Infrared problem for the Nelson model on static space-times

TL;DR

This paper studies the Nelson model on static space-times with variable coefficients, focusing on the existence of ground states and ultraviolet cutoff removal, especially addressing the infrared problem when boson mass vanishes at infinity.

Contribution

It extends the Nelson model to static space-times with variable coefficients and analyzes ground state existence under infrared conditions.

Findings

Established conditions for ground state existence in variable coefficient Nelson models.

Analyzed the impact of vanishing boson mass at infinity on the model's spectral properties.

Provided insights into the ultraviolet cutoff removal in curved space-time settings.

Abstract

We consider the Nelson model with variable coefficients and investigate the problem of existence of a ground state and the removal of the ultraviolet cutoff. 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. A physical example is obtained by quantizing the Klein-Gordon equation on a static space-time coupled with a non-relativistic particle. We investigate the existence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass tends to 0 at infinity.