Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions

TL;DR

This paper derives Feynman-Kac formulas for fiber Hamiltonians in a two-dimensional relativistic Nelson model, extending previous results to fixed total momenta and providing an alternative derivation for the full Hamiltonian.

Contribution

It presents new Feynman-Kac formulas for fiber Hamiltonians in a relativistic Nelson model, building on prior work and offering a self-contained derivation.

Findings

Derived Feynman-Kac formulas for fiber Hamiltonians at fixed momenta.

Provided an alternative derivation for the full translation invariant Hamiltonian.

Extended previous results to a two-dimensional relativistic setting.

Abstract

In this proceeding we consider a translation invariant Nelson type model in two spatial dimensions modeling a scalar relativistic particle in interaction with a massive radiation field. As is well-known, the corresponding Hamiltonian can be defined with the help of an energy renormalization. First, we review a Feynman-Kac formula for the semigroup generated by this Hamiltonian proven by the authors in a recent preprint (where several matter particles and exterior potentials are treated as well). After that, we employ a few technical key relations and estimates obtained in our preprint to present an otherwise self-contained derivation of new Feynman-Kac formulas for the fiber Hamiltonians attached to fixed total momenta of the translation invariant system. We conclude by inferring an alternative derivation of the Feynman-Kac formula for the full translation invariant Hamiltonian.