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Non-Fock Ground States in the Translation-Invariant Nelson Model Revisited Non-Perturbatively

David Hasler, Benjamin Hinrichs, Oliver Siebert

TL;DR

The paper addresses the infrared problem in the translation-invariant Nelson model by constructing dressed one-particle states in a non-Fock representation and proving their existence non-perturbatively for almost all total momenta. The authors develop a general compactness framework in Fock spaces via a Fréchet–Kolmogorov–Riesz approach, and establish a dressing transform that yields norm-resolvent convergence to an infrared-renormalized operator. They show that, in a physically relevant momentum region, the infrared-renormalized Hamiltonian admits a ground state, even in the infrared-critical case where the original Hamiltonian lacks one, while in the infrared-regular case the transformed and original models are unitarily equivalent. The work also provides analytic perturbation results for the mass shell in the massive regime, infrared bounds for the resolvent, and a rigorous passage from massive to massless bosons, culminating in a robust non-perturbative construction of non-Fock ground states with implications for scattering theory and infrared renormalization. Overall, the results validate the dressed-electron picture non-perturbatively across non-relativistic and semi-relativistic Nelson-type models and establish a solid theoretical foundation for infrared-renormalized ground states and their momentum-continuity properties.

Abstract

The Nelson model, describing a quantum mechanical particle linearly coupled to a bosonic field, exhibits the infrared problem in the sense that no ground state exists at arbitrary total momentum. However, passing to a non-Fock representation, one can prove the existence of so-called dressed one-particle states. In this article, we give a simple non-perturbative proof for the existence of such one-particle states at arbitrary coupling strength and for almost all total momenta in a physically motivated momentum region. Our results hold both for the non- and the semi-relativistic Nelson model.

Non-Fock Ground States in the Translation-Invariant Nelson Model Revisited Non-Perturbatively

TL;DR

The paper addresses the infrared problem in the translation-invariant Nelson model by constructing dressed one-particle states in a non-Fock representation and proving their existence non-perturbatively for almost all total momenta. The authors develop a general compactness framework in Fock spaces via a Fréchet–Kolmogorov–Riesz approach, and establish a dressing transform that yields norm-resolvent convergence to an infrared-renormalized operator. They show that, in a physically relevant momentum region, the infrared-renormalized Hamiltonian admits a ground state, even in the infrared-critical case where the original Hamiltonian lacks one, while in the infrared-regular case the transformed and original models are unitarily equivalent. The work also provides analytic perturbation results for the mass shell in the massive regime, infrared bounds for the resolvent, and a rigorous passage from massive to massless bosons, culminating in a robust non-perturbative construction of non-Fock ground states with implications for scattering theory and infrared renormalization. Overall, the results validate the dressed-electron picture non-perturbatively across non-relativistic and semi-relativistic Nelson-type models and establish a solid theoretical foundation for infrared-renormalized ground states and their momentum-continuity properties.

Abstract

The Nelson model, describing a quantum mechanical particle linearly coupled to a bosonic field, exhibits the infrared problem in the sense that no ground state exists at arbitrary total momentum. However, passing to a non-Fock representation, one can prove the existence of so-called dressed one-particle states. In this article, we give a simple non-perturbative proof for the existence of such one-particle states at arbitrary coupling strength and for almost all total momenta in a physically motivated momentum region. Our results hold both for the non- and the semi-relativistic Nelson model.
Paper Structure (20 sections, 39 theorems, 141 equations)

This paper contains 20 sections, 39 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Fock Space Notions
  4. General Nelson Models
  5. Main Results
  6. Compactness Theorems for Fock Spaces
  7. The Transformed Operators
  8. Analysis of Ground States
  9. Simple Properties of the Mass Shell
  10. Uniqueness of Ground States
  11. The Massive Case
  12. Existence of Ground States
  13. Analytic Perturbation Theory for the Total Momentum
  14. Infrared Bounds
  15. Approximation by Massive Models
  16. ...and 5 more sections

Key Result

Lemma 2.1

The operator $H(\Theta,\omega,v,P)$ given on the domain ${\mathcal{D}}(\Theta(P_{{\mathsf f}}))\cap {\mathcal{D}}({\mathsf d}\Gamma(\omega))$ by def:general is selfadjoint and lower-semibounded for any $P\in{\mathbb R}^d$.

Theorems & Definitions (93)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • proof
  • ...and 83 more