Abstract:
The Bogoliubov-Dirac-Fock (BDF) model is the mean-field approximation of no-photon Quantum Electrodynamics. The present paper is devoted to the study of the minimization of the BDF energy functional under a charge constraint. An associated minimizer, if it exists, will usually represent the ground state of a system of $N$ electrons interacting with the Dirac sea, in an external electrostatic field generated by one or several fixed nuclei. We prove that such a minimizer exists when a binding (HVZ-type) condition holds. We also derive, study and interpret the equation satisfied by such a minimizer. Finally, we provide two regimes in which the binding condition is fulfilled, obtaining the existence of a minimizer in these cases. The first is the weak coupling regime for which the coupling constant $\alpha$ is small whereas $\alpha Z$ and the particle number $N$ are fixed. The second is the non-relativistic regime in which the speed of light tends to infinity (or equivalently $\alpha$ tends to zero) and $Z$, $N$ are fixed. We also prove that the electronic solution converges in the non-relativistic limit towards a Hartree-Fock ground state.

Abstract:
We study the Bogoliubov-Dirac-Fock model which is a mean-field approximation of QED. It allows to consider relativistic electrons interacting with the Dirac sea. We study the system of two electrons in the vacuum: it has been shown in a previous work that an electron alone can bind due to the vacuum polarization, under some technical assumptions. Here we prove the absence of binding for the system of two electrons: the response of the vacuum is not sufficient to counterbalance the repulsion of the electrons.

Abstract:
The Bogoliubov-Dirac-Fock (BDF) model is a no-photon, mean-field approxi- mation of quantum electrodynamics. It describes relativistic electrons in the Dirac sea. In this model, a state is fully characterized by its one-body density matrix, an infinite rank nonnegative operator. We prove the existence of the positronium, the bound state of an electron and a positron, represented by a critical point of the energy functional in the absence of external field. This state is interpreted as the ortho-positronium, where the two particles have parallel spins.

Abstract:
We study the mean-field approximation of Quantum Electrodynamics, by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normal-ordering or choice of bare electron/positron subspaces. Neglecting photons, we define properly this Hamiltonian in a finite box $[-L/2;L/2)^3$, with periodic boundary conditions and an ultraviolet cut-off $\Lambda$. We then study the limit of the ground state (i.e. the vacuum) energy and of the minimizers as $L$ goes to infinity, in the Hartree-Fock approximation. In case with no external field, we prove that the energy per volume converges and obtain in the limit a translation-invariant projector describing the free Hartree-Fock vacuum. We also define the energy per unit volume of translation-invariant states and prove that the free vacuum is the unique minimizer of this energy. In the presence of an external field, we prove that the difference between the minimum energy and the energy of the free vacuum converges as $L$ goes to infinity. We obtain in the limit the so-called Bogoliubov-Dirac-Fock functional. The Hartree-Fock (polarized) vacuum is a Hilbert-Schmidt perturbation of the free vacuum and it minimizes the Bogoliubov-Dirac-Fock energy.

Abstract:
The electrical {\em dc}-conductivity of disordered, non-interacting electrons is calculated in the asymptotic limit of high lattice dimensions $d\to \infty$. To go beyond the lowest-order contribution in the expansion parameter $1/d$ of the single bubble diagram, vertex corrections are calculated from an asymptotic expression for the two-particle vertex. A mean-field approximation for the dc-conductivity containing the leading high-dimensional vertex corrections is proposed which is free of spurious non-analyticities, i.e. the conductivity is non-negative and shows no unphysical behavior in $d\geq 3$.

Abstract:
The Bogoliubov-Dirac-Fock (BDF) model is a no-photon approximation of quantum electrodynamics. It allows to study relativistic electrons in interaction with the Dirac sea. A state is fully characterized by its one-body density matrix, an infinite rank nonnegative projector. We prove the existence of the para-positronium, the bound state of an electron and a positron with antiparallel spins, in the BDF model represented by a critical point of the energy functional in the absence of external field. We also prove the existence of the dipositronium, a molecule made of two electrons and two positrons that also appears as a critical point. More generally, for any half integer $j\in \tfrac{1}{2}+\mathbb{Z}_+$, we prove the existence of a critical point of the energy functional made of $2j+1$ electrons and $2j+1$ positrons.

