Canonical transformation method in classical electrodynamics

The solutions of Maxwell's equations in the parabolic equation approximation is obtained on the basis of the canonical transformation method. The Hamiltonian form of the equations for the field in an anisotropic stratified medium is also examined. The perturbation theory for the calculation of the wave reflection and transmission coefficients is developed.     

Shear free solutions of Maxwell's equations

A twisting shear free solution of Maxwell's equations is obtained by transforming to a complex coordinate system in which the corresponding solution is complex but twist free. The equations in this system are easily solved, and, by transforming back to the original coordinates, a twisting shear free solution of Maxwell's equations is obtained.     

Free‐Electron Gas Spectrum Uniqueness in the Mixture of Noble Gases

Gas mixtures can reach the Maxwell's specter shape in case of low‐ionized mono‐atomic mixtures in the weak electric field. The parameters pertaining to the Maxwell spectrum of free electrons' gas straightforwardly settle on the insulating characteristics of the examined gas mixture at the fundamental level. In this paper, a condition for breakdown has been accomplished taking as a starting point the ionization coefficients derived accordingly, as well as the conditions for breakdown in keeping with the Townsend mechanism. The dc breakdown voltage value of noble gases mixture has been measured in the experimental part of the paper. The hypothesis that the free‐electron gas spectrum is unique in the noble gas mixture and is of Maxwell's type has been verified. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vacuum electrodynamics of accelerated systems: Nonlocal Maxwell's equations

The nonlocal electrodynamics of accelerated systems is discussed in connection with the development of Lorentz‐invariant nonlocal field equations. Nonlocal Maxwell's equations are presented explicitly for certain linearly accelerated systems. In general, the field equations remain nonlocal even after accelerated motion has ceased.     

Many‐body Green's function theory for thin ferromagnetic films: exact treatment of the single‐ion anisotropy

A theory for the magnetization of ferromagnetic films is formulated within the framework of many‐body Green's function theory which considers all components of the magnetization. The model Hamiltonian includes a Heisenberg term, an external magnetic field, a second‐ and fourth‐order uniaxial single‐ion anisotropy, and the magnetic dipole‐dipole coupling. The single‐ion anisotropy terms can be treated exactlyby introducing higher‐order Green's functions and subsequently taking advantage of relations between products of spin operators which leads to an automatic closure of the hierarchy of the equations of motion for the Green's functions with respect to the anisotropy terms. This is an improvement on the method of our previous work, which treated the corresponding terms only approximately by decoupling them at the level of the lowest‐order Green's functions. RPA‐like approximations are used to decouple the exchange interaction terms in both the low‐order and higher‐order Green's functions. As a first numerical example we apply the theory to a monolayer for spin S = 1 in order to demonstrate the superiority of the present treatment of the anisotropy terms over the previous approximate decouplings.     

Electromagnetic and Force Field Equations

In classical physics the electromagnetic equations are described by Maxwell's equations. Maxwell's equations proved to be invariant under gauge, or Lorentz transformations. Also, Einstein's equations of the special theory of relativity are invariant under Lorentz transformations. On the other hand classical mechanics and quantum mechanics laws are invariant under Galilean transformations. This means that, there are two different dynamical structures describing our universe. Einstein's unified field theory failled in putting our universe in one dynamical structure. New electromagnetic and force field equations are going to be derived. They have the same shape like Maxwell's equations, but with different dynamical structure. Those equations are invariant under Galilean transformations and in the density matrix formalism of quantum mechanics.     

Semiclassical expansion of quantum characteristics for many‐body potential scattering problem

In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star‐product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many‐body potential scattering problem simplifies to a statistical‐mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in ?. We present semiclassical expansion of quantum characteristics for many‐body scattering problem and provide tools for calculation of average values of time‐dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non‐locality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum‐mechanical extensions of the Liouville theorem on conservation of the phase‐space volume and the Poincaré theorem on conservation of 2p‐forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior.     

Implicit DG Method for Time Domain Maxwell's Equations Involving Metamaterials

An implicit discontinuous Galerkin method is introduced to solve the time-domain Maxwell's equations in metamaterials. The Maxwell's equations in metamaterials are represented by integral-differential equations. Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain. The fully discrete numerical scheme is proved to be unconditionally stable. When polynomial of degree at most $p$ is used for spatial approximation, our scheme is verified to converge at a rate of $\mathcal{O}(τ^2+h^{p+1/2})$. Numerical results in both 2D and 3D are provided to validate our theoretical prediction.     

