On the Vacuum Polarization Density Caused by an External Field

TL;DR

This paper rigorously proves that a sufficiently strong external potential can cause vacuum polarization leading to spontaneous electron-positron pair creation, with the vacuum becoming charged when the potential's strength exceeds a critical threshold.

Contribution

It provides a rigorous mathematical proof of vacuum polarization and pair creation phenomena induced by external fields in the Dirac model.

Findings

Vacuum remains neutral for small potential strength.

When the potential strength exceeds a critical value, the vacuum becomes charged with 2e.

Large external potentials can produce electron-positron pairs.

Abstract

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.