On a Direct Description of Pseudorelativistic Nelson Hamiltonians

TL;DR

This paper extends interior-boundary conditions to define Hamiltonians for pseudorelativistic Nelson models, providing explicit descriptions of their domains and connecting to renormalisation, avoiding ultraviolet cutoffs.

Contribution

It introduces a novel application of IBCs to pseudorelativistic particles, explicitly constructs Hamiltonians, and links the approach to traditional renormalisation techniques.

Findings

Constructed Hamiltonians for pseudorelativistic Nelson models.

Provided explicit domain descriptions for these Hamiltonians.

Connected IBC approach with established renormalisation methods.

Abstract

Abstract interior-boundary conditions (IBC's) allow for the direct description of the domain and the action of Hamiltonians for a certain class of ultraviolet-divergent models in Quantum Field Theory. The method was recently applied to models where nonrelativistic scalar particles are linearly coupled to a quantised field, the best known of which is the Nelson model. Since this approach avoids the use of ultraviolet-cutoffs, there is no need for a renormalisation procedure. Here, we extend the IBC method to pseudorelativistic scalar particles that interact with a real bosonic field. We construct the Hamiltonians for such models via abstract boundary conditions, describing their action explicitly. In addition, we obtain a detailed characterisation of their domain and make the connection to renormalisation techniques. As an example, we apply the method to two relativistic variants of Nelson's model, which have been renormalised for the first time by J. P. Eckmann and A. D. Sloan in 1970 and 1974, respectively.