Validity of One-Dimensional QED for a System with Spatial Symmetry

TL;DR

This paper evaluates the accuracy of one-dimensional quantum electrodynamics in predicting forces and charges in symmetric three-dimensional systems, finding it correctly predicts Coulomb forces but slightly underestimates quantum corrections.

Contribution

It demonstrates that a 1D QED approach can accurately predict Coulomb forces in symmetric 3D systems, with minor discrepancies in quantum corrections.

Findings

1D QED correctly predicts Coulomb force law for symmetric systems.

Quantum corrections are slightly underestimated in 1D compared to 3D.

The approach simplifies calculations while maintaining reasonable accuracy.

Abstract

We examine the accuracy of an intrinsically one-dimensional quantum electrodynamics to predict accurately the forces and charges of a three-dimensional system that has a high degree of symmetry and therefore depends effectively only on a single coordinate. As a test case we analyze two charged capacitor plates that are infinitely extended along two coordinate directions. Using the lowest-order fine structure correction to the photon propagator we compute the vacuum's induced charge polarization density and show that the force between the charged plates is increased. Although a one-dimensional theory cannot take the transverse character of the virtual (force-mediating) photons into account, nevertheless it predicts, in lowest order of the fine-structure constant, the Coulomb force law between the plates correctly. However, the quantum correction to the classical result is slightly different between the 1d and 3d theories with the polarization charge density induced from the vacuum underestimated by the 1d approach.