Mean-field limits of particles in interaction with quantized radiation fields
arXiv:1806.10843 · doi:10.1007/978-3-030-01602-9_9
Abstract
We report on a simple strategy to treat mean-field limits of quantum mechanical systems in which a large number of particles weakly couple to a second-quantized radiation field. Extending the method of counting, introduced in [Lett. Math. Phys. 97, 151-164], with ideas inspired by [http://www.mathematik.uni-muenchen.de/%7Ebohmmech/theses/Matulevicius_Vytautas_MA.pdf] and [J. Math. Phys. 54(1), 012303] leads to a technique that can be seen as a combination of the method of counting and the coherent state approach. It is similar to the coherent state approach but might be slightly better suited to systems in which a fixed number of particles couple to radiation. The strategy is effective and provides explicit error bounds. As an instructional example we derive the Schrödinger-Klein-Gordon system of equations from the Nelson model with ultraviolet cutoff. Furthermore, we derive explicit bounds on the rate of convergence of the one-particle reduced density matrix of the non-relativistic particles in Sobolev norm. More complicated models like the Pauli-Fierz Hamiltonian can be treated in a similar manner [arXiv:1609.01545].
arXiv commentA final version (improved presentation and many more references) will appear in "Macroscopic Limits of Quantum Systems", Springer Proceedings in Mathematics and Statistics
References in corpus (23)
- Mean-Field Dynamics: Singular Potentials and Rate of Convergence
- Erratum: Exact Ground State Energy of the Strong-Coupling Polaron
- Rate of Convergence Towards Hartree Dynamics
- Quantitative Derivation of the Gross-Pitaevskii Equation
- Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction
- Quantum many-body fluctuations around nonlinear Schrödinger dynamics
- Bogoliubov corrections and trace norm convergence for the Hartree dynamics
- Dynamical Collapse of Boson Stars
- Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions
- Classical limit of the Nelson model with cut off
- Derivation of the Hartree equation for compound Bose gases in the mean field limit
- Mean-field quantum dynamics for a mixture of Bose-Einstein condensates
- Dynamics of a strongly coupled polaron
- Multivariate Central Limit Theorem in Quantum Dynamics
- Derivation of an effective evolution equation for a strongly coupled polaron
- On the dynamics of polarons in the strong-coupling limit
- A simple proof of convergence to the Hartree dynamics in Sobolev trace norms
- Derivation of the Time Dependent Two Dimensional Focusing NLS Equation
- Derivation of the Maxwell-Schrödinger Equations from the Pauli-Fierz Hamiltonian
- Mean-field quantum dynamics with magnetic fields
- Effective N-body dynamics for the massless Nelson model and adiabatic decoupling without spectral gap
- Bohr's correspondence principle in quantum field theory and classical renormalization scheme: the Nelson model
- The NLS limit for bosons in a quantum waveguide
Cited by in corpus (6)
- Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions
- Derivation of the Maxwell-Schrödinger Equations from the Pauli-Fierz Hamiltonian
- Derivation of the Landau-Pekar equations in a many-body mean-field limit
- Mean-field Dynamics for the Nelson Model with Fermions
- Bogoliubov Dynamics and Higher-order Corrections for the Regularized Nelson Model
- Asymptotic analysis of the weakly interacting Bose gas: A collection of recent results and applications