Maxwell-Dirac System Research Articles

Bilinear Fourier restriction estimates related to the 2d wave equation

We study bilinear $L^2$ Fourier restriction estimates which are related to the 2d wave equation in the sense that we restrict to subsets of thickened null cones. In an earlier paper we studied the corresponding 3d problem, obtaining several refinements of the Klainerman-Machedon-type estimates. The latter are bilinear generalizations of the $L^4$ estimate of Strichartz for the 3d wave equation. In 2d there is no $L^4$ estimate for solutions of the wave equation, but, as we show here, one can nevertheless obtain $L^2$ bilinear estimates for thickened null cones, which can be viewed as analogs of the 3d Klainerman-Machedon estimates. We then prove a number of refinements of these estimates. The application we have in mind is the Maxwell-Dirac system.

Global well-posedness of the Maxwell–Dirac system in two space dimensions

In recent work, Grünrock and Pecher proved that the Dirac–Klein–Gordon system in 2d is globally well-posed in the charge class (data in L 2 for the spinor and in a suitable Sobolev space for the scalar field). Here we obtain the analogous result for the full Maxwell–Dirac system in 2d. Making use of the null structure of the system, found in earlier joint work with Damiano Foschi, we first prove local well-posedness in the charge class. To extend the solutions globally we build on an idea due to Colliander, Holmer and Tzirakis. For this we rely on the fact that MD is charge subcritical in two space dimensions, and make use of the null structure of the Maxwell part.

Global charge solutions of Maxwell–Dirac equations in

We prove a global well-posedness in charge space of the Maxwell–Dirac system in one space dimension by using an explicit representation of its solution in terms of initial data. A decay of local charge and asymptotic behaviors of the field can be shown directly.

A simple and accurate mixed [formula omitted] solver for the Maxwell–Dirac equations

In this paper we propose and analyze a simple but accurate numerical method for solving the Maxwell–Dirac system for relativistic particles submitted to an external ultrashort intense pulse. Maxwell’s equations are solved using a second order finite element method in space and time. The Dirac equation solver is based on a method of characteristics with a P 0 formulation of the Dirac spinor field. Numerical simulations are provided to show the efficiency of the method.

Null structure and almost optimal local well-posedness of the Maxwell-Dirac system

We uncover the full null structure of the Maxwell-Dirac system in Lorenz gauge. This structure, which cannot be seen in the individual component equations, but only when considering the system as a whole, is expressed in terms of tri- and quadrilinear integral forms with cancellations measured by the angles between spatial frequencies. In the 3D case, we prove frequency-localized $L^2$ space-time estimates for these integral forms at the scale invariant regularity up to a logarithmic loss, hence we obtain almost optimal local well-posedness of the system by iteration.

A numerical method with particle conservation for the Maxwell–Dirac system

A numerical method is presented for solving the Maxwell–Dirac systems. The Maxwell equations with particle and current densities as the source terms are discretized explicitly. To guarantee the particle conservation, the Dirac equations coupled electromagnetic potentials are discretized by the time-splitting method and implicit finite difference. These numerical schemes are conservative in particle density and have second-order accuracy in time and space. One-dimensional numerical results are given to validate the accuracy and the conservation and three-dimensional examples are presented to describe dynamical behaviors of the Maxwell–Dirac system with several external potentials.

Convergence of the Dirac–Maxwell System to the Vlasov–Poisson System

We study the behavior of solutions of the (mixed state) Dirac–Maxwell system in the simultaneous non-relativistic and semi-classical limits, i.e., as the speed of light c tends to infinity and the Planck constant ℏ tends to zero. Under a constraint of the form , where M is a sufficiently large integer, and with suitable conditions on the initial data, we prove that the scalar Wigner transform of the Dirac spinors converges weakly to a solution of the classical Vlasov–Poisson system. Our key technique is a blend of null form bilinear estimates with the machinery of the Wigner transform.

