Surprises of Klein paradox

Abstract

We consider second quantized $1+1$ dimensional Dirac fermions with mass $m$ in background electric fields. This theory mimics the following situations. We consider two parallel capacitors with opposite directions of the electric field $E$ in them. The potential energy difference in one of them is equal to $V_0 > 2m$ and in the other to $V+V_0$, with $0<V<2m$. According to general physical understanding of the Schwinger effect, each capacitor separately will create pairs and there will be oppositely directed currents, which will not be equal to each other. The total current will not vanish. In this paper we show that this is too naive an observation: the limiting expectation value of the current operator, as $t\to +\infty$, is actually zero, if the above form of the potential with $0<V<2m$ is instantaneously turned on. The non-vanishing current is obtained only if $V>2m$. We also consider the same theory in the presence of the infinite wall and a deep well $V_0>2m$. We show that in such a potential the limiting expectation value of the current is zero in the Gaussian approximation. We discuss the relation of these observations to the Schwinger effet and to the decay of supercritical nuclei.