The contribution of theC-field to the action functional

It is shown that the action of the usual field theory requires the inclusion of two terms in order to be equivalent, in the macroscopic case, to the action proposed by Hoyle & Narlikar (1964c). These actions correspond only to a modified form of Maxwell's equations, which, in consequence, lose their property of conformal invariance. It is also demonstrated how theC-field and electromagnetic field can be brought into unison by an appropriate re-definition of the vector potential. Both field theories can thus be described in terms of one vector Green's function.     

The uniqueness of the Einstein-Maxwell field equations

The most general Lagrange density (which is a concomitant of the metric tensor together with a vector field and its first derivatives) for which the associated Euler-Lagrange equations are precisely Maxwell's equations is obtained. Although it is more general than the Lagrangian which is commonly used, it still has essentially the same energy momentum tensor.     

Electro-geometrodynamics

By distinguishing between the metric of a Riemannian geometry and the interval defining function it is demonstrated that both Einstein's gravitational field equations and Maxwell's electromagnetic field equations can be generated from a single geometry.     

Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity

Microdisk lasers are investigated for their thresholds characteristics. We present a novel approach for studying the threshold gains of the whispering-gallery (WG) and other modes based on solving the boundary value problem for the Maxwell's equations. The novelty is that we consider the real-value pairs of frequencies and material gains as eigenvalues. In the two-dimensional (2D) approximation this problem is reduced to the set of independent transcendental equations. A Newton's method is further used to calculate the thresholds and natural frequencies numerically.     

On causality in nonlinear vacuum electrodynamics of the Plebański class

We investigate the Plebański class of electrodynamical theories, i.e., theories of nonlinear vacuum electrodynamics that derive from a Lorentz‐invariant Lagrangian (or Hamiltonian). In any such theory the light rays are the lightlike geodesics of two optical metrics that depend on the electromagnetic background field. A set of necessary and sufficient conditions is found whose fulfillment secures that the optical metrics are causal in the sense that the light rays are lightlike or timelike with respect to the underlying space‐time metric. Thereupon we derive conditions on the Lagrangian, or the Hamiltonian, of the theory such that the causality conditions are satisfied for all allowed background fields. (The allowed values of the field strength tensor are those for which the excitation tensor is finite and real.) The general results are illustrated with several examples.

     

On the calculation of <H 2> molecular integrals

The density matrix equations of motion arising in the triplet mechanism of chemically induced electron spin polarization are solved exactly without the imposition of the Redfield approximation. It is shown that the triplet spin relaxation time occurring in the final expression is not the true relaxation time because the spectral density involved depends both on the rotational correlation time and on the quenching rate. The effective spin relaxation time differs only slightly from the true time. The equations are extended to the case where the initial triplet passes on its polarization to the secondary triplet and exact solutions for the polarizations of the latter's doublets are obtained in the form Π_B = cΠ_A; an explicit expression for c is presented. The consequences of the secondary triplet being able to pass back its polarization to the initial triplet are explored and a ‘coherence effect’ on the polarization on the first triplet's doublets is analysed.     

Equations of motion for a (non‐linear) scalar field model as derived from the field equations

The problem of derivation of the equations of motion from the field equations is considered. Einstein's field equations have a specific analytical form: They are linear in the second order derivatives and quadratic in the first order derivatives of the field variables. We utilize this particular form and propose a novel algorithm for the derivation of the equations of motion from the field equations. It is based on the condition of the balance between the singular terms of the field equation. We apply the algorithm to a non‐linear Lorentz invariant scalar field model. We show that it results in the Newton law of attraction between the singularities of the field moved on approximately geodesic curves. The algorithm is applicable to the N‐body problem of the Lorentz invariant field equations.     

Third‐order nonlinear and linear time‐dependent dynamical diffraction of X‐rays in crystals

For the first time the third‐order nonlinear time‐dependent Takagi's equations of X‐rays in crystals are obtained and investigated. The third‐order nonlinear and linear time‐dependent dynamical diffraction of X‐rays spatially restricted in the diffraction plane pulses in crystals is investigated theoretically. A method of solving the linear and the third‐order nonlinear time‐dependent Takagi's equations is proposed. Based on this method, results of analytical and numerical calculations for both linear and nonlinear diffraction cases are presented and compared.