectively described by a nonlinear Hartree equation, providing a rigorous link between quantum many-body systems and boson star models.

Contribution

It establishes the mean field limit for relativistic bosons with Coulomb interactions, deriving the nonlinear Hartree equation from the N-body Schrödinger equation.

Findings

The one-particle density evolution converges to the relativistic nonlinear Hartree equation as N approaches infinity.

The work rigorously connects quantum many-body dynamics with classical boson star models.

Provides mathematical validation for the use of the Hartree equation in boson star physics.

Abstract

We consider a quantum mechanical system of N bosons with relativistic dispersion interacting through a mean field Coulomb potential (attractive or repulsive). We choose the initial wave function to describe a condensate, where the N bosons are all in the same one-particle state. Starting from the N-body Schroedinger equation, we prove that, in the limit N goes to infinity, the time evolution of the one-particle density is governed by the relativistic nonlinear Hartree equation. This equation is used to describe the dynamics of boson stars (Chandrasekhar theory). The corresponding static problem was rigorously solved by Lieb and Yau.