Dirac Particle in the Coulomb Field on the Background of Hyperbolic Lobachevsky Model

The known systems of radial equations describing the relativistic hydrogen atom on the base of the Dirac equation in Lobachevsky hyperbolic space is solved. The relevant 2-nd order differential equation has six regular singular points, its solutions of Frobenius type are constructed explicitly. To produce the quantization rule for energy values we have used the known condition for determination of the transcendental Frobenius solutions. This defines the energy spectrum which is physically interpretable and similar to the spectrum arising for the scalar Klein-Fock-Gordon equation in Lobachevsky space. In the present paper, exact analytical solutions referring to this spectrum are constructed. Convergence of the series involved is proved analytically and numerically. Squared integrability of the solutions is demonstrated numerically. It is shown that the spectrum coincides precisely with that previously found within the semi-classical approximation.

Magnetic dominance of axion electrodynamics: photon capture effect and anisotropy of Coulomb potential

For magnetic fields larger than the characteristic scale linked to axion-electrodynamics, quantum vacuum fluctuations due to axion-like fields can dominate over those associated with the electron-positron fields. This conjecture is explored by investigating both the axion-modified photon capture by a strong magnetic field and the Coulomb potential of a static pointlike charge. We show that in magnetic fields characteristic of neutron stars $$\sim 10^{13}$$ ∼ 10 13 –$$10^{15}\;\mathrm{G}$$ 10 15 G , the capture of gamma photons prior to the production of a pair can prevent the existence of an electron-positron plasma, essential for explaining the pulsar radiation mechanism. This incompatibility is used to limit the axion parameter space. Our bounds improve existing outcomes in the region of mass $$m\sim 10^{-10}$$ m ∼ 10 - 10 –$$10^{-5}\;{\mathrm{eV}}$$ 10 - 5 eV . The effect of capture, known in QED as relating to gamma-quanta, is extended in axion electrodynamics to include X-ray photons with the result that a specially polarized part of the heat radiation from the surface is canalized along the magnetic field. Besides, we find that in the regime in which the dominance takes place, the running QED coupling depends on the field strength and the modified Coulomb potential is of Yukawa-type in the direction perpendicular to the magnetic field at distances much smaller than the axion Compton wavelength, while along the field it follows approximately the Coulomb law at any length scale. Despite the Coulomb singularity manifested in the latter case, we argue that the ground-state energy of a non-relativistic hydrogen atom placed in a strong magnetic field turns out to be bounded due to the nonrenormalizable feature of axion-electrodynamics.

Symmetries of Relativistic Hydrogen Atom

The Dirac equation in the external Coulomb field is proved to possess the symmetry determined by 31 operators, which form the 31-dimensional algebra. Two different fermionic realizations of the SO(1,3) algebra of the Lorentz group are found. Two different bosonic realizations of this algebra are found as well. All generators of the above-mentioned algebras commute with the operator of the Dirac equation in an external Coulomb field and, therefore, determine the algebras of invariance of such Dirac equation. Hence, the spin s = (1, 0) Bose symmetry of the Dirac equation for the free spinor field, proved recently in our papers, is extended here for the Dirac equation interacting with an external Coulomb field. A relativistic hydrogen atom is modeled by such Dirac equation. We are able to prove for the relativistic hydrogen atom both the fermionic and bosonic symmetries known from our papers in the case of a non-interacting spinor field. New symmetry operators are found on the basis of new gamma matrix representations of the Clifford and SO(8) algebras, which are known from our recent papers as well. Hidden symmetries were found both in the canonical Foldy–Wouthuysen and covariant Dirac representations. The found symmetry operators, which are pure matrix ones in the Foldy–Wouthuysen representation, become non-local in the Dirac model.

An accurate parameterization for scattering factors, electron densities and electrostatic potentials for neutral atoms that obey all physical constraints

An efficient procedure and computer program are outlined for fitting numerical X-ray and electron scattering factors with the correct inclusion of all physical constraints. The numerical electron scattering factors have been parameterized using five analytic non-relativistic hydrogen electron scattering factors as basis functions for 103 neutral atoms of the periodic table. The inclusion of the correct physical constraints in the electron scattering factor and its derived quantities allows the use of the new parameterization in different fields. In terms of quality of the fit, the proposed parameterization of the electron scattering factor is one order of magnitude better than the previous analytic fittings.