Absence of Ground States in the Renormalized Massless Translation-Invariant Nelson Model

TL;DR

This paper proves that the fiber operators of the renormalized massless Nelson model lack ground states for any non-zero coupling and total momentum, extending recent methods to models with degenerate ground states.

Contribution

It provides a new generalization of methods to prove the absence of ground states, applicable to models with degenerate eigenspaces.

Findings

Fiber operators do not have ground states for any non-zero coupling and momentum.

The method can be applied to models with degenerate ground state eigenspaces.

Results hold for the renormalized fiber operators in the Nelson model.

Abstract

We consider a model for a massive uncharged non-relativistic particle interacting with a massless bosonic field, widely referred to as the Nelson model. It is well known, that an ultraviolet renormalized Hamilton operator exists in this case. Further, due to translation-invariance, it decomposes into fiber operators. In this paper, we treat the renormalized fiber operators. We give a description of the operator and form domains and prove that the fiber operators do not have a ground state. Our results hold for any non-zero coupling constant and arbitrary total momentum. Our proof for the absence of ground states is a new generalization of methods recently applied to related models. A major enhancement we provide, is that the method can be applied to models with degenerate ground state eigenspaces.