Schwinger mechanism revisited
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Abstract
The vacuum persistence probability, ${P}_{\mathrm{vac}}(t)$, for a system of charged fermions in a fixed, external, and spatially homogeneous electric field, was derived long ago by Schwinger; $w\ensuremath{\equiv}\ensuremath{-}\mathrm{log}[{P}_{\mathrm{vac}}(t)]/Vt$ has often been identified as the rate at which fermion-antifermion pairs are produced per unit volume due to the electric field. In this paper, we separately compute exact expressions for both $w$ and for the rate of fermion-antifermion pair production per unit volume, $\ensuremath{\Gamma}$, and show that they differ. While $w$ is given by the standard Schwinger mechanism result of $w=\frac{(qE{)}^{2}}{4{\ensuremath{\pi}}^{3}}\ensuremath{\sum}_{n=1}^{\ensuremath{\infty}}\frac{1}{{n}^{2}}\mathrm{exp}(\ensuremath{-}\frac{n\ensuremath{\pi}{m}^{2}}{qE})$, the pair production rate, $\ensuremath{\Gamma}$, is just the first term of that series. Our calculation is done for a system with periodic boundary conditions in the ${A}_{0}=0$ gauge but the result should hold for any consistent set of boundary conditions. We discuss, the physical reason why the rates $w$ and $\ensuremath{\Gamma}$ differ.
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