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The weak coupling limit of the Pauli-Fierz model

Fumio Hiroshima

TL;DR

This work provides a rigorous treatment of the weak coupling limit for the Pauli-Fierz Hamiltonian in non-relativistic QED, and analyzes how coupling to the quantized field renormalizes the particle dynamics through an effective mass. By employing a fiber decomposition in total momentum, Feynman-Kac representations, and generating operator techniques based on generalized Hermite polynomials, the authors derive precise limiting objects as the coupling scale $\kappa\to\infty$, including a limiting semigroup and resolvent expressed through an effective Hamiltonian $H_{\rm eff}$ with mass $m_*$. A central finding is that the ground-state energy satisfies $E_\kappa(0)=\kappa^2\mathcal{E}$ and that, under infrared integrability conditions, the dispersion obeys $E_\kappa(p)-E_\kappa(0) \to p^2/(2m_*)$, with the limit of the full dynamics captured by projecting onto the field ground state via a unitary transform $\mathscr{U}$. The results extend the dipole-approximation insights to the full Pauli-Fierz model, establishing a rigorous link between WCL and mass renormalization and offering a robust framework for understanding radiative corrections in NRQED. The methods have potential implications for dispersion relations, ground-state properties, and transport in quantum electrodynamical settings. $

Abstract

We investigate the weak coupling limit of the Pauli- Fierz Hamiltonian within a mathematically rigorous framework. Furthermore, we establish the asymptotic behavior of the effective mass in this regime.

The weak coupling limit of the Pauli-Fierz model

TL;DR

This work provides a rigorous treatment of the weak coupling limit for the Pauli-Fierz Hamiltonian in non-relativistic QED, and analyzes how coupling to the quantized field renormalizes the particle dynamics through an effective mass. By employing a fiber decomposition in total momentum, Feynman-Kac representations, and generating operator techniques based on generalized Hermite polynomials, the authors derive precise limiting objects as the coupling scale , including a limiting semigroup and resolvent expressed through an effective Hamiltonian with mass . A central finding is that the ground-state energy satisfies and that, under infrared integrability conditions, the dispersion obeys , with the limit of the full dynamics captured by projecting onto the field ground state via a unitary transform . The results extend the dipole-approximation insights to the full Pauli-Fierz model, establishing a rigorous link between WCL and mass renormalization and offering a robust framework for understanding radiative corrections in NRQED. The methods have potential implications for dispersion relations, ground-state properties, and transport in quantum electrodynamical settings. $

Abstract

We investigate the weak coupling limit of the Pauli- Fierz Hamiltonian within a mathematically rigorous framework. Furthermore, we establish the asymptotic behavior of the effective mass in this regime.
Paper Structure (23 sections, 32 theorems, 224 equations)

This paper contains 23 sections, 32 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. The Pauli-Fierz Hamiltonian
  3. Boson Fock space
  4. Second quantizations
  5. The Pauli-Fierz Hamiltonian
  6. The weak coupling limit and the singular coupling limit
  7. Main theorems
  8. Scaling of the Pauli-Fierz Hamiltonian and its difficulties
  9. Effective mass of the Pauli-Fierz Hamiltonian
  10. Summary of main results
  11. The Pauli-Fierz Hamiltonian with total momentum $p\in{\fam\msyfam R}^d$
  12. Dipole approximation
  13. Generating operators
  14. Ground state energy of $H_{\rm dip,\kappa}(p)$ by path measure
  15. Uniform lower bound of $H_\kappa(p)-\kappa^2\mathcal{E}$ and the WCL of $H_\kappa(p)$
  16. ...and 8 more sections

Key Result

Proposition 1.4

Let $V=0$. Suppose Assumption a2. Then $H(p)$ is self-adjoint on ${\rm D}(H_{\rm f})\cap {\rm D}({P_{\rm f}}^2)$ and it follows that

Theorems & Definitions (37)

  • Proposition 1.4
  • Theorem 1.5: The WCL of effective mass
  • Theorem 1.6: The WCL of the Pauli-Fierz Hamiltonian
  • Lemma 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 2.6: Feynman-Kac formula
  • Remark 2.7
  • ...and 27 more