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Derivation of the Maxwell-Schrödinger and Vlasov-Maxwell Equations from Non-Relativistic QED

Nikolai Leopold

Abstract

We study the spinless Pauli-Fierz Hamiltonian in a semiclassical mean-field limit of many fermions. For appropriate initial conditions, we prove, in the trace norm topology of reduced density matrices, that the many-body quantum state converges to a tensor product of a semiclassically structured Slater determinant and a coherent photon state. These evolve according to a fermionic variant of the Maxwell-Schrödinger equations. By combining this result with [arXiv:2308.16074] through a suitable regularization of the initial data, we further show that, in the limit of large particle number, the dynamics of the Pauli-Fierz Hamiltonian can be approximately described by the non-relativistic Vlasov--Maxwell system for extended charges.

Derivation of the Maxwell-Schrödinger and Vlasov-Maxwell Equations from Non-Relativistic QED

Abstract

We study the spinless Pauli-Fierz Hamiltonian in a semiclassical mean-field limit of many fermions. For appropriate initial conditions, we prove, in the trace norm topology of reduced density matrices, that the many-body quantum state converges to a tensor product of a semiclassically structured Slater determinant and a coherent photon state. These evolve according to a fermionic variant of the Maxwell-Schrödinger equations. By combining this result with [arXiv:2308.16074] through a suitable regularization of the initial data, we further show that, in the limit of large particle number, the dynamics of the Pauli-Fierz Hamiltonian can be approximately described by the non-relativistic Vlasov--Maxwell system for extended charges.

Paper Structure

This paper contains 28 sections, 23 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. The Pauli--Fierz Hamiltonian
  3. Scaling and Assumptions on the Electron Density
  4. The (fermionic) Maxwell--Schrödinger equations
  5. The non-relativistic Vlasov--Maxwell equations
  6. Notation
  7. Main results
  8. Convegence to the Vlasov--Maxwell system with extended charges
  9. Comparison with the Literature
  10. Proof of the main results
  11. Proof of Theorem \ref{['theorem:main theorem']}
  12. Proof of Theorem \ref{['theorem:derivation of the Vlasov-Maxwell equations']}
  13. Properties of the Maxwell--Schrödinger Solutions
  14. Proofs of Inequality \ref{['eq:bound for the h-1 norm of alpha']}
  15. Proof of Lemma \ref{['lemma: semiclassical structure']}
  16. ...and 13 more sections

Key Result

Proposition II.1

Let $\kappa$ satisfy Assumption assumption:cutoff function. For all $(p_0, \alpha_0) \in \textfrak{S}_{+}^{2,1} (L^2(\mathbb{R}^3)) \times \mathfrak{h}_{1/2} \cap \dot{\mathfrak{h}}_{-1/2}$ the Cauchy problem for the Maxwell--Schrödinger system eq:Maxwell-Schroedinger equations associated with $(p_0 and mass of the system are conserved, i.e. Under the additional assumption $\alpha_0 \in \mathfrak

Theorems & Definitions (54)

  • Proposition II.1
  • proof
  • Theorem II.2
  • Remark II.3
  • Remark II.4
  • Remark II.5
  • Remark II.6
  • Proposition II.7
  • Theorem II.8
  • Remark II.9
  • ...and 44 more