Ultraviolet Renormalization of the Nelson Hamiltonian through Functional Integration

TL;DR

This paper demonstrates ultraviolet renormalization of the Nelson Hamiltonian via functional integration, establishing a self-adjoint operator in the zero cutoff limit and deriving an effective Yukawa interaction.

Contribution

It introduces a method to perform ultraviolet renormalization of the Nelson Hamiltonian using Feynman-Kac representation, identifying the divergent term and establishing a weak coupling limit.

Findings

Existence of a self-adjoint renormalized Hamiltonian after subtracting a divergent term.

Derivation of an effective Yukawa potential in the weak coupling limit.

Validation of the renormalization procedure through functional integration techniques.

Abstract

Starting from the N-particle Nelson Hamiltonian defined by imposing an ultraviolet cutoff, we perform ultraviolet renormalization by showing that in the zero cutoff limit a self-adjoint operator exists after a logarithmically divergent term is subtracted from the original Hamiltonian. We obtain this term as the diagonal part of a pair interaction appearing in the density of a Gibbs measure derived from the Feynman-Kac representation of the Hamiltonian. Also, we show existence of a weak coupling limit of the renormalized Hamiltonian and derive an effective Yukawa interaction potential between the particles.