Absence of ground state for the Nelson model on static space-times

TL;DR

This paper proves that the Nelson model on static space-times with variable boson mass tending to zero at infinity lacks a ground state, using path space techniques to analyze the infrared problem.

Contribution

It demonstrates the absence of a ground state for the Nelson model on static space-times with position-dependent boson mass decaying at infinity, extending previous results to more general geometries.

Findings

No ground state exists if boson mass decays faster than |x|^{-μ} with μ>1.

Path space techniques effectively analyze the infrared problem in curved space-times.

Results apply to Nelson models with variable coefficients and static metrics.

Abstract

We consider the Nelson model on some static space-times and investigate the problem of absence of a ground state. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We investigate the absence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass $m(x)$ tends to $0$ at spatial infinity. Using path space techniques, we show that if $m(x)\leq C |x|^{-\mu}$ at infinity for some $C>0$ and $\mu>1$ then the Nelson Hamiltonian has no ground state.