203 research outputs found
Renormalization and asymptotic expansion of Dirac's polarized vacuum
We perform rigorously the charge renormalization of the so-called reduced
Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac
operator, describes atoms and molecules while taking into account vacuum
polarization effects. We consider the total physical density including both the
external density of a nucleus and the self-consistent polarization of the Dirac
sea, but no `real' electron. We show that it admits an asymptotic expansion to
any order in powers of the physical coupling constant \alphaph, provided that
the ultraviolet cut-off behaves as \Lambda\sim e^{3\pi(1-Z_3)/2\alphaph}\gg1.
The renormalization parameter $
Low Density Limit of BCS Theory and Bose-Einstein Condensation of Fermion Pairs
We consider the low density limit of a Fermi gas in the BCS approximation. We
show that if the interaction potential allows for a two-particle bound state,
the system at zero temperature is well approximated by the Gross-Pitaevskii
functional, describing a Bose-Einstein condensate of fermion pairs.Comment: LaTeX2e, 17 page
Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields
Using the Pauli-Villars regularization and arguments from convex analysis, we
construct solutions to the classical time-independent Maxwell equations in
Dirac's vacuum, in the presence of small external electromagnetic sources. The
vacuum is not an empty space, but rather a quantum fluctuating medium which
behaves as a nonlinear polarizable material. Its behavior is described by a
Dirac equation involving infinitely many particles. The quantum corrections to
the usual Maxwell equations are nonlinear and nonlocal. Even if photons are
described by a purely classical electromagnetic field, the resulting vacuum
polarization coincides to first order with that of full Quantum
Electrodynamics.Comment: Final version to appear in Arch. Rat. Mech. Analysi
A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case
This article is concerned with the derivation and the mathematical study of a
new mean-field model for the description of interacting electrons in crystals
with local defects. We work with a reduced Hartree-Fock model, obtained from
the usual Hartree-Fock model by neglecting the exchange term. First, we recall
the definition of the self-consistent 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 details 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, S\'er\'e and Solovej, based on ideas
from Chaix and Iracane. This enables us to define the ground state of the
self-consistent Fermi sea in the presence of a defect. We end the paper by
proving that our model is in fact the thermodynamic limit of the so-called
supercell model, widely used in numerical simulations.Comment: Final version, to appear in Comm. Math. Phy
On Blowup for time-dependent generalized Hartree-Fock equations
We prove finite-time blowup for spherically symmetric and negative energy
solutions of Hartree-Fock and Hartree-Fock-Bogoliubov type equations, which
describe the evolution of attractive fermionic systems (e. g. white dwarfs).
Our main results are twofold: First, we extend the recent blowup result of
[Hainzl and Schlein, Comm. Math. Phys. \textbf{287} (2009), 705--714] to
Hartree-Fock equations with infinite rank solutions and a general class of
Newtonian type interactions. Second, we show the existence of finite-time
blowup for spherically symmetric solutions of a Hartree-Fock-Bogoliubov model,
where an angular momentum cutoff is introduced. We also explain the key
difficulties encountered in the full Hartree-Fock-Bogoliubov theory.Comment: 24 page
Scale Dependence of the Retarded van der Waals Potential
We study the ground state energy for a system of two hydrogen atoms coupled
to the quantized Maxwell field in the limit together with the
relative distance between the atoms increasing as , . In particular we determine explicitly the crossover function from the
van der Waals potential to the retarded van der Waals
potential, which takes place at scale .Comment: 19 page
Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation
We consider a generalized Dirac-Fock type evolution equation deduced from
no-photon Quantum Electrodynamics, which describes the self-consistent
time-evolution 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 Hartree-Fock type
approximation, the elements describing the physical state of the electrons are
infinite rank projectors. Using the Bogoliubov-Dirac-Fock formalism, introduced
by Chaix-Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989), and recently
established by Hainzl-Lewin-Sere, we prove the existence of global-in-time
solutions of the considered evolution equation.Comment: 12 pages; more explanations added, some final (minor) corrections
include
Non-perturbative embedding of local defects in crystalline materials
We present a new variational model for computing the electronic first-order
density matrix of a crystalline material in presence of a local defect. A
natural way to obtain variational discretizations of this model is to expand
the difference Q between the density matrix of the defective crystal and the
density matrix of the perfect crystal, in a basis of precomputed maximally
localized Wannier functions of the reference perfect crystal. This approach can
be used within any semi-empirical or Density Functional Theory framework.Comment: 13 pages, 4 figure
Self-consistent solution for the polarized vacuum in a no-photon QED model
We study the Bogoliubov-Dirac-Fock model introduced by Chaix and Iracane
({\it J. Phys. B.}, 22, 3791--3814, 1989) which is a mean-field theory deduced
from no-photon 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 self-consistent equation.
In a recent paper math-ph/0403005, we proved the convergence of the iterative
fixed-point scheme naturally associated with this equation to a global
minimizer of the BDF functional, under some restrictive conditions on the
external potential, the ultraviolet cut-off and the bare fine
structure constant . In the present work, we improve this result by
showing the existence of the minimizer by a variational method, for any cut-off
and without any constraint on the external field.
We also study the behaviour of the minimizer as goes to infinity
and show that the theory is "nullified" in that limit, as predicted first by
Landau: the vacuum totally kills the external potential. Therefore the limit
case of an infinite cut-off 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 , on a
simplified model where the exchange term is neglected.Comment: Final version, to appear in J. Phys. A: Math. Ge
Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms
We study the reduced Bogoliubov-Dirac-Fock (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 mean-field approximation of
Quantum Electrodynamics (QED) where photons and the so-called exchange term are
neglected. A state of the system is described by its one-body density matrix,
an infinite rank self-adjoint 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 . Our
result covers neutral and positively charged molecules, provided that the
positive charge is not large enough to create electron-positron pairs. We also
prove that the density of any minimizer is an function and compute the
effective charge of the system, recovering the usual renormalization of charge:
the physical coupling constant is related to by the formula
, where
is the ultraviolet cut-off. We eventually prove an estimate on the
highest number of electrons which can be bound by a nucleus of charge . In
the nonrelativistic limit, we obtain that this number is , recovering
a result of Lieb.
This work is based on a series of papers by Hainzl, Lewin, Sere and Solovej
on the mean-field approximation of no-photon QED.Comment: 37 pages, 1 figur
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