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

Abstract

According to Dirac's ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator D 0 . In the presence of an external field, these virtual particles react and the vacuum becomes polarized.In this paper, following Chaix and Iracane (J. Phys. B, 22, 3791-3814, 1989), we consider the Bogoliubov-Dirac-Fock model, which is derived from no-photon QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is solution of a self-consistent equation.We show the existence of a unique minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum.

Construction of the Pauli–Villars-Regulated Dirac Vacuum in Electromagnetic Fields

“…In this paper we have considered a second-quantized Dirac field which only interacts with a classical electromagnetic field. There is another way to arrive at exactly the same Lagrangian action (2.28), which is closer to our previous works [22,23,25,24,18,19]. We start with Coulombgauge QED with quantized transverse photons.…”

“…For A ≡ 0, the electrostatic stability of the free Dirac vacuum was pointed out first by Chaix, Iracane and Lions [4,5] and proved later in full generality in [1,22,23]. It is possible to include the exchange term and even establish the global stability of the free Dirac vacuum [22,23,24].…”

“…The function M is positive and satisfies the uniform estimate 22) where Λ was defined previously in (2.14).…”

“…The latter was computed in the Physics literature in [28] and our task will be to estimate it. The higher order terms are then bounded in a rather crude way, following techniques of [22]. The main difficulty in our work is to verify that the Pauli-Villars conditions (2.12) induce the appropriate cancellations in the few first order terms, and to estimate them using the L 2 -norm of the electromagnetic fields and nothing else.…”

“…It is possible to include the exchange term and even establish the global stability of the free Dirac vacuum [22,23,24]. Dealing with magnetic fields is more complicated and, so far, we are only able to prove local stability, using the Pauli-Villars regularization.…”