Vacuum polarization near boundaries

TL;DR

This paper investigates how boundary conditions affect vacuum polarization in two-dimensional charged scalar fields, revealing boundary-dependent screening effects and a divergence at critical field strength for Neumann conditions.

Contribution

It demonstrates the impact of Dirichlet and Neumann boundary conditions on vacuum polarization and identifies a divergence phenomenon under Neumann conditions at critical fields.

Findings

Dirichlet conditions lead to zero charge density at the boundary.

Neumann conditions produce maximum charge density at the boundary.

Vacuum polarization diverges at a critical field strength for Neumann boundary conditions.

Abstract

We study the effect of boundary conditions on vacuum polarization for charged scalar fields in two space-time dimensions. We find that both Dirichlet and Neumann boundary conditions lead to screening. In the Dirichlet case, the vacuum polarization charge density vanishes at the boundary, whereas it attains its maximum there for Neumann boundary conditions. At a critical field strength, the vacuum polarization diverges for Neumann boundary conditions, an effect due to the instability of the lowest energy mode in the presence of the external field.