On the Vacuum Polarization Density Caused by an External Field

We consider an external potential, $-\lambda \phi$, due to one or more nuclei. Following the Dirac picture such a potential polarizes the vacuum. The polarization density as derived in physics literature, after a well known renormalization procedure, depends decisively on the strength of $\lambda$. For small $\lambda$, more precisely as long as the lowest eigenvalue, $e_1(\lambda)$, of the corresponding Dirac operator stays in the gap of the essential spectrum, the integral over the density vanishes. In other words the vacuum stays neutral. But as soon as $e_1(\lambda)$ dives into the lower continuum the vacuum gets spontaneously charged with charge $ 2e$. Global charge conservation implies that two positrons were emitted out of the vacuum, this is, a large enough external potential can produce electron-positron pairs. We give a rigorous proof of that phenomenon.