A Lower Bound on the Renormalized Nelson Model

TL;DR

This paper establishes explicit lower bounds for the ground-state energy of the renormalized Nelson model, depending on the coupling constant and particle number, providing the first such bounds with clear parameter dependence.

Contribution

It presents the first fully explicit lower bounds for the Nelson model's ground-state energy with reasonable dependence on parameters, including the coupling constant and particle number.

Findings

Derived bounds of the form -Cα^2 N^3 log^2(α N) for large α

Established bounds of the form -Cα^2 N^3 log^2 N for small α

Provided bounds for N=1 and N≥2 cases

Abstract

We provide explicit lower bounds for the ground-state energy of the renormalized Nelson model in terms of the coupling constant $\alpha$ and the number of particles $N$, uniform in the meson mass and valid even in the massless case. In particular, for any number of particles $N$ and large enough $\alpha$ we provide a bound of the form $-C\alpha^2 N^3\log^2(\alpha N)$, where $C$ is an explicit positive numerical constant; and if $\alpha$ is sufficiently small, we give one of the form $-C\alpha^2 N^3\log^2 N$ for $N \geq 2$, and $-C\alpha^2$ for $N = 1$. Whereas it is known that the renormalized Hamiltonian of the Nelson model is bounded below (as realized by E. Nelson) and implicit lower bounds have been given elsewhere (as in a recent work by Gubinelli, Hiroshima, and L\"{o}rinczi), ours seem to be the first fully explicit lower bounds with a reasonable dependence on $\alpha$ and $N$. We emphasize that the logarithmic term in the bounds above is probably an artifact in our calculations, since one would expect that the ground-state energy should behave as $-C\alpha^2 N^3$ for large $N$ or $\alpha$, as in the polaron model of H. Fr\"{o}hlich.