# Particle Creation at a Point Source by Means of Interior-Boundary Conditions

## TL;DR

This paper introduces a rigorous method for defining quantum Hamiltonians with particle creation at point sources using interior-boundary conditions, avoiding ultraviolet regularization and ensuring self-adjointness.

## Contribution

It proves that Hamiltonians based on interior-boundary conditions for particle creation can be rigorously constructed without regularization and are self-adjoint, connecting to renormalization procedures.

## Key findings

- Hamiltonian defined via IBC is self-adjoint.
- No ultraviolet regularization needed for the model.
- Renormalization with cutoff removal yields the IBC Hamiltonian.

## Abstract

We consider a way of defining quantum Hamiltonians involving particle creation and annihilation based on an interior-boundary condition (IBC) on the wave function, where the wave function is the particle-position representation of a vector in Fock space, and the IBC relates (essentially) the values of the wave function at any two configurations that differ only by the creation of a particle. Here we prove, for a model of particle creation at one or more point sources using the Laplace operator as the free Hamiltonian, that a Hamiltonian can indeed be rigorously defined in this way without the need for any ultraviolet regularization, and that it is self-adjoint. We prove further that introducing an ultraviolet cut-off (thus smearing out particles over a positive radius) and applying a certain known renormalization procedure (taking the limit of removing the cut-off while subtracting a constant that tends to infinity) yields, up to addition of a finite constant, the Hamiltonian defined by the IBC.