Abstract:
We are commenting on the article Phys. Rev. {\bf B 65}, 155112 (2002) by G. Biroli and G. Kotliar in which they make a comparison between two cluster techniques, the {\it Cellular Dynamical Mean Field Theory} (CDMFT) and the {\it Dynamical Cluster Approximation} (DCA). Based upon an incorrect implementation of the DCA technique in their work, they conclude that the CDMFT is a faster converging technique than the DCA. We present the correct DCA prescription for the particular model Hamiltonian studied in their article and conclude that the DCA, once implemented correctly, is a faster converging technique for the quantities averaged over the cluster. We also refer to their latest response to our comment where they argue that instead of averaging over the cluster, local observables should be calculated in the bulk of the cluster which indeed makes them converge much faster in the CDMFT than in the DCA. We however show that in their original work, the authors themselves use the cluster averaged quantities to draw their conclusions in favor of using the CDMFT over the DCA.

Abstract:
From the observations by the PIXIE and UVI cameras on board the Polar satellite, we derive global maps of the precipitating electron energy spectra from less than 1keV to 100keV. Based on the electron spectra, we generate instantaneous global maps of Hall and Pedersen conductances. The UVI camera provides good coverage of the lower electron energies contributing most to the Pedersen conductance, while PIXIE captures the high energy component of the precipitating electrons affecting the Hall conductance. By characterizing the energetic electrons from some tens of keV and up to about 100keV using PIXIE X-ray measurements, we will, in most cases, calculate a larger electron flux at higher energies than estimated from a simple extrapolation of derived electron spectra from UVI alone. Instantaneous global conductance maps derived with and without inclusion of PIXIE data have been implemented in the Assimilative Mapping of Ionospheric Electrodynamics (AMIE) procedure, to study the effects of energetic electrons on electrodynamical parameters in the ionosphere. We find that the improved electron spectral characterization using PIXIE data most often results in a larger Hall conductance and a smaller inferred electric field. In some localized regions the increase in the Hall conductance can exceed 100%. On the contrary, the Pedersen conductance remains more or less unaffected by the inclusion of the PIXIE data. The calculated polar cap potential drop may decrease more than 10%, resulting in a reduction of the estimated Joule heating integrated over the Northern Hemisphere by up to 20%. Locally, Joule heating may decrease more than 50% in some regions. We also find that the calculated energy flux by precipitating electrons increases around 5% when including the PIXIE data. Combined with the reduction of Joule heating, this results in a decrease in the ratio between Joule heating and energy flux, sometimes exceeding 25%. An investigation of the relationship between Joule heating and the AE index shows a nearly linear correspondence between the two quantities, in accordance with previous studies. However, we find lower proportionality factors than reported by others when taking geomagnetic conditions into account, ranging between 0.13 and 0.23GW/nT. We also find that the contribution from auroral particles to the energy budget is more important than most previous studies have reported. Key words. Ionosphere (auroral ionosphere; particle precipitation) – Magnetospheric physics (storms and substorms)

Abstract:
Pairing of valence electrons can lead to energy minimization of a crystal. It can be proved by use of representation of the valence electrons as plane waves in periodic potential of the crystal.

Abstract:
We study the Bogoliubov-Dirac-Fock (BDF) model, a no-photon, mean-field approxi- mation of quantum electrodynamics that allows to study relativistic electrons interacting with the vacuum. It is a variational model in which states are represented by Hilbert- Schmidt operators. We prove a charge renormalisation formula that holds close to the non-relativistic limit: the density of a ground state is shown to be integrable although such a state is known not to be trace-class. We prove that we can take the non-relativistic limit by keeping track of the vacuum polarisation. We get an altered Hartree-Fock model due to the screening effect.