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Feynman-Kac formula and asymptotic behavior of the minimal energy for the relativistic Nelson model in two spatial dimensions

Benjamin Hinrichs, Oliver Matte

TL;DR

The paper develops a rigorous Feynman–Kac framework for the renormalized relativistic Nelson model in two dimensions, replacing Brownian motion by appropriate Lévy processes to model the relativistic particle paths. It proves a ultraviolet-cutoff FK formula and establishes norm-resolvent convergence to a renormalized Hamiltonian, while providing ergodicity and weighted $L^p\to L^q$ bounds for the associated semigroup under relativistic Kato-type potentials. It derives sharp two-sided bounds and asymptotics for the minimal energy in translation-invariant settings and analyzes the leading behavior in three regimes: large particle number, strong matter-field coupling, and vanishing boson mass. The results combine probabilistic representations, exponential moment controls of the complex action, and variational/trial-state arguments to illuminate the spectral properties and ground-state behavior of the model with ultraviolet renormalization.

Abstract

We consider the renormalized relativistic Nelson model in two spatial dimensions for a finite number of spinless, relativistic quantum mechanical matter particles in interaction with a massive scalar quantized radiation field. We find a Feynman-Kac formula for the corresponding semigroup and discuss some implications such as ergodicity and weighted $L^p$ to $L^q$ bounds, for external potentials that are Kato decomposable in the suitable relativistic sense. Furthermore, our analysis entails upper and lower bounds on the minimal energy for all values of the involved physical parameters when the Pauli principle for the matter particles is ignored. In the translation invariant case (no external potential) these bounds permit to compute the leading asymptotics of the minimal energy in the three regimes where the number of matter particles goes to infinity, the coupling constant for the matter-radiation interaction goes to infinity and the boson mass goes to zero.

Feynman-Kac formula and asymptotic behavior of the minimal energy for the relativistic Nelson model in two spatial dimensions

TL;DR

The paper develops a rigorous Feynman–Kac framework for the renormalized relativistic Nelson model in two dimensions, replacing Brownian motion by appropriate Lévy processes to model the relativistic particle paths. It proves a ultraviolet-cutoff FK formula and establishes norm-resolvent convergence to a renormalized Hamiltonian, while providing ergodicity and weighted bounds for the associated semigroup under relativistic Kato-type potentials. It derives sharp two-sided bounds and asymptotics for the minimal energy in translation-invariant settings and analyzes the leading behavior in three regimes: large particle number, strong matter-field coupling, and vanishing boson mass. The results combine probabilistic representations, exponential moment controls of the complex action, and variational/trial-state arguments to illuminate the spectral properties and ground-state behavior of the model with ultraviolet renormalization.

Abstract

We consider the renormalized relativistic Nelson model in two spatial dimensions for a finite number of spinless, relativistic quantum mechanical matter particles in interaction with a massive scalar quantized radiation field. We find a Feynman-Kac formula for the corresponding semigroup and discuss some implications such as ergodicity and weighted to bounds, for external potentials that are Kato decomposable in the suitable relativistic sense. Furthermore, our analysis entails upper and lower bounds on the minimal energy for all values of the involved physical parameters when the Pauli principle for the matter particles is ignored. In the translation invariant case (no external potential) these bounds permit to compute the leading asymptotics of the minimal energy in the three regimes where the number of matter particles goes to infinity, the coupling constant for the matter-radiation interaction goes to infinity and the boson mass goes to zero.
Paper Structure (50 sections, 52 theorems, 272 equations)

This paper contains 50 sections, 52 theorems, 272 equations.

Table of Contents

  1. Introduction
  2. General introduction
  3. The relativistic Nelson model in two spatial dimensions
  4. Main results
  5. Important operators on Fock space
  6. Lévy processes for particle paths
  7. Lévy processes for relativistic particles
  8. Lévy measures
  9. Infinitesimal generator
  10. Associated Poisson point processes of jumps and stochastic integrals
  11. The Feynman--Kac formula
  12. Relativistic Kato class for $N$ particles
  13. The processes appearing in the creation/annihilation terms
  14. The complex action
  15. Feynman--Kac integrand and formula
  16. ...and 35 more sections

Key Result

Theorem 2.1

Let $\Psi\in L^2(\mathbb{R}^{2N},\mathcal{F})$ and $t>0$. Then $\mathrm{e}^{-tH}\Psi$ has a unique continuous representative which for every $x\in\mathbb{R}^{2N}$ is given by

Theorems & Definitions (110)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 100 more