CANONICAL AND GRAVITATIONAL STRESS-ENERGY TENSORS

We deal with the question, under which circumstances the canonical Noether stress-energy tensor is equivalent to the gravitational (Hilbert) tensor for general matter fields under the influence of gravity. In the framework of general relativity, the full equivalence is established for matter fields that do not couple to the metric derivatives. Spinor fields are included into our analysis by reformulating general relativity in terms of tetrad fields, and the case of Poincaré gauge theory, with an additional, independent Lorentz connection, is also investigated. Special attention is given to the flat limit, focusing on the expressions for the matter field energy (Hamiltonian). The Dirac–Maxwell system is investigated in detail, with special care given to the separation of free (kinetic) and interaction (or potential) energy. Moreover, the stress-energy tensor of the gravitational field itself is briefly discussed.

A time-splitting spectral scheme for the Maxwell–Dirac system

We present a time-splitting spectral scheme for the Maxwell–Dirac system and similar time-splitting methods for the corresponding asymptotic problems in the semi-classical and the non-relativistic regimes. The scheme for the Maxwell–Dirac system conserves the Lorentz gauge condition is unconditionally stable and highly efficient as our numerical examples show. In particular, we focus in our examples on the creation of positronic modes in the semi-classical regime and on the electron–positron interaction in the non-relativistic regime. Furthermore, in the non-relativistic regime, our numerical method exhibits uniform convergence in the small parameter δ, which is the ratio of the characteristic speed and the speed of light.

ON THE ASYMPTOTIC ANALYSIS OF THE DIRAC–MAXWELL SYSTEM IN THE NONRELATIVISTIC LIMIT

We study the behavior of solutions of the Dirac–Maxwell system (DM) in the nonrelativistic limit c → ∞, where c is the speed of light. DM is a nonlinear system of PDEs obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the moving charge of the spinor. The limit c → ∞, sometimes also called post-Newtonian, yields a Schrödinger–Poisson system, where the spin and magnetic field no longer appear. We prove that DM is locally well-posed for H1 data (for fixed c), and that as c → ∞ the existence time grows at least as fast as log(c), provided the data are uniformly bounded in H1. Moreover, if the datum for the Dirac spinor converges in H1, then the solution of DM converges, modulo a phase correction, in C([0,T];H1) to a solution of a Schrödinger–Poisson system. Our results also apply to a mixed state formulation of DM, and give also a convergence result for the Pauli equation as the "semi-nonrelativistic" limit. The proof relies on modifications of the bilinear null form estimates of Klainerman and Machedon, and extends our previous work on the nonrelativistic limit of the Klein–Gordon–Maxwell system.

An efficient and stable numerical method for the Maxwell–Dirac system

In this paper, we present an explicit, unconditionally stable and accurate numerical method for the Maxwell–Dirac system (MD) and use it to study dynamics of MD. As preparatory steps, we take the three-dimensional (3D) Maxwell–Dirac system, scale it to obtain a two-parameter model and review plane wave solution of free MD. Then we present a time-splitting spectral method (TSSP) for MD. The key point in the numerical method is based on a time-splitting discretization of the Dirac system, and to discretize nonlinear wave-type equations by pseudospectral method for spatial derivatives, and then solving the ordinary differential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between different time intervals. The method is explicit, unconditionally stable, time reversible, time transverse invariant, and of spectral-order accuracy in space and second-order accuracy in time. Moreover, it conserves the particle density exactly in discretized level and gives exact results for plane wave solution of free MD. Extensive numerical tests are presented to confirm the above properties of the numerical method. Furthermore, the tests also suggest the following meshing strategy (or ε-resolution) is admissible in the `nonrelativistic' limit regime (0< ε≪1): spatial mesh size h=O( ε) and time step △ t=O( ε 2), where the parameter ε is inversely proportional to the speed of light.

Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations

We prove uniqueness of solutions to the Maxwell-Dirac system in the energy space, namely . We also give a proof for uniqueness of finite energy solutions to the Maxwell-Klein-Gordon equations, which is simpler than that given in [16].

