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31
The meanfield approximation in quantum electrodynamics. The nophoton case
 Comm. Pure Applied Math
"... We study the meanfield approximation of Quantum Electrodynamics, by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normalordering or choice of bare electron/positron subspaces. Neglecting photons, we define properly this Hamiltonian in a fi ..."
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Cited by 24 (11 self)
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We study the meanfield approximation of Quantum Electrodynamics, by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normalordering 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 cutoff Λ. 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 HartreeFock approximation. In case with no external field, we prove that the energy per volume converges and obtain in the limit a translationinvariant projector describing the free HartreeFock vacuum. We also define the energy per unit volume of translationinvariant 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 socalled BogoliubovDiracFock functional. The HartreeFock (polarized) vacuum is a HilbertSchmidt perturbation of the free vacuum and it minimizes the BogoliubovDiracFock energy. 1
A new approach to the modelling of local defects in crystals: the reduced HartreeFock case
 Commun. Math. Phys
"... Abstract. This article is concerned with the derivation and the mathematical study of a new meanfield model for the description of interacting electrons in crystals with local defects. We work with a reduced HartreeFock model, obtained from the usual HartreeFock model by neglecting the exchange t ..."
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Cited by 17 (10 self)
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Abstract. This article is concerned with the derivation and the mathematical study of a new meanfield model for the description of interacting electrons in crystals with local defects. We work with a reduced HartreeFock model, obtained from the usual HartreeFock model by neglecting the exchange term. First, we recall the definition of the selfconsistent Fermi sea of the perfect crystal, which is obtained as a minimizer of some periodic problem, as was shown by Catto, Le Bris and Lions. We also prove some of its properties which were not mentioned before. Then, we define and study in detail a nonlinear model for the electrons of the crystal in the presence of a defect. We use formal analogies between the Fermi sea of a perturbed crystal and the Dirac sea in Quantum Electrodynamics in the presence of an external electrostatic field. The latter was recently studied by Hainzl, Lewin, Séré and Solovej, based on ideas from Chaix and Iracane. This enables us to define the ground state of the selfconsistent Fermi sea in the presence of a defect.
Selfconsistent solution for the polarized vacuum in a nophoton QED model
, 2005
"... We study the BogoliubovDiracFock model introduced by Chaix and Iracane (J. Phys. B., 22, 3791–3814, 1989) which is a meanfield theory deduced from nophoton QED. The associated functional is bounded from below. In the presence of an external field, a minimizer, if it exists, is interpreted as t ..."
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Cited by 15 (10 self)
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We study the BogoliubovDiracFock model introduced by Chaix and Iracane (J. Phys. B., 22, 3791–3814, 1989) which is a meanfield theory deduced from nophoton QED. The associated functional is bounded from below. In the presence of an external field, a minimizer, if it exists, is interpreted as the polarized vacuum and it solves a selfconsistent equation. In a recent paper, we proved the convergence of the iterative fixedpoint scheme naturally associated with this equation to a global minimizer of the BDF functional, under some restrictive conditions on the external potential, the ultraviolet cutoff Λ and the bare fine structure constant α. In the present work, we improve this result by showing the existence of the minimizer by a variational method, for any cutoff Λ and without any constraint on the external field. We also study the behaviour of the minimizer as Λ goes to infinity and show that the theory is “nullified ” in that limit, as predicted first by Landau: the vacuum totally cancels the external potential. Therefore the limit case of an infinite cutoff makes no sense both from a physical and mathematical point of view. Finally, we perform a charge and density renormalization scheme applying simultaneously to all orders of the fine structure constant α, on a simplified model where the exchange term is neglected.
A Minimization Method for Relativistic Electrons in a MeanField Approximation of Quantum Electrodynamics
 Phys. Rev. A
"... Abstract. We study a meanfield relativistic model which is able to describe both the behavior of finitely many spin1/2 particles like electrons and of the Dirac sea which is selfconsistently polarized in the presence of the real particles. The model is derived from the QED Hamiltonian in Coulomb ..."
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Cited by 13 (8 self)
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Abstract. We study a meanfield relativistic model which is able to describe both the behavior of finitely many spin1/2 particles like electrons and of the Dirac sea which is selfconsistently polarized in the presence of the real particles. The model is derived from the QED Hamiltonian in Coulomb gauge neglecting the photon field. All our results are nonperturbative and mathematically rigorous. Contents
Some bound state problems in quantum mechanics
 Proc. Symp. Pure Math., 76.1, American Mathematical Society , in “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday
, 2007
"... We give a review of semiclassical estimates for bound states and their eigenvalues for Schrödinger operators. Motivated by the classical results, we discuss their recent improvements for single particle Schrödinger operators as well as some applications of these semiclassical bounds to multipart ..."
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Cited by 11 (0 self)
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We give a review of semiclassical estimates for bound states and their eigenvalues for Schrödinger operators. Motivated by the classical results, we discuss their recent improvements for single particle Schrödinger operators as well as some applications of these semiclassical bounds to multiparticle systems, in particular, large atoms and the stability of matter.
