Semiclassical equivalence of two white dwarf models as ground states of the relativistic Hartree-Fock and Vlasov-Poisson energies

TL;DR

This paper demonstrates that in the semi-classical limit, the ground states of relativistic Hartree-Fock energies for white dwarfs converge to those of the Vlasov-Poisson model, establishing a connection between quantum and classical models.

Contribution

It proves the convergence of relativistic Hartree-Fock ground states to Vlasov-Poisson ground states as Planck's constant approaches zero, linking quantum and classical white dwarf models.

Findings

Convergence of ground energies as Planck's constant tends to zero

Equivalence of quantum and classical ground states in the semi-classical limit

Validation of classical Vlasov-Poisson model as a limit of quantum Hartree-Fock model

Abstract

We are concerned with the semi-classical limit for ground states of the relativistic Hartree-Fock energies (HF) under a mass constraint, which are considered as the quantum mean-field model of white dwarfs \cite{LeLe}. In Jang and Seok \cite{JS}, fermionic ground states of the relativistic Vlasov-Poisson energy (VP) are constructed as a classical mean-field model of white dwarfs, and are shown to be equivalent to the classical Chandrasekhar model. In this paper, we prove that as the reduced Planck constant $\hbar$ goes to the zero, the $\hbar$-parameter family of the ground energies and states of (HF) converges to the fermionic ground energy and state of (VP) with the same mass constraint.