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Presented in this paper is a new matrix representation of classical electromagnetic theory. The basis of this representation is a space-time, eight- by-eight differential matrix operator. This matrix operator is initially formulated from the differential form of the Maxwell field equations in vacuum. The resulting matrix formulation of Maxwell's equations allows simple and direct derivation of the electromagnetic wave and charge continuity equation s, the Lorentz conditions and definition of the electromagnetic potentials, the Lorentz and Coulomb gauges, the electromagnetic potential wave equations, and Poynting's conservation of energy theorem. A four-dimensional Fourier transform of the matrix equations casts them into an eight-dimensional transfer theorem. The transfer function has an inverse, and this allows the equations to be inverted. This inversion expresses the fields directly in terms of the charge and current source distributions, i. e., without the need for calculating intermediary potentials. This inversion formula is new, for the general scenario of space- and time-dependent sources. A simple pedagogical example is included illustrating use of the formulation.

The link between the longitudinal electromagnetic waves and the system of Maxwell equations is demonstrated. The longitudinal wave component of the electric field strength vector is found as the exact solution of the standard Maxwell equations with specific gradient-type case of electric current and charge densities.

We derive Huygens' principle for electrodynamics in terms of 4-vector potentials defined as distributions supported on a surface surrounding the charge-current density. By combining the Pauli algebra with distribution theory, a compact and conceptually simple derivation of the Stratton-Chu and Kottler-Franz equations is obtained. These are extended to freely moving integration surfaces, so that the fields due to charge distributions in arbitrary motion are represented. A further generalization is obtained to multiple surfaces, which can be used to enclose clusters of transmitters, scatterers and receivers.

In the direct action form of classical EM we give the equation of motion for a classical massless bare charge without self-interaction in the presence of an external field. That equation permits superluminal speeds and time-reversals, and so is a realization of the Stueckelberg-Feynman view of electrons and positrons as different segments of a single trajectory. We give a particular solution to a one body problem, and briefly discuss some aspects of the two-body problem. There is some discussion of the historical context of this effort, including the direct action and absorber theories, and some speculation on how the massless bare charge may acquire mass, and how these findings impact the problem of singular self-action.

We extend our previous work (see arXiv:quant-ph/0501026), which compared the predictions of quantum electrodynamics concerning radiation reaction with those of the Abraham-Lorentz-Dirac theory for a charged particle in linear motion. Specifically, we calculate the predictions for the change in position of a charged scalar particle, moving in three-dimensional space, due to the effect of radiation reaction in the one-photon-emission process in quantum electrodynamics. The scalar particle is assumed to be accelerated for a finite period of time by a three-dimensional electromagnetic potential dependent only on one of the spacetime coordinates. We perform this calculation in the $\hbar\to 0$ limit and show that the change in position agrees with that obtained in classical electrodynamics with the Lorentz-Dirac force treated as a perturbation. We also show for a time-dependent but space-independent electromagnetic potential that the forward-scattering amplitude at order $e^2$ does not contribute to the position change in the $\hbar \to 0$ limit after the mass renormalization is taken into account.

Equation of motion of Sommerfeld sphere in the one-dimensional potential hole, produced by two equal charges on some distance from each other, is numerically investigated. Two types of solutions are found: (i) damping oscillations, (ii) oscillations without damping (radiationless motion). Solutions with growing amplitude ("climbing-up-the-wall solution") for chosen initial conditions were not founded.

We calculate the first order maximal acceleration corrections to the classical electrodynamics of a particle in external electromagnetic fields. These include additional dissipation terms, the presence of a critical electric field, a correction to the cyclotron frequency of an electron in a constant magnetic field and the power radiated by the particle. The electric effects are sizeble at the fields that are considered attainable with ultrashort $TW$ laser pulses on plasmas.

Real time rearrangement of particle spectra is studied numerically in a U(1) Gauge+Higgs system, in the unitary gauge and in three spatial dimensions. The cold system starts from the symmetric phase. Evolution of the partial energy densities and pressures reveal well-defined equations of state for the longitudinal and transversal gauge fields very early. Longitudinal modes are excited more efficiently and thermalize the slowest. Hausdorff-dimension of the Higgs-defect manifold, eventually seeding vortex excitations is thoroughly discussed. Scaling dependence of the vortex density on the characteristic time of the symmetry breaking transition is established.

There are three electromagnetic integrals of motion that can be interpreted as the energy. These are the background energy, the elastic energy and the integral in the torsion field commonly referred to as the energy of the electromagnetic field. The integral in the torsion field gains the meaning of the energy insomuch as it is concerned with the mechanical energy of a charged particle.

Standard formulae of classical electromagnetism for the forces between electric charges in motion derived from retarded potentials are compared with those obtained from a recently developed relativistic classical electrodynamic theory with an instantaneous inter-charge force. Problems discussed include small angle Rutherford scattering, Jackson's recent `torque paradox' and circular Keplerian orbits. Results consistent with special relativity are obtained only with an instantaneous interaction. The impossiblity of stable circular motion with retarded fields in either classical electromagnetism or Newtonian gravitation is demonstrated.

