Abstract
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.
Highlights
The possibility of creating matter in the presence of a strong external field is a remarkable feature of relativistic quantum fields
In Ref. [10] the system was studied using the instanton formalism, and it was argued that for small compactification radii the expression of the rate of pair production changes significantly from the one found in the noncompact case
Before investigating case of a compact spatial dimension, here we review the real-time analysis of the Schwinger effect in 1 þ 1 noncompact dimensions
Summary
The possibility of creating matter in the presence of a strong external field is a remarkable feature of relativistic quantum fields. Already in the noncompact case, a direct computation of the rate of pair production in the in-in formalism gives a nonvanishing result even in the decoupling limit where the mass m of the produced particles diverges This unphysical behavior is taken care of by using the formalism of the Bogolyubov coefficients, where twopoint functions are computed in terms of the normalordered ladder operators defined in the far future. The field is discharged by the effect of pair production in a stepwise fashion While this is inconsistent with the assumption of a time-translation invariant background that would call for a uniform rate, we will argue in Sec. VA that this behavior is not unusual for a process of particle creation.
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