Abstract.
We consider an external potential, −λ φ, due to one or more nuclei. Following the Dirac picture such a potential polarizes the vacuum. The polarization density, \(\rho _{{\text{vac}}}^\lambda ,\) as derived in physics literature, after a well-known renormalization procedure, depends decisively on the strength of λ. For small λ, more precisely as long as the lowest eigenvalue, e1(λ), of the corresponding Dirac operator stays in the gap of the essential spectrum, the integral over the density \(\rho _{{\text{vac}}}^\lambda ,\) vanishes. In other words the vacuum stays neutral. But as soon as e1(λ) 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.
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Communicated by Rafael D. Benguria
submitted 18/12/03, accepted 08/04/04
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Hainzl, C. On the Vacuum Polarization Density Caused by an External Field. Ann. Henri Poincaré 5, 1137–1157 (2004). https://doi.org/10.1007/s00023-004-0194-4
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DOI: https://doi.org/10.1007/s00023-004-0194-4