LAUSR

We compute the discharging rate of a uniform electric field due to Schwinger pair production in ($1+1$)-dimensional scalar electrodynamics with a compact dimension of radius $R$. Our calculation is performed in real time, using the in-in formalism. For large compactification radii, $R\ensuremath{\rightarrow}\ensuremath{\infty}$, we recover the standard noncompact space result. However, other ranges of values of $R$ and of the mass $m$ of the charged scalar give rise to a richer set of behaviors. For $R\ensuremath{\gtrsim}\mathcal{O}(1/m)$ with $m$ large enough, the electric field oscillates in time, whereas for $R\ensuremath{\rightarrow}0$ it decreases in steps. We discuss the origin of these results.