Existence of globalintime solutions to a generalized DiracFock type evolution equation
"... Abstract. We consider a generalized DiracFock type evolution equation deduced from nophoton Quantum Electrodynamics, which describes the selfconsistent timeevolution of relativistic electrons, the observable ones as well as those filling up the Dirac sea. This equation has been originally introdu ..."
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Cited by 11 (8 self)
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Abstract. We consider a generalized DiracFock type evolution equation deduced from nophoton Quantum Electrodynamics, which describes the selfconsistent timeevolution of relativistic electrons, the observable ones as well as those filling up the Dirac sea. This equation has been originally introduced by Dirac in 1934 in a simplified form. Since we work in a HartreeFock type approximation, the elements describing the physical state of the electrons are infinite rank projectors. Using the BogoliubovDiracFock formalism, introduced by ChaixIracane (J. Phys. B., 22, 3791–3814, 1989), and recently established by HainzlLewinSéré, we prove the existence of globalintime solutions of the considered evolution equation. 1.
Existence of Atoms and Molecules in the MeanField Approximation of NoPhoton Quantum Electrodynamics
, 2008
"... The BogoliubovDiracFock (BDF) model is the meanfield approximation of nophoton 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 s ..."
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Cited by 9 (4 self)
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The BogoliubovDiracFock (BDF) model is the meanfield approximation of nophoton 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 (HVZtype) 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 α is small whereas αZ and the particle number N are fixed. The second is the nonrelativistic regime in which the speed of light tends to infinity (or equivalently α tends to zero) and Z, N are fixed. We also prove that the electronic solution converges in the nonrelativistic limit towards a HartreeFock ground state.
M.: The dielectric permittivity of crystals in the reduced HartreeFock approximation
, 2010
"... Abstract. In a recent article (Cancès, Deleurence and Lewin, Commun. Math. Phys. 281 (2008), pp. 129–177), we have rigorously derived, by means of bulk limit arguments, a new variational model to describe the electronic ground state of insulating or semiconducting crystals in the presence of local d ..."
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Cited by 8 (6 self)
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Abstract. In a recent article (Cancès, Deleurence and Lewin, Commun. Math. Phys. 281 (2008), pp. 129–177), we have rigorously derived, by means of bulk limit arguments, a new variational model to describe the electronic ground state of insulating or semiconducting crystals in the presence of local defects. In this socalled reduced HartreeFock model, the ground state electronic density matrix is decomposed as γ = γ0 per + Qν,εF, where γ0 per is the ground state density matrix of the host crystal and Qν,ε F the modification of the electronic density matrix generated by a modification ν of the nuclear charge of the host crystal, the Fermi level εF being kept fixed. The purpose of the present article is twofold. First, we study more in details the mathematical properties of the density matrix Qν,ε F (which is known to be a selfadjoint HilbertSchmidt operator on L 2 (R 3)). We show in particular that if ´ R 3 ν ̸ = 0, Qν,ε F is not traceclass. Moreover, the associated density of charge is not in L 1 (R 3) if the crystal exhibits anisotropic dielectric properties. These results are obtained by analyzing, for a small defect ν, the linear and nonlinear terms of the resolvent expansion of Qν,ε F Second, we show that, after an appropriate rescaling, the potential generated by the microscopic total charge (nuclear plus electronic contributions) of the crystal in the presence of the defect, converges to a homogenized electrostatic potential solution to a Poisson equation involving the macroscopic dielectric permittivity of the crystal. This provides an alternative (and rigorous) derivation of the AdlerWiser formula. Contents
Variational methods in relativistic quantum mechanics
, 2008
"... This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. Contrary to the Laplacian appearing in the equations of nonrelativistic ..."
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Cited by 6 (3 self)
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This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the EulerLagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R³, the PalaisSmale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems. In the first part, we consider the fixed eigenvalue problem for models of a
Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms
, 2007
"... We study the reduced BogoliubovDiracFock (BDF) energy which allows to describe relativistic electrons interacting with the Dirac sea, in an external electrostatic potential. The model can be seen as a meanfield approximation of Quantum Electrodynamics (QED) where photons and the socalled exchange ..."
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Cited by 6 (4 self)
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We study the reduced BogoliubovDiracFock (BDF) energy which allows to describe relativistic electrons interacting with the Dirac sea, in an external electrostatic potential. The model can be seen as a meanfield approximation of Quantum Electrodynamics (QED) where photons and the socalled exchange term are neglected. A state of the system is described by its onebody density matrix, an infinite rank selfadjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator (the Dirac sea). We study the minimization of the reduced BDF energy under a charge constraint. We prove the existence of minimizers for a large range of values of the charge, and any positive value of the coupling constant α. Our result covers neutral and positively charged molecules, provided that the positive charge is not large enough to create electronpositron pairs. We also prove that the density of any minimizer is an L 1 function and compute the effective charge of the system, recovering the usual renormalization of charge: the physical coupling constant is related to α by the formula αphys ≃ α(1 + 2α/(3π) log Λ) −1, where Λ is the ultraviolet cutoff. We eventually prove an estimate on the highest number of electrons which can be bound by a nucleus of charge Z. In the nonrelativistic limit, we obtain that this number is ≤ 2Z, recovering a result of Lieb. This work is based on a series of papers by Hainzl, Lewin, Séré and Solovej on the meanfield approximation of nophoton QED.