Interaction between two point-like charges in nonlinear electrostatics

We consider two point-like charges in electrostatic interaction between them within the framework of a nonlinear model, associated with QED, that provides finiteness of their field energy. We find the common field of the two charges in a dipole-like approximation, where the separation between them $R$ is much smaller than the observation distance $r:$ with the linear accuracy with respect to the ratio $R/r$, and in the opposite approximation, where $R\gg r,$ up to the the term quadratic in the ratio $r/R$. The consideration fulfilled proposes the law $a+b R^{1/3}$ for the energy, when the charges are close to one another, $R\rightarrow 0$. This leads to the singularity of the force between them to be $R^{-2/3}$, which is weaker than Coulomb law $R^{-2}$.Comment: 20 pages, 1 figur

The Dirac equation in an external electromagnetic field: symmetry algebra and exact integration

Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are not obtained by separation of variables for several external electromagnetic fields. We have considered an example of crossed electric and magnetic fields of a special type for which the Dirac equation admits a nonlocal symmetry operato

Interaction between two point-like charges in nonlinear electrostatics

We consider two point-like charges in electrostatic interaction within the framework of a nonlinear model, associated with QED, that provides finiteness of their field energy. We find the common field of the two charges in a dipole-like approximation, where the separation between them R is much smaller than the observation distance r : with the linear accuracy with respect to the ratio R / r, and in the opposite approximation, where R≫r, up to the term quadratic in the ratio r / R. The consideration proposes the law a+bR1/3 for the energy, when the charges are close to one another, R→0 . This leads to the singularity of the force between them to be R−2/3 , which is weaker than the Coulomb law, R−2

Harmonic oscillator coherent states from the orbit theory standpoint

We study the known coherent states of a quantum harmonic oscillator from the standpoint of the original developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schr\"odinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution of a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation.Comment: 16 page

Symmetry operators and separation of variables in the (2+1)(2+1)-dimensional Dirac equation with external electromagnetic field

We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a $(2+1)$-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a $(2+1)$-dimensional Minkowski (flat) space. For each of the sets we carry out a complete separation of variables in the Dirac equation and find a corresponding electromagnetic potential permitting separation of variables.Comment: 24 pages, version accepted for publication in Int. J. Geom. Methods Mod. Phy

Resonant entanglement of photon beams by a magnetic field

In spite of the fact that photons do not interact with an external magnetic field, the latter field may indirectly affect photons in the presence of a charged environment. This opens up an interesting possibility to continuously control the entanglement of photon beams without using any crystalline devices. We study this possibility in the framework of an adequate QED model. In an approximation it was discovered that such entanglement has a resonant nature, namely, a peak behavior at certain magnetic field strengths, depending on characteristics of photon beams direction of the magnetic field and parameters of the charged medium. Numerical calculations illustrating the above-mentioned resonant behavior of the entanglement measure and some concluding remarks are presented.Comment: 20 pages, 2 figure

Vacuum instability in time-dependent electric fields. New example of exactly solvable case

A new exactly solvable case in strong-field quantum electrodynamics with a time-dependent external electric field is presented. The corresponding field is given by an analytic function, which is asymmetric (in contrast to Sauter-like electric field) with respect to the time instant, where it reaches its maximum value, that is why we call it the analytic asymmetric electric field. We managed to exactly solve the Dirac equation with such a field, which made it possible to calculate characteristics of the corresponding vacuum instability nonperturbatively. We construct the so-called in- and out-solutions and with their help calculate mean differential and total numbers of created charged particles, probability of the vacuum to remain a vacuum, vacuum mean values of current density and energy-momentum tensor of the particles. We study the vacuum instability in regimes of rapidly and slowly changing analytic asymmetric electric field, and compare the obtained results with corresponding ones obtained earlier for the case of the symmetric Sauter-like electric field. We also compare exact results in the regime of slowly changing field with corresponding results obtained within the slowly varying field approximation recently proposed by two of the authors, thus demonstrating the effectiveness of such an approximation.Comment: 27 pages, 7 figures, some minor changes introduce