# The Massless Nelson Hamiltonian and its Domain

## TL;DR

This paper applies a novel interior-boundary conditions method to define the massless Nelson Hamiltonian explicitly, avoiding renormalization and cutoffs, advancing the mathematical understanding of quantum field models.

## Contribution

It extends the interior-boundary conditions approach to the massless Nelson model, providing a rigorous Hamiltonian definition without renormalization or cutoffs.

## Key findings

- Explicit Hamiltonian and domain characterized
- No ultraviolet or infrared cutoffs needed
- Method successfully applied to massless case

## Abstract

In the theory of point interactions, one is given a formal expression for a quantum mechanical Hamiltonian. The interaction terms of the Hamiltonian are singular: they can not be rigorously defined as a perturbation (in the operator or form sense) of an unperturbed free operator. A similar situation occurs in Quantum Field Theory, where it is known as the ultraviolet problem. Recently, it was shown that some of the tools used in the context of point interactions can be adapted to solve the problem of directly defining a Hamiltonian for the Nelson model. This model provides a well studied example of a bosonic quantum field that is linearly coupled to nonrelativistic particles. The novel method employs so called abstract interior-boundary conditions to explicitly characterise the action and the domain of the Hamiltonian without the need for a renormalisation procedure. Here, for the first time, the method of interior-boundary conditions is applied to the massless Nelson model. Neither ultraviolet nor infrared cutoffs are needed.