Explicit solutions of the inhomogeneous dirac equation

In this paper we establish a general principle which may be used to construct many explicit solutions to special inhomogeneous Dirac equations with distributional right-hand side. These solutions are presented as series of products of Clifford algebra valued functions which themselves satisfy Dirac equations in a lower dimension. We also present several special examples, including plane waves, zonal functions, Cauchy kernels and electromagnetic fields.     

Neutral Fermion Having Electric and Magnetic Moments in an External Electromagnetic Field

We show that in 2+1 dimensions, the Dirac equation for a neutral fermion possessing electric and magnetic dipole moments in an external electromagnetic field reduces to the Dirac equation for a charged fermion in a external field characterized by a certain 3-pseudo-vector potential. The effective charge of the neutral fermion is determined by its dipole moments. The effects of coupling electric and magnetic moments of the neutral fermion to the external electromagnetic field seem to be inseparable in physical experiments of any type. We find an exact solution of the Dirac equation for a massive neutral fermion with electric and magnetic dipole moments in a external plane-wave electromagnetic field. We derive expressions for the fermionic vacuum current induced by neutral fermions in the presence of external electromagnetic fields.     

Scattering of Dirac Particles by Electromagnetic Fields with Small Support in Two Dimensions and Effect from Scalar Potentials

We study the asymptotic behavior of scattering amplitudes for the scattering of Dirac particles in two dimensions when electromagnetic fields with small support shrink to point-like fields. The result is strongly affected by perturbations of scalar potentials and the asymptotic form changes discontinuously at half-integer fluxes of magnetic fields even for small perturbations. The analysis relies on the behavior at low energy of resolvents of magnetic Schrödinger operators with resonance at zero energy. The magnetic scattering of relativistic particles appears in the interaction of cosmic string with matter. We discuss this closely related subject as an application of the obtained results. Communicated by Bernard Helffersubmitted 05/05/03, accepted 31/07/03     

Symmetry and Classification of the Dirac–Fock Equation

We consider the properties of the Dirac–Fock equation with differential operators of the first-order symmetry. For a relativistic particle in an electromagnetic field, we describe the covariant properties of the Dirac equation in an arbitrary Riemannian space V₄ with the signature (?1,?1,?1, 1). We present a general form of the differential operator with a first-order symmetry and characterize the pair of such commuting operators. We list the spaces where the free Dirac equation admits at least one differential operator with a first-order symmetry. We perform a symmetry classification of electromagnetic field tensors and construct complete sets of symmetry operators.     

Radiative transport limit of Dirac equations with random electromagnetic field

Abstract

This paper concerns the kinetic limit of the Dirac equation with a random electromagnetic field. We give a detailed mathematical analysis of the radiative transport limit for the phase space energy density of solutions to the Dirac equation. Our derivation is based on a martingale method and a perturbed test function expansion. This requires the electromagnetic field to be a Markovian space-time random field. The main mathematical tool in the derivation of the kinetic limit is the matrix-valued Wigner transform of the vector-valued Dirac solution. The major novelty compared with the scalar (Schrödinger) case is the proof of the weak convergence of cross modes to zero. The propagating modes are shown to converge in an appropriate probabilistic sense to their deterministic limit.     

Fiber Bundles, Connections, General Relativity, and the Einstein-Cartan Theory – Part II

The formalism of tetrads and spin connections is presented, and with it, the Einstein-Cartan (E-C) equations are derived from the usual actions. It is shown how torsion vanishes in the case of pure gravity. Then, the equations in the presence of the Dirac and the Maxwell fields are derived. A possibility of choosing a locally inertial coordinate system in the presence of a totally antisymmetric torsion is proved. We discuss a conflict between torsion and the local gauge invariance of the electromagnetic field. Finally, we study Lorentz and Poincaré gauge invariance of general relativity and the E-C theory.     

Dispersion Relations for Threshold Reactions in an External Electromagnetic Field

Nonlinear threshold phenomena in quantum systems in a strong external electromagnetic field are investigated. A development of traditional quantum electrodynamics (QED) and a modified QED with new universal parameter, fundamental mass, are used. Analysis of the threshold phenomena in an external magnetic field is based on the modified Dirac equation in an external field. The analysis shows that the fundamental mass determines a bound on the external field strength. Dispersion relations are proved for some threshold reactions in rather general external electromagnetic fields.     

Discrete spectra of the Dirac Hamiltonian in Coulomb and Aharonov-Bohm potentials in 2+1 dimensions

We find all self-adjoint Dirac Hamiltonians in Coulomb and Aharonov-Bohm potentials in 2+1 dimensions with the fermion spin taken into account. We obtain implicit equations for the spectra and construct eigenfunctions for all self-adjoint Dirac Hamiltonians in the indicated external fields. We find explicit solutions of the equations for the spectra in some cases.     

Asymptotics of the scattering phase for the Dirac operator: High energy, semi-classical and non-relativistic limits

In this paper we prove several results for the scattering phase (spectral shift function) related with perturbations of the electromagnetic field for the Dirac operator in the Euclidean space. Many accurate results are now available for perturbations of the Schrödinger operator, in the high energy regime or in the semi-classical regime. Here we extend these results to the Dirac operator. There are several technical problems to overcome because the Dirac operator is a system, its symbol is a 4×4 matrix, and its continuous spectrum has positive and negative values. We show that we can separate positive and negative energies to prove high energy asymptotic expansion and we construct a semi-classical Foldy-Wouthuysen transformation in the semi-classical case. We also prove an asymptotic expansion for the scattering phase when the speed of light tends to infinity (non-relativistic limit).     

Strichartz and Smoothing Estimates for Dispersive Equations with Magnetic Potentials

We prove global smoothing and Strichartz estimates for the Schrödinger, wave, Klein–Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the magnetic part, while the electric part can be large. The decay and regularity assumptions on the coefficients are close to critical.