On the Vacuum Polarization Density Caused by an External Field

Abstract

Abstract. We consider an external potential, −λϕ, due to one or more nuclei. Following the Dirac picture such a potential polarizes the vacuum. The polarization density, ρ λ vac , 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, e 1 (λ), of the corresponding Dirac operator stays in the gap of the essential spectrum, the integral over the density ρ λ vac vanishes. In other words the vacuum stays neutral. But as soon as e 1 (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.

Existence of a Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation

“…This procedure has recently been clarified by Hainzl and Siedentop in [29]. Some interesting features of ρ αϕ ren , in the case of strong external fields, were obtained by Hainzl in [27]. We do not want to give a precise definition of ρ αϕ ren here and we refer the reader to [29,27].…”

“…Some interesting features of ρ αϕ ren , in the case of strong external fields, were obtained by Hainzl in [27]. We do not want to give a precise definition of ρ αϕ ren here and we refer the reader to [29,27]. It would be tempting, instead of using a cut-off, to renormalize ρ Q a priori in equation (6), as in [29].…”

The mean‐field approximation in quantum electrodynamics: The no‐photon case

“…where C is the charge conjugation operator acting on H L Λ , defined by Cf := iβα 2 f , and Ψ(f ) has been introduced in (24). It is then easy to see that the following relations hold…”

Semi-Classical Dirac Vacuum Polarisation in a Scalar Field

“…Using various methods of regularisation, these have been studied rigorously by many authors, especially from a variational point of view [3,4,1,19,14,15,17,16]. The charge density of the vacuum was derived in a perturbative regime by Hainzl [13].…”