Minimization method for relativistic electrons in a mean-field approximation of quantum electrodynamics (original) (raw)
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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 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 Hartree-Fock approximation.
Archive for Rational Mechanics and Analysis, 2009
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.
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arXiv: Mathematical Physics, 2009
We consider a U(1)-invariant nonlinear Dirac equation in dimension n=3n=3n=3, interacting with itself via the mean field mechanism. We analyze the long-time asymptotics of solutions and prove that, under certain generic assumptions, each finite charge solution converges as ttopminftyt\to\pm\inftyttopminfty to the two-dimensional set of all "nonlinear eigenfunctions" of the form phi(x)esp−iomegat\phi(x)e\sp{-i\omega t}phi(x)esp−iomegat. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. The research is inspired by Bohr's postulate on quantum transitions and Schr\"odinger's identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled U(1)-invariant Maxwell-Schr\"odinger and Maxwell-Dirac equations.
Variational methods in relativistic quantum mechanics
Bulletin of the American Mathematical Society, 2008
10 1.3. Existence of solutions by perturbation theory 11 1.4. Nonlinear Dirac equations in the Schwarzschild metric 12 1.5. Solutions of the Maxwell-Dirac equations 12 1.6. Nonlinear Dirac evolution problems 17 2. Linear Dirac equations for an electron in an external field 19 2.1. A variational characterization of the eigenvalues of D c + V 20 2.2. Numerical method based on the min-max formula 25 2.3. New Hardy-like inequalities 27 2.4. The nonrelativistic limit 27 2.5. Introduction of a constant external magnetic field 28 3. The Dirac-Fock equations for atoms and molecules 30 3.1. The (non-relativistic) Hartree-Fock equations 30 3.2. Existence of solutions to the Dirac-Fock equations 31 3.3. Nonrelativistic limit and definition of the Dirac-Fock "ground state" 34 4. The mean-field approximation in Quantum Electrodynamics 36 4.1. Definition of the free vacuum 40 4.2. The Bogoliubov-Dirac-Fock model 42 4.3. Global minimization of E ν BDF : the polarized vacuum. 45 4.4. Minimization of E ν BDF in charge sectors 47 4.5. Neglecting Vacuum Polarization: Mittleman's conjecture 50 References 53
Mean motion of Dirac electrons in a gravitational field
Annals of Physics, 1977
A sequence of Foldy-Wouthuysen transformations applied to the Dirac equation coupled to a background gravitational field is used to obtain evolution equations for the mean position, mean momentum, and mean spin operators. These equations are compared with the analogous Papapetrou equations for a classical gravitationally coupled pole-dipole particle.
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Conceptual analogies among statistical mechanics and classical (or quantum) mechanics often appeared in the literature. For classical two-body mean field models, an analogy develops into a proper identification between the free energy of Curie-Weiss type magnetic models and the Hamilton-Jacobi action for a one dimensional mechanical system. Similarly, the partition function plays the role of the wave function in quantum mechanics and satisfies the heat equation that plays, in this context, the role of the Schrödinnger equation in quantum mechanics.
Self-consistent theory of mean-field electrodynamics
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Mean-field electrodynamics, including both α and β effects while accounting for the effects of small-scale magnetic fields, is derived for incompressible magnetohydrodynamics. The principal result is α=(α 0 +β 0 R⋅∇×R)/(1+R 2 ), β=β 0 ; where α 0 ,β 0 are conventional kinematic ...