Vacuum polarization in a one-dimensional effective quantum-electrodynamics model

TL;DR

This paper investigates a simplified one-dimensional effective QED model of a hydrogen-like atom to understand vacuum polarization effects, calculating key quantities like the Lamb shift and exploring basis convergence issues.

Contribution

It provides a detailed mathematical formulation of a 1D effective QED model, computes vacuum polarization and Lamb shift, and discusses basis convergence improvements, aiding future 3D QED atom/molecule studies.

Findings

Vacuum-polarization density is difficult to converge in finite basis.

Proposed methods improve convergence of vacuum polarization calculations.

Results extend previous work on Lamb shift in simplified QED models.

Abstract

With the aim of progressing toward a practical implementation of an effective quantum-electrodynamics (QED) theory of atoms and molecules, which includes the effects of vacuum polarization through the creation of virtual electron-positron pairs but without the explicit photon degrees of freedom, we study a one-dimensional effective QED model of the hydrogen-like atom with delta-potential interactions. This model resembles the three-dimensional effective QED theory with Coulomb interactions while being substantially simpler. We provide some mathematical details about the definition of this model, calculate the vacuum-polarization density, and the Lamb-type shift of the bound-state energy, correcting and extending results of previous works. We also study the approximation of the model in a finite plane-wave basis, and in particular we discuss the basis convergence of the bound-state energy and eigenfunction, of the vacuum-polarization density, and of the Lamb-type shift of the bound-state energy. We highlight the difficulty of converging the vacuum-polarization density in a finite basis and we propose a way to improve it. The present work could give hints on how to perform similar calculations for the three-dimensional effective QED theory of atoms and molecules.