THE PROBLEM OF MOTION IN THE KALUZA THEORY

It is shown that the equations of motion for a charged massive particle are consequences of the field equations in Kaluza unification theory of gravitation and electromagnetism, i.e., the equations of motion for the particle can be deduced from Kaluza field equations, just as that in Einstein's theory of motion of general relativity the equations of motion for a massive particle are consequences of the Einstein equations. Furthermore, the Lorentz equations for a particle maving in the Maxwell electromagnetic field on the Minkowskian space-time can also be obtained from the Maxwell equations by means of the Kaluze mechanism of the Maxwell theory.     

Kaluza理论的运动定律

本文指出,如同在广义相对论中粒子运动方程是场方程的推论一样,在引力场与电磁场的Kaluza统一理论中,粒子的运动方程也是场方程的一个推论,即带电粒子在引力场和电磁场中的运动方程可以从Kaluza统一理论中的场方程推导出来。本文进而在Minkowski时空的条件下,借助Maxwell理论的Kaluza形式,得到Maxwell方程也包含了带电粒子运动方程的结论。 关键词：     

关于麦克斯韦方程组的讨论

本文讨论了麦克斯韦方程组的来龙去脉、形式和地位，对麦克斯韦主程组的教学具有一定的参考价值。     

Maxwell Equations in the Spinning Cosmic String Spacetimes

Maxwell equations are explicitly written in spin weighted form by using NP formalism in general flat spacetimes. Corresponding Maxwell equations are obtained in spinning and straight cosmic string backgrounds.     

Extremal Kähler metrics and Bach–Merkulov equations

In this paper, we study a coupled system of equations on oriented compact 4-manifolds which we call the Bach–Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein–Maxwell equations. Inspired by the work of C. LeBrun on Einstein–Maxwell equations on compact Kähler surfaces, we give a variational characterization of solutions to Bach–Merkulov equations as critical points of the Weyl functional. We also show that extremal Kähler metrics are solutions to these equations, although, contrary to the Einstein–Maxwell analogue, they are not necessarily minimizers of the Weyl functional. We illustrate this phenomenon by studying the Calabi action on Hirzebruch surfaces.     

Can the macroscopic Maxwell equations be obtained from the microscopic Maxwell-Lorentz equations by performing averages?

It is shown that the usual procedures of obtaining the macroscopic Maxwell equations from the microscopic Maxwell-Lorentz equations by performing averages contain an arbitrary choice of gauge. By a suitable different choice of the gauge the so-obtained Maxwell equations can be cast back to the form of the starting Maxwell-Lorentz equations. Therefore one cannot consider the Maxwell equations to be obtainable from the Maxwell-Lorentz equations by simply performing averages. The implication of this result is that besides the electromagnetic fields produced by the moving electric charges, as given by the Maxwell-Lorentz equations, there may be some other agents that cannot be identified as some kind of motion of the electric charges and that participate in the production of the electromagnetic fields.     

On a relation between Maxwell and Dirac theories

It is shown that the Dirac equation can be written in a form similar to Maxwell equations, where the Maxwell tensor is written as a bilinear expression of the Dirac field and the current is a simple function of the external potential and the Dirac field. Similarly, the Maxwell equations can be written as a self-coupled Dirac equation where the potential is a simple function of the Dirac field itself. It is illustrated by examples how the new formalism helps to find solutions of the coupled field equations.     

Comments on the unification of electromagnetism and gravitation through “Generalized Einstein manifolds”

Akbar-Zadeh [J. Geom. Phys. 17 (1995) 342] has recently proposed a new geometric formulation of Einstein-Maxwell system with source in terms of what are called “Generalized Einstein manifolds”. We show that, contrary to the claim, Maxwell equations have not been derived in this formulation, and that the assumed equations can be identified only as source-free Maxwell equations in the proposed geometric setup. A genuine derivation of source-free Maxwell equations is presented within the same framework. We draw a conclusion that the proposed unification scheme can pertain only to source-free situations.     

Nonlinear Maxwell Equations in Inhomogeneous Media

We study the basic properties of the Maxwell equations for nonlinear inhomogeneous media. Assuming the classical nonlinear optics representation for the nonlinear polarization as a power series, we show that the solution exists and is unique in an appropriate space if the excitation current is not too large. The solution to the nonlinear Maxwell equations is represented as a power series in terms of the solution of the corresponding linear Maxwell equations. This representation holds at least for the time period inversely proportional to the appropriate norm of the solution to the linear Maxwell equation. We derive recursive formulas for the terms of the power series for the solution including an explicit formula for the first significant term attributed to the nonlinearity.     

Group analysis of the nonlinear electrodynamics equations

The symmetry of the Maxwell equations is investiated in this paper in a nonlinear isotropic medium without currents and charges. A group-theoretic classification is given of the material equations and partially invariant solutions. Exact solutions of the Maxwell equations are found.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 27–31, July, 1990.     

