# Mean-Field Dynamics for the Nelson Model with Fermions

@article{Leopold2018MeanFieldDF,
title={Mean-Field Dynamics for the Nelson Model with Fermions},
author={Nikolai Leopold and Soren Petrat},
journal={Annales Henri Poincar{\'e}},
year={2018},
volume={20},
pages={3471 - 3508}
}
• Published 18 July 2018
• Physics
• Annales Henri Poincaré
We consider the Nelson model with ultraviolet cutoff, which describes the interaction between non-relativistic particles and a positive or zero mass quantized scalar field. We take the non-relativistic particles to obey Fermi statistics and discuss the time evolution in a mean-field limit of many fermions. In this case, the limit is known to be also a semiclassical limit. We prove convergence in terms of reduced density matrices of the many-body state to a tensor product of a Slater determinant…
11 Citations
• Physics
Archive for Rational Mechanics and Analysis
• 2021
The Fröhlich Hamiltonian is considered in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field and it is shown that the dynamics of the system is approximately described by the Landau–Pekar equations.
• Mathematics
Reviews in Mathematical Physics
• 2022
We study the time evolution of the Nelson model in a mean-field limit in which N nonrelativistic bosons weakly couple (w.r.t. the particle number) to a positive or zero mass quantized scalar field.
We study the time evolution of the Fr¨ohlich Hamiltonian in a mean-ﬁeld limit in which many particles weakly couple to the quantized phonon ﬁeld. Assuming that the particles are initially in a
• Physics
Pure and Applied Analysis
• 2021
We consider the Frohlich Hamiltonian with large coupling constant $\alpha$. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding
• Physics
• 2022
. The purpose of this article is twofold: • On one hand, we rigorously derive the Newton–Maxwell equation in the Coulomb gauge from ﬁrst principles of quantum electrodynamics in agreement with the
• Physics, Mathematics
Journal of Mathematical Physics
• 2023
We slightly extend prior results about the derivation of the Maxwell–Schrödinger equations from the bosonic Pauli–Fierz Hamiltonian. More concretely, we show that the findings from Leopold and Pickl
• Physics
Analysis & PDE
• 2021
We prove an adiabatic theorem for the Landau-Pekar equations. This allows us to derive new results on the accuracy of their use as effective equations for the time evolution generated by the Frohlich
• Mathematics
• 2019
We prove an optimal semiclassical bound on the trace norm of the following commutators [1(−∞,0](H~), x], [1(−∞,0](H~),−i~∇] and [1(−∞,0](H~), e], where H~ is a Schrödinger operator with a
• Mathematics
• 2019
We prove an optimal semiclassical bound on the trace norm of the following commutators $[\boldsymbol{1}_{(-\infty,0]}(H_\hbar),x]$, $[\boldsymbol{1}_{(-\infty,0]}(H_\hbar),-i\hbar\nabla]$ and
• Materials Science
Letters in Mathematical Physics
• 2020
We prove an optimal semiclassical bound on the trace norm of the following commutators [1(-∞,0](Hħ),x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}

## References

SHOWING 1-10 OF 56 REFERENCES

• Physics
• 2004
Abstract.We consider the Nelson model which describes a quantum system of nonrelativistic identical particles coupled to a possibly massless scalar Bose field through a Yukawa type interaction. We
• Physics, Mathematics
• 2014
The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems.
In this paper we analyze the classical limit of the Nelson model with cutoff, when both non-relativistic and relativistic particles number goes to infinity. We prove convergence of quantum
• Physics
• 2016
We study the many body Schrödinger evolution of weakly coupled fermions interacting through a Coulomb potential. We are interested in a joint mean field and semiclassical scaling, that emerges
• Physics
• 2013
We extend the derivation of the time-dependent Hartree-Fock equation recently obtained by Benedikter et al. [“Mean-field evolution of fermionic systems,” Commun. Math. Phys. (to be published)] to
• Physics, Mathematics
• 1979
We study the classical field limit of non relativistic many-boson theories in space dimensionn≧3, extending the results of a previous paper to more singular interactions. We prove the expected
We consider the many-body dynamics of fermions with Coulomb interaction in a mean-field scaling limit where the kinetic and potential energy are of the same order for large particle numbers. In the
Abstract. The Schrödinger equation for N particles interacting through effective pair potentials is derived from the massless Nelson model with ultraviolet cutoffs. We consider a scaling limit where
We demonstrate the mathematical existence of a meson theory with nonrelativistic nucleons. A system of Schrodinger particles is coupled to a quantized relativistic scalar field. If a cutoff is put on
We consider a system of N fermions in the mean-field regime interacting though an inverse power law potential $V(x)=1/|x|^{\alpha}$, for $\alpha\in(0,1]$. We prove the convergence of a solution of