19 research outputs found
Charge renormalisation in a mean-field approximation of QED
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
Absence of binding in a mean-field approximation of quantum electrodynamics
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
The positronium in a mean-field approximation of quantum electrodynamics
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
The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations
The derivation of effective evolution equations is central to the study of
non-stationary quantum many-body sytems, and widely used in contexts such as
superconductivity, nuclear physics, Bose-Einstein condensation and quantum
chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of
reduced density matrices, and apply it to fermionic and bosonic many-body
systems. We obtain the Bogoliubov-de-Gennes and Hartree-Fock-Bogoliubov
equations, respectively. While we do not prove quantitative error estimates,
our formulation does show that the approximation is optimal within the class of
quasifree states. Furthermore, we prove well-posedness of the
Bogoliubov-de-Gennes equations in energy space and discuss conserved
quantities.Comment: 46 pages, 1 figure; v2: simplified proof of conservation of particle
number, additional references; v3: minor clarification
Spectral flow for Dirac operators with magnetic links
This paper is devoted to the study of the spectral properties of Dirac
operators on the three-sphere with singular magnetic fields supported on
smooth, oriented links. As for Aharonov-Bohm solenoids in Euclidean
three-space, the flux carried by an oriented knot features a -periodicity
of the associated operator. For a given link one thus obtains a family of Dirac
operators indexed by a torus of fluxes. We study the spectral flow of paths of
such operators corresponding to loops in this torus. The spectral flow is in
general non-trivial. In the special case of a link of unknots we derive an
explicit formula for the spectral flow of any loop on the torus of fluxes. It
is given in terms of the linking numbers of the knots and their writhes
On symmetry of traveling solitary waves for dispersion generalized NLS
We consider dispersion generalized nonlinear Schr\"odinger equations (NLS) of
the form , where denotes a
(pseudo)-differential operator of arbitrary order. As a main result, we prove
symmetry results for traveling solitary waves in the case of powers . The arguments are based on Steiner type rearrangements in Fourier
space. Our results apply to a broad class of NLS-type equations such as
fourth-order (biharmonic) NLS, fractional NLS, square-root Klein-Gordon and
half-wave equations.Comment: 17 pages. Any comments are welcome
Etude d'un modèle de champ moyen en électrodynamique quantique
In QED, mean-field models appear in the modelling of the electron clouds of heavy atoms. This modelling plays a increasing role in physics and in quantum chemistry: relativistic effects cannot be neglected in these atoms. In relativistic quantum physics the vacuum is a polarizable medium that can react to the presence of an electromagnetic field.We consider the so-called Bogoliubov-Dirac-Fock (BDF) model, a variational model which is a mean-field approximation of no-photon QED (in particular the interactions are purely electrostatic).We point out that an ultraviolet regularisation is necessary to properly define the BDF model. The vacuum polarisation leads to a \emph{renormalisation} phenomenon, the "observed" charge of the electron depends on its "bare" charge and the regularisation parameter. We rigorously study both the problem of charge renormalisation and mass renormalisation. This last one is linked to the existence of ground state in the case of an electron in the vacuum, without any external field. In contrast, we show there is no ground state in the case of two electrons.Finally we exhibit some critical points of the BDF energy which are interpreted as vacuum excited states. In particular, there are the positronium (a metastable system constituted by an electron and its antiparticle called the positron) and the dipositronium (a metastable molecule constituted by two electrons and two positrons).The methods that we use are variational (concentration-compactness, Borwein and Preiss's Lemma).Les modèles de champ moyen en QED apparaissent naturellement dans la modélisation du nuage électronique des atomes lourds. Cette modélisation joue un rôle croissant en physique et chimie quantique, les effets relativistes ne pouvant pas être négligés pour ces atomes. En physique quantique