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Existence of a stable polarized vacuum in the BogoliubovDiracFock approximation
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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 41 (18 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
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 27 (9 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
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 22 (12 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.
Spacetime algebra and electron physics
 Advances in Imaging and Electron Physics
, 1996
"... This paper surveys the application of geometric algebra to the physics of electrons. It first appeared in 1996 and is reproduced here with only minor modifications. Subjects covered include nonrelativistic and relativistic spinors, the Dirac equation, operators and monogenics, the Hydrogen atom, pr ..."
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Cited by 19 (11 self)
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This paper surveys the application of geometric algebra to the physics of electrons. It first appeared in 1996 and is reproduced here with only minor modifications. Subjects covered include nonrelativistic and relativistic spinors, the Dirac equation, operators and monogenics, the Hydrogen atom, propagators and scattering theory, spin precession, tunnelling times, spin measurement, multiparticle quantum mechanics, relativistic multiparticle wave equations, and semiclassical mechanics.
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 19 (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.
The Thermodynamic Universe
 World Scientific, Singapoure
, 2008
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Nonperturbative Mass and Charge Renormalization in Relativistic Nophoton QED
 Commun. Math. Phys
"... Abstract. Starting from a formal Hamiltonian as found in the physics literature – omitting photons – we define a renormalized Hamiltonian through charge and mass renormalization. We show that the restriction to the oneelectron subspace is welldefined. Our construction is nonperturbative and does n ..."
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Cited by 16 (8 self)
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Abstract. Starting from a formal Hamiltonian as found in the physics literature – omitting photons – we define a renormalized Hamiltonian through charge and mass renormalization. We show that the restriction to the oneelectron subspace is welldefined. Our construction is nonperturbative and does not use a cutoff. The Hamiltonian is relevant for the description of the Lamb shift in muonic atoms. 1.
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.
The Blackholic energy and the canonical GammaRay Burst
 in XIIth Brazilian School of Cosmology and Gravitation, edited by
"... Abstract. GammaRay Bursts (GRBs) represent very likely “the ” most extensive computational, theoretical and observational effort ever carried out successfully in physics and astrophysics. The extensive campaign of observation from space based Xray and γray observatory, such as the Vela, CGRO, Bep ..."
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Cited by 10 (9 self)
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Abstract. GammaRay Bursts (GRBs) represent very likely “the ” most extensive computational, theoretical and observational effort ever carried out successfully in physics and astrophysics. The extensive campaign of observation from space based Xray and γray observatory, such as the Vela, CGRO, BeppoSAX, HETEII, INTEGRAL, Swift, RXTE, Chandra, XMM satellites, have been matched by complementary observations in the radio wavelength (e.g. by the VLA) and in the optical band (e.g. by VLT, Keck, ROSAT). The net result is unprecedented accuracy in the received data allowing the determination of the energetics, the time variability and the spectral properties of these GRB sources. The very fortunate situation occurs that these data can be confronted with a mature theoretical development. Theoretical interpretation of the above data allows progress in three different frontiers of knowledge: a) the ultrarelativistic regimes of a macroscopic source moving at Lorentz gamma factors up to ∼ 400; b) the occurrence of vacuum polarization process verifying some of the yet untested regimes of ultrarelativistic quantum field theories; and c) the first evidence for extracting, during the process of gravitational collapse leading to the formation of a black hole, amounts of energies up to 10 55 ergs of blackholic energy — a new form of energy in physics and astrophysics. We outline how this progress leads to the confirmation of three interpretation paradigms for GRBs proposed in July 2001. Thanks mainly to the observations by Swift and the optical observations by VLT, the outcome