Hamiltonians Without Ultraviolet Divergence for Quantum Field Theories

TL;DR

This paper introduces a new class of Hamiltonians for quantum field theories, called IBC Hamiltonians, which are mathematically well-defined and ultraviolet finite without the need for renormalization or particle smearing.

Contribution

It presents a novel approach to defining quantum field theory Hamiltonians using interior-boundary conditions, avoiding ultraviolet divergences and renormalization procedures.

Findings

IBC Hamiltonians are mathematically well-defined and ultraviolet finite.

They agree with renormalized Hamiltonians when both are known.

Explicit examples of IBC Hamiltonians are provided.

Abstract

We propose a way of defining Hamiltonians for quantum field theories without any renormalization procedure. The resulting Hamiltonians, called IBC Hamiltonians, are mathematically well-defined (and in particular, ultraviolet finite) without an ultraviolet cut-off such as smearing out the particles over a nonzero radius; rather, the particles are assigned radius zero. These Hamiltonians agree with those obtained through renormalization whenever both are known to exist. We describe explicit examples of IBC Hamiltonians. Their definition, which is best expressed in the particle-position representation of the wave function, involves a kind of boundary condition on the wave function, which we call an interior-boundary condition (IBC). The relevant configuration space is one of a variable number of particles, and the relevant boundary consists of the configurations with two or more particles at the same location. The IBC relates the value (or derivative) of the wave function at a boundary point to the value of the wave function at an interior point (here, in a sector of configuration space corresponding to a lesser number of particles).