Vacuum polarization in $d+{1/2}$ dimensions

TL;DR

This paper analyzes the one-loop vacuum polarization in spinor QED within half-integer dimensions with boundary conditions, revealing boundary-induced effects like localized transverse currents in 2+1 dimensions.

Contribution

It provides an exact expression for the induced current in half-integer dimensions with boundary conditions, highlighting boundary-induced parity breaking effects.

Findings

Exact expression for vacuum polarization in half-integer dimensions.

Discovery of boundary-localized transverse current in 2+1 dimensions.

Identification of boundary-induced parity breaking effects.

Abstract

We study the main properties of the one-loop vacuum polarization function ($\Pi_{\alpha \beta}$) for spinor $QED$ in `$d + {1/2}$ dimensions', i.e., with fields defined on ${\mathcal M} \subset {\mathbb R}^{d+1}$ such that ${\mathcal M} = \{(x_0,...,x_d) | x_{d}\geq 0 \}$, with bag-like boundary conditions on the boundary $\partial{\mathcal M} = \{(x_0,...,x_d) | x_{d}= 0 \}$. We obtain an exact expression for the induced current due to an external constant electric field normal to the boundary. We show that, for the particular case of 2+1 dimensions, there is a transverse component for the induced current, which is localized on a region close to $\partial{\mathcal M}$. This current is a parity breaking effect purely due to the boundary.