Semiclassical asymptotics for the Maxwell–Dirac system

We study the coupled system of Maxwell and Dirac equations from a semiclassical point of view. A rigorous nonlinear WKB-analysis, locally in time, for solutions of (critical) order O(ε) is performed, where the small semiclassical parameter ε≪1 denotes the microscopic/macroscopic scale ratio.

The stationary Maxwell–Dirac equations

The Maxwell–Dirac equations are the equations for electronic matter, the 'classical' theory underlying QED. The system combines the Dirac equations with the Maxwell equations sourced by the Dirac current. A stationary Maxwell–Dirac system has ψ = e−iEt, with independent of t. The system is said to be isolated if the dependent variables obey quite weak regularity and decay conditions. In this paper, we prove the following strong localization result for isolated, stationary Maxwell–Dirac systems, there are no embedded eigenvalues in the essential spectrum, i.e. −m ≤ E ≤ m; if |E| < m then the Dirac field decays exponentially as |x| → ∞; if |E| = m then the system is 'asymptotically' static and decays exponentially if the total charge is non-zero.

Sur quelques limites de la physique des particules chargées vers la (magnéto)hydrodynamique

We discuss the connection between different scalings limits of the quantum-relativistic Dirac–Maxwell system. In particular we give rigorous results for the quasi-neutral/non-relativistic limit of the Vlasov–Maxwell system: we obtain a magneto-hydro-dynamic system when we consider the magnetic field as a non-relativistic effect and we obtain the Euler equation when we see it as a relativistic effect. A mathematical key is the modulated energy method. To cite this article: Y. Brenier et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 239–244.

The Selfconsistent Pauli Equation

We deal with consistent first order non-relativistic corrections (i.e. in the small parameter \(\), where c is the speed of light) of the Dirac–Maxwell system. We discuss a selfconsistent modeling of the Pauli equation as the O(ɛ) approximation of the Dirac equation. We suggest a coupling to the “magnetostatic”O(ɛ) approximation of the Maxwell equations consisting of Poisson equations for the four components of the potential. We sketch the semiclassical/nonrelativistic limits of this model.

Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations

The Maxwell-Dirac system describes the interaction of an electron with its own electromagnetic field. We prove the existence of soliton-like solutions of Maxwell-Dirac in (3+1)-Minkowski space-time. The solutions obtained are regular, stationary in time, and localized in space. They are found by a variational method, as critical points of an energy functional. This functional is strongly indefinite and presents a lack of compactness. We also find soliton-like solutions for the Klein-Gordon-Dirac system, arising in the Yukawa model.

A solitary wave solution of the Maxwell-Dirac equations

We investigate a class of localized, stationary, particular numerical solutions to the Maxwell-Dirac system of classical nonlinear field equations. The solutions are discrete energy eigenstates bound predominantly by the self-produced electric field.

Fields Connected with Particles in Terms of Complex-Riemannian Geometry

Abstract A complex analytical method of solving the generalised Dirac-Maxwell system has recently been proposed by two of us for a certain class of complex Riemannian metrics. The Dirac equation without the field potential in such a metric appeared to be equivalent to the Dirac-Maxwell system including the field potentials produced by the currents of a particle in question. The method proposed is connected with applying the Fourier transform with respect to the electric charge treated as a variable, with the consideration of the mass as an eigenvalue, and with solving suitable convolution equations. In the present research an explicit calculation based on linearization of the spinor connections is given. The conditions for the motion are interpreted as a starting point to seek selection rules for curved space-times corresponding to actually existing particles. Then the same method is applied to solids. Namely, by a suitable transformation of the configuration space in terms of elements of the interaction matrix corresponding to the Coulomb, exchange, and dipole integrals, the interaction term in the hamiltonian becomes zero, thus leading to experimentally verificable formulae for the autocorrelation time

Combining relativity and quantum mechanics: Schrödinger's interpretation of ψ

The incongruence between quantum theory and relativity theory is traced to the probability interpretation of the former. The classical continium interpretation of ψ removes the difficulty. How quantum properties of matter and light, and in particular the radiative problems, like spontaneous emission and Lamb shift, may be accounted in a first quantized Maxwell-Dirac system is discussed.