On the semigroup generated by the renormalized Nelson Hamiltonian

TL;DR

This paper proves that the exponential of the renormalized Nelson Hamiltonian is positivity improving for all momenta and positive times, providing insights into its spectral properties in quantum field theory.

Contribution

It establishes the positivity improving property of the semigroup generated by the renormalized Nelson Hamiltonian at fixed momentum.

Findings

Positivity improving property holds for all momenta and times.

The result applies in the Fock representation.

Provides a foundation for spectral analysis of the model.

Abstract

Let us consider the renormalized Nelson model at a fixed total momentum $P$: $H_{\mathrm{ren}}(P)$; The Hamiltonian $H_{\mathrm{ren}}(P)$ is defined through an infinite energy renormalization. We prove that $e^{-\beta H_{\mathrm{ren}}(P)}$ is positivity improving for all $P\in \mathbb{R}^3$ and $\beta >0$ in the Fock representation.