relativiste, le vide est un milieu polarisable, susceptible de réagir à la présence de champ électromagnétique.On se place dans le cadre du modèle variationnel de Bogoliubov-Dirac-Fock (BDF) qui est une approximation de champ moyen de la QED sans photon (en particulier, les interactions considérées sont purement électrostatiques).Il est à noter que pour donner un sens au modèle BDF, il est nécessaire d'introduire une régularisation ultra-violette. Il se produit un phénomène de renormalisation de charge due à la polarisation du vide : la charge de l'électron observée dépend de la charge « nue » de l'électron et du paramètre de régularisation. On étudie rigoureusement ce phénomène ainsi que le problème de la renormalisation de la masse. Cette dernière est en lien avec l'existence d'un état fondamental pour le système d'un électron dans le vide, en l'absence de tout champ extérieur. En revanche, on montre l'absence de minimiseurs dans le cas de deux électrons.Enfin, on exhibe des points critiques de l'énergie BDF, interprétés comme des états excités du vide. On met en évidence le positronium, système métastable d'un électron et de son antiparticule le positron, ainsi que le dipositronium, molécule métastable constituée de deux électrons et de deux positrons.Les méthodes utilisées sont variationnelles (concentration-compacité, lemme de Borwein et Preiss)
Study of a mean-field model in quantum electrodynamics
Les modèles de champ moyen en QED apparaissent naturellement dans la modélisation du nuage électronique des atomes lourds. Cette modélisation joue un rôle croissant en physique et chimie quantique, les effets relativistes ne pouvant pas être négligés pour ces atomes. En physique quantique relativiste, le vide est un milieu polarisable, susceptible de réagir à la présence de champ électromagnétique.On se place dans le cadre du modèle variationnel de Bogoliubov-Dirac-Fock (BDF) qui est une approximation de champ moyen de la QED sans photon (en particulier, les interactions considérées sont purement électrostatiques).Il est à noter que pour donner un sens au modèle BDF, il est nécessaire d'introduire une régularisation ultra-violette. Il se produit un phénomène de renormalisation de charge due à la polarisation du vide : la charge de l'électron observée dépend de la charge « nue » de l'électron et du paramètre de régularisation. On étudie rigoureusement ce phénomène ainsi que le problème de la renormalisation de la masse. Cette dernière est en lien avec l'existence d'un état fondamental pour le système d'un électron dans le vide, en l'absence de tout champ extérieur. En revanche, on montre l'absence de minimiseurs dans le cas de deux électrons.Enfin, on exhibe des points critiques de l'énergie BDF, interprétés comme des états excités du vide. On met en évidence le positronium, système métastable d'un électron et de son antiparticule le positron, ainsi que le dipositronium, molécule métastable constituée de deux électrons et de deux positrons.Les méthodes utilisées sont variationnelles (concentration-compacité, lemme de Borwein et Preiss).In QED, mean-field models appear in the modelling of the electron clouds of heavy atoms. This modelling plays a increasing role in physics and in quantum chemistry: relativistic effects cannot be neglected in these atoms. In relativistic quantum physics the vacuum is a polarizable medium that can react to the presence of an electromagnetic field.We consider the so-called Bogoliubov-Dirac-Fock (BDF) model, a variational model which is a mean-field approximation of no-photon QED (in particular the interactions are purely electrostatic).We point out that an ultraviolet regularisation is necessary to properly define the BDF model. The vacuum polarisation leads to a \emph{renormalisation} phenomenon, the "observed" charge of the electron depends on its "bare" charge and the regularisation parameter. We rigorously study both the problem of charge renormalisation and mass renormalisation. This last one is linked to the existence of ground state in the case of an electron in the vacuum, without any external field. In contrast, we show there is no ground state in the case of two electrons.Finally we exhibit some critical points of the BDF energy which are interpreted as vacuum excited states. In particular, there are the positronium (a metastable system constituted by an electron and its antiparticle called the positron) and the dipositronium (a metastable molecule constituted by two electrons and two positrons).The methods that we use are variational (concentration-compactness, Borwein and Preiss's Lemma)