Boundary Conditions that Remove Certain Ultraviolet Divergences

TL;DR

This paper introduces interior-boundary conditions (IBCs) as a novel method to eliminate ultraviolet divergences in quantum field theory Hamiltonians, avoiding traditional renormalization procedures.

Contribution

It presents IBCs as a new boundary condition approach that directly defines Hamiltonians free of UV divergences, offering an alternative to renormalization.

Findings

IBC method removes UV divergences without renormalization.

IBCs relate wave function values at configurations connected by particle creation/annihilation.

Provides a new framework for defining quantum Hamiltonians with improved mathematical properties.

Abstract

In quantum field theory, Hamiltonians contain particle creation and annihilation terms that are usually ultraviolet (UV) divergent. It is well known that these divergences can sometimes be removed by adding counter-terms and taking limits in which an UV cut-off tends to infinity. Here, I review a novel way of removing UV divergences: by imposing a kind of boundary condition on the wave function. These conditions, called interior-boundary conditions (IBCs), relate the values of the wave function at two configurations linked by the creation or annihilation of a particle. They allow for a direct definition of the Hamiltonian without renormalization or limiting procedures. In the last section, I review another boundary condition that serves for determining the probability distribution of detection times and places on a timelike 3-surface.