Finite automata over algebraic structures: models and some methods of analysis

In this paper some results of research in two new trends of finite automata theory are presented. For understanding the value and the aim of these researches some short retrospective analysis of development of finite automata theory is given. The first trend deals with families of finite automata defined via recurrence relations on algebraic structures over finite rings. The problem of design of some algorithm that simulates with some accuracy any element of given family of automata is investigated. Some general scheme for design of families of hash functions defined by outputless automata is elaborated. Computational security of these families of hash functions is analyzed. Automata defined on varieties with some algebra are presented and their homomorphisms are characterized. Special case of these automata, namely automata on elliptic curves, are investigated in detail. The second trend deals with quantum automata. Languages accepted by some basic models of quantum automata under supposition that unitary operators associated with input alphabet commute each with the others are characterized

Lepton pair production by high-energy neutrino in an external electromagnetic field

The process of the lepton pair production by a neutrino propagating in an external electromagnetic field is investigated in the framework of the Standard Model. Relatively simple exact expression for the probability as the single integral is obtained, which is suitable for a quantitative analysis.Comment: 9 pages, LATEX, 2 PS figures, submitted to Modern Physics Letters

Photon-pair conversion into neutrinos in a strong magnetic field

A general analysis of the three-vertex loop amplitude in a strong magnetic field, based on the asymptotic form of the electron propagator in the field, is performed. In order to investigate the photon-neutrino process gamma gamma -> nu bar nu, the vertex combinations of the scalar-vector-vector (SVV), pseudoscalar-vector-vector (PVV), 3-vector (VVV), and axial-vector-vector-vector (AVV) types are considered. It is shown that only the SVV amplitude grows linearly with the magnetic field strength, while in the other amplitudes, PVV, VVV, and AVV, the linearly growing terms are cancelled. The process gamma gamma -> nu bar nu is investigated in the left-right-symmetric extension of the standard model of electroweak interaction, where the effective scalar nu nu e e coupling could exist. Possible astrophysical manifestations of the considered process are discussed.Comment: 9 pages, LaTeX, to appear in Modern Physics Letters

Critical nucleus charge in a superstrong magnetic field: effect of screening

A superstrong magnetic field stimulates the spontaneous production of positrons by naked nuclei by diminishing the value of the critical charge Z_{cr} . The phenomenon of screening of the Coulomb potential by a superstrong magnetic field which has been discovered recently acts in the opposite direction and prevents the nuclei with Z52 for a nucleus to become critical stronger B are needed than without taking screening into account.Comment: 13 pages, 2 figures, version to be published in Physical Review

Real and virtual photons in an external constant electromagnetic field of most general form

The photon behavior in an arbitrary superposition of constant magnetic and electric fields is considered on most general grounds basing on the first principles like Lorentz- gauge- charge- and parity-invariance. We make model- and approximation-independent, but still rather informative, statements about the behavior that the requirement of causal propagation prescribes to massive and massless branches of dispersion curves, and describe the way the eigenmodes are polarized. We find, as a consequence of Hermiticity in the transparency domain, that adding a smaller electric field to a strong magnetic field in parallel to the latter causes enhancement of birefringence. We find the magnetic field produced by a point electric charge far from it (a manifestation of magneto-electric phenomenon). We establish degeneracies of the polarization tensor that (under special kinematic conditions) occur due to space-time symmetries of the vacuum left after the external field is imposed.Comment: 30 pages, 1 figure, 57 equations, reference list of 38 item

Electric field of a pointlike charge in a strong magnetic field and ground state of a hydrogenlike atom

In an external constant magnetic field, so strong that the electron Larmour length is much shorter than its Compton length, we consider the modification of the Coulomb potential of a point charge owing to the vacuum polarization. We establish a short-range component of the static interaction in the Larmour scale, expressed as a Yukawa-like law, and reveal the corresponding "photon mass" parameter. The electrostatic force regains its long-range character in the Compton scale: the tail of the potential follows an anisotropic Coulomb law, decreasing away from the charge slower along the magnetic field and faster across. In the infinite-magnetic-field limit the potential is confined to an infinitely thin string passing though the charge parallel to the external field. This is the first evidence for dimensional reduction in the photon sector of quantum electrodynamics. The one-dimensional form of the potential on the string is derived that includes a delta-function centered in the charge. The nonrelativistic ground-state energy of a hydrogenlike atom is found with its use and shown not to be infinite in the infinite-field limit, contrary to what was commonly accepted before, when the vacuum polarization had been ignored. These results may be useful for studying properties of matter at the surface of extremely magnetized neutron stars.Comment: 45 pages, 6 figures, accepted to Phys. Rev.

Atomic levels in superstrong magnetic fields and D=2 QED of massive electrons: screening

The photon polarization operator in superstrong magnetic fields induces the dynamical photon "mass" which leads to screening of Coulomb potential at small distances $z\ll 1/m$, $m$ is the mass of an electron. We demonstrate that this behaviour is qualitatively different from the case of D=2 QED, where the same formula for a polarization operator leads to screening at large distances as well. Because of screening the ground state energy of the hydrogen atom at the magnetic fields $B \gg m^2/e^3$ has the finite value $E_0 = -me^4/2 \ln^2(1/e^6)$.Comment: 12 pages, 2 figure

Influence of the photon - neutrino processes on magnetar cooling

The photon-neutrino processes $\gamma e^{\pm} \to e^{\pm} \nu \bar \nu$, $\gamma \to \nu \bar \nu$ and $\gamma \gamma \to \nu \bar \nu$ are investigated in the presence of a strongly magnetized and dense electron-positron plasma. The amplitudes of the reactions $\gamma e^{\pm} \to e^{\pm} \nu \bar \nu$ and $\gamma \gamma \to \nu \bar \nu$ are obtained. In the case of a cold degenerate plasma contributions of the considering processes to neutrino emissivity are calculated. It is shown that contribution of the process $\gamma \gamma \to \nu \bar \nu$ to neutrino emissivity is supressed in comparision with the contributions of the processes $\gamma e^{\pm} \to e^{\pm} \nu \bar \nu$ and $\gamma \to \nu \bar \nu$. The constraint on the magnetic field strength in the magnetar outer crust is obtained.Comment: 8 pages, LaTeX, 2 PS figures, based on the talk presented by D.A. Rumyantsev at the XV International Seminar Quarks'2008, Sergiev Posad, Moscow Region, May 23-29, 2008, to appear in the Proceeding

Modification of Coulomb law and energy levels of the hydrogen atom in a superstrong magnetic field

We obtain the following analytical formula which describes the dependence of the electric potential of a point-like charge on the distance away from it in the direction of an external magnetic field B: \Phi(z) = e/|z| [ 1- exp(-\sqrt{6m_e^2}|z|) + exp(-\sqrt{(2/\pi) e^3 B + 6m_e^2} |z|) ]. The deviation from Coulomb's law becomes essential for B > 3\pi B_{cr}/\alpha = 3 \pi m_e^2/e^3 \approx 6 10^{16} G. In such superstrong fields, electrons are ultra-relativistic except those which occupy the lowest Landau level (LLL) and which have the energy epsilon_0^2 = m_e^2 + p_z^2. The energy spectrum on which LLL splits in the presence of the atomic nucleus is found analytically. For B > 3 \pi B_{cr}/\alpha, it substantially differs from the one obtained without accounting for the modification of the atomic potential.Comment: version to be published in Physical Review D (incorrect "Keywords" in previous version have been cancelled