Scaling limits with a removal of ultraviolet cutoffs for semi-relativistic particles system coupled to a scalar Bose field

TL;DR

This paper investigates the scaling limits of a semi-relativistic particle system coupled to a scalar Bose field, deriving effective potentials like Yukawa and Coulomb through a renormalized Hamiltonian approach.

Contribution

It introduces a method to obtain effective potentials in semi-relativistic particle systems by analyzing the strong resolvent limit of a renormalized Hamiltonian.

Findings

Derivation of Yukawa potential as an effective interaction.

Derivation of Coulomb potential as an effective interaction.

Application of an abstract scaling limit theory to quantum field models.

Abstract

The system of semi-relativistic particles coupled to a scalar Bose field is considered. A scaled total Hamiltonian for the system is a self-adjoint operator on a tensor product of a square-integrable function space and a boson Fock space. We consider the strong resolvent limit of a renormalized Hamiltonian, which is defined by subtracting a divergent term from the scaled total Hamiltonian. Applying an abstract scaling limit theory and a unitary transformation, we derive the Yukawa potential and Coulomb potential as effective potentials.