Mean-field limit and Semiclassical Expansion of a Quantum Particle System

TL;DR

This paper analyzes the semiclassical expansion of a quantum many-particle system with mean-field interactions, showing that each term converges to the Hartree equation as the number of particles grows large.

Contribution

It establishes the correspondence between the semiclassical expansion terms of the quantum system and the Hartree equation in the mean-field limit using the Wigner formalism.

Findings

Each term of the semiclassical expansion converges to the Hartree equation as N→∞.

The Wigner formalism effectively captures the classical limit of the quantum system.

The propagation of chaos is consistent with the semiclassical limit.

Abstract

We consider a quantum system constituted by $N$ identical particles interacting by means of a mean-field Hamiltonian. It is well known that, in the limit $N\to\infty$, the one-particle state obeys to the Hartree equation. Moreover, propagation of chaos holds. In this paper, we take care of the $\hbar$ dependence by considering the semiclassical expansion of the $N$-particle system. We prove that each term of the expansion agrees, in the limit $N\to\infty$, with the corresponding one associated with the Hartree equation. We work in the classical phase space by using the Wigner formalism, which seems to be the most appropriate for the present problem.