Vacuum polarization in $d+1/2$ dimensions

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 $\mathcal{M}=\{(x_0,...,x_d)|x_d\ge 0\}$, 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.

11 pages, no figures