Avoiding Ultraviolet Divergence by Means of Interior-Boundary Conditions

TL;DR

This paper introduces a new method for defining Hamiltonians in quantum field theories using interior-boundary conditions, potentially avoiding the need for ultraviolet cut-offs or renormalization, thus offering a mathematically rigorous approach.

Contribution

It proposes a novel interior-boundary condition framework for Hamiltonians in QFTs that can be well-defined and self-adjoint without traditional renormalization.

Findings

Potentially well-defined, self-adjoint Hamiltonians for some QFTs

Avoids ultraviolet cut-offs and renormalization

Provides a new mathematical foundation for QFT Hamiltonians

Abstract

We describe here a novel way of defining Hamiltonians for quantum field theories (QFTs), based on the particle-position representation of the state vector and involving a condition on the state vector that we call an "interior-boundary condition." At least for some QFTs (and, we hope, for many), this approach leads to a well-defined, self-adjoint Hamiltonian without the need for an ultraviolet cut-off or renormalization.