Octonionic Lorenz-like condition

In this study, the octonion algebra and its general properties are defined by the Cayley–Dickson’s multiplication rules for octonion units. The field equations, potential equations and Maxwell equations for electromagnetism are investigated with the octonionic equations and these equations can be compared with their vectorial representations. The potential and wave equations for fields with sources are also provided. By using Maxwell equations, a Lorenz-like condition is newly suggested for electromagnetism. The existing equations including the photon mass provide the most acknowledged Lorenz condition for the magnetic monopole and the source.     

Numerical solutions to 2D Maxwell–Bloch equations

Rare-earth-doped crystals contain inhomogeneously broadened two-level atoms. Optical propagation and nonlinear interaction in the crystals can be described by the Maxwell–Bloch equations. We show a consistent numerical approach that solves Maxwell’s equations by using the FFT-finite difference beam propagation method and the Bloch equations by using the finite difference method. Numerical simulation results are given for an off-axis 3-pulse photon echo.     

Numerical modelling of the two-dimensional Fourier transformed Vlasov–Maxwell system

The two-dimensional Vlasov–Maxwell system, for a plasma with mobile, magnetised electrons and ions, is investigated numerically. A previously developed method for solving the two-dimensional electrostatic Vlasov equation, Fourier transformed in velocity space, for mobile electrons and with ions fixed in space, is generalised to the fully electromagnetic, two-dimensional Vlasov–Maxwell system for mobile electrons and ions. Special attention is paid to the conservation of the divergences of the electric and magnetic fields in the Maxwell equations. The Maxwell equations are rewritten, by means of the Lorentz potentials, in a form which conserves these divergences. Linear phenomena are investigated numerically and compared with theory and with previous numerical results.     

Towards the deformation quantization of linearized gravity

We present a first attempt to apply the approach of deformation quantization to linearized Einstein's equations. We use the analogy with Maxwell equations to derive the field equations of linearized gravity from a modified Maxwell Lagrangian which allows the construction of a Hamiltonian in the standard way. The deformation quantization procedure for free fields is applied to this Hamiltonian. As a result we obtain the complete set of quantum states and the discrete energy spectrum of linearized gravity.     

Fractional dual solutions to the Maxwell equations in chiral nihility medium

Solutions which seem as the dual solutions to the original solution of the Maxwell equations for chiral nihility medium are listed. Using operators composed of fractional curl, solutions to the Maxwell equations which may be regarded as intermediate step between the original solution and dual to the original solution are determined. Dual solutions which are not valid have been pointed out.     

Spinor and electromagnetic fields in space with torsion

A combined system of Einstein-Cartan, Ivanenko-Heisenberg, and Maxwell equations is reduced to a combined system of Einstein, Dirac, and Maxwell equations in Riemann space. A solution in which metric singularities are absent is found.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 31–35, November, 1978.     

Scaling theory for homogenization of the Maxwell equations

A scaling theory for homogenization of the Maxwell equations is developed upon the representation of any field as a sum of its dipole, quadrupole, and magnetic dipole moments. This representation is exact and is connected neither with multipole expansion nor with the Helmholtz theorem. A chain of hierarchical equations is derived to calculate the moments. It is shown that the resulting macroscopic fields are governed by the homogenized Maxwell equations. Generally, these fields differ from the mean values of microscopic fields.     

Numerical Simulation of Antennae by Discrete Exterior Calculus

Numerical simulation of antennae is a topic in computational lectromagnetism, which is concerned with the numerical study of Maxwell equations. By discrete exterior calculus and the lattice gauge theory with coefficient R, weobtain the Bianchi identity on prism lattice. By defining an inner product of discrete differential forms, we derive the source equation and continuity equation. Those equations compose the discrete Maxwell equations in vacuum case on discrete manifold, which are implemented on Java development platform to simulate the Gaussian pulse radiation on antennaes.     

Determination of the Electromagnetic Lagrangian from a System of Poisson Brackets

The Lagrangian and Hamiltonian formulations of electromagnetism are reviewed and the Maxwell equations are obtained from the Hamiltonian for a system of many electric charges. It is shown that three of the equations which were obtained from the Hamiltonian, namely the Lorentz force law and two Maxwell equations, can be obtained as well from a set of postulated Poisson brackets. It is shown how the results derived from these brackets can be used to reconstruct the original Lagrangian for the theory aided by some reasoning based on physical concepts.     

Dirac and Maxwell equations in the Clifford and spin-Clifford bundles

We show how to write the Dirac and the generalized Maxwell equations (including monopoles) in the Clifford and spin-Clifford bundles (of differential forms) over space-time (either of signaturep=1,q=3 orp=3,q=1). In our approach Dirac and Maxwell fields are represented by objects of the same mathematical nature and the Dirac and Maxwell equations can then be directly compared. We show also that all presentations of the Maxwell equations in (matrix) Dirac-like spinor form appearing in the literature can be obtained by choosing particular global idempotents in the bundles referred to above. We investigate also the transformation laws under the action of the Lorentz group of Dirac and Maxwell fields (defined as algebraic spinor sections of the Clifford or spin-Clifford bundles), clearing up several misunderstandings and misconceptions found in the literature. Among the many new results, we exhibit a factorization of the Maxwell field into two-component spinor fields (Weyl spinors), which is important.