Mean-Field Limits of Particles in Interaction with Quantized Radiation Fields

2018/01/01 by Nikolai Leopold, Peter Pickl · 10 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Boson #Cold Atom Physics and Bose-Einstein Condensates #Convergence (economics) #Cutoff #Density matrix #Field (mathematics) #Massless particle #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Norm (philosophy) #Physics #Pure mathematics #Quantum #Quantum field theory #Quantum mechanics #Scalar field #Sobolev space #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:35Q40 #msc:81Q05 #msc:82C10

paper · pdf · doi:10.1007/978-3-030-01602-9_9

published in Springer proceedings in mathematics & statistics, 185-214 (Springer International Publishing) · A final version (improved presentation and many more references) will appear in "Macroscopic Limits of Quantum Systems", Springer Proceedings in Mathematics and Statistics

openalex publication_date 2018/01/01 · arxiv created 2018/06/29 · arxiv updated 2021/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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].

Citations

• Derivation of the Maxwell-Schrödinger Equations from the Pauli-Fierz Hamiltonian
• The classical limit for quantum mechanical correlation functions
• Coherent and Incoherent States of the Radiation Field
• Mean-field quantum dynamics for a mixture of Bose–Einstein condensates
• Derivation of the Time Dependent Gross–Pitaevskii Equation in Two Dimensions
• On the dynamics of polarons in the strong-coupling limit
• Derivation of the Hartree equation for compound Bose gases in the mean field limit