We report on some conceptual changes in our present understanding of Quantum Field Theory and muse about possible consequences for the understanding of $v>c$ signals.

We report on some conceptual changes in our present understanding of Quantum Field Theory and muse about possible consequences for the understanding of $v>c$ signals.

The new derivation of the equation of the spin precession is given for a particle possessing electric and magnetic dipole moments. Contributions from classical electrodynamics and from the Thomas effect are explicitly separated. A fully covariant approach is used. The final equation is expressed in a very simple form in terms of the fields in the instantaneously accompanying frame. The Lorentz transformations of the electric and magnetic dipole moments and of the spin are derived from basic equations of classical electrodynamics. For this purpose, the Maxwell equations in matter are used and the result is confirmed by other methods. An antisymmetric four-tensor is correctly constructed from the electric and magnetic dipole moments.

The behavior of a classical charged point particle under the influence of only a Coulombic binding potential and classical electromagnetic zero-point radiation, is shown to yield agreement with the probability density distribution of Schroedinger's wave equation for the ground state of hydrogen. These results, obtained without any fitting parameters, again raise the possibility that the main tenets of stochastic electrodynamics (SED) are correct, thereby potentially providing a more fundamental basis of quantum mechanics. The present methods should help propel yet deeper investigations into SED.

The Maxwell electromagnetic and the Lorentz type force equations are derived in the framework of the R. Feynman proper time paradigm and the related vacuum field theory approach. The electron inertia problem is analyzed within the Lagrangian and Hamiltonian formalisms and the related pressure-energy compensation principle. The modified Abraham- Lorentz damping radiation force is derived, the electromagnetic electron mass origin is argued.

We first write down a very general description of nonlinear classical electrodynamics, making use of generalized constitutive equations and constitutive tensors. Our approach includes non-Lagrangian as well as Lagrangian theories, allows for electromagnetic fields in the widest possible variety of media (anisotropic, piroelectric, chiral and ferromagnetic), and accommodates the incorporation of nonlocal effects. We formulate electric-magnetic duality in terms of the constitutive tensors. We then propose a supersymmetric version of the general constitutive equations, in a superfield approach.

This book is a manual for the course of electrodynamics and theory of relativity. It is recommended primarily for studentsof mathematical departments. This de�nes its style: I use elements of vectorial and tensorial analysis, di_erential geometry, and theory of distributions in it.

de la Pe\~na 1980 and Puthoff 1987 show that circular orbits in the hydrogen problem of Stochastic Electrodynamics are stable. Though the Cole-Zou 2003 simulations support the stability, our recent numerics always lead to self-ionisation. Here the de la Pe\~na-Puthoff argument is extended to elliptic orbits. For very eccentric orbits with energy close to zero and angular momentum below some not-small value, there is on the average a net gain in energy for each revolution, which explains the self-ionisation. Next, an $1/r^2$ potential is added, which could stem from a dipolar deformation of the nuclear charge by the electron at its moving position. This shape retains the analytical solvability. When it is enough repulsive, the ground state of this modified hydrogen problem is predicted to be stable. The same conclusions hold for positronium.

de la Pe\~na 1980 and Puthoff 1987 show that circular orbits in the hydrogen problem of Stochastic Electrodynamics are stable. Though the Cole-Zou 2003 simulations support the stability, our recent numerics always lead to self-ionisation. Here the de la Pe\~na-Puthoff argument is extended to elliptic orbits. For very eccentric orbits with energy close to zero and angular momentum below some not-small value, there is on the average a net gain in energy for each revolution, which explains the self-ionisation. Next, an $1/r^2$ potential is added, which could stem from a dipolar deformation of the nuclear charge by the electron at its moving position. This shape retains the analytical solvability. When it is enough repulsive, the ground state of this modified hydrogen problem is predicted to be stable. The same conclusions hold for positronium.

The Maxwell equations are formulated in a generally covariant and metric-free way in 1+3 and subsequently in 4 dimensions. For this purpose, we use the excitations $\cal D$, $\cal H$ and the field strengths $E,B$. A local and linear constitutive law between excitations and field strengths is assumed, with a constitutive tensor $\chi^{ijkl}=-\chi^{jikl}=-\chi^{ijlk}$ of 36 components. The properties of this tensor are discussed. In particular, we address the validity of the Post constraint, a subject that is very much under discussion. In this connection, the Tellegen gyrator, the axion field, and the ``perfect electromagnetic conductor'' of Lindell & Sihvola are compared with each other.

This is the reply to `COMMENT ON ``HELMHOLTZ THEOREM AND THE V-GAUGE IN THE PROBLEM OF SUPERLUMINAL AND INSTANTANEOUS SIGNALS IN CLASSICAL ELECTRODYNAMICS" BY A. CHUBYKALO ET AL' BY J. A. HERAS [FOUND. PHYS. LETT. vol. 19(6) p. 579 (2006)]

We prove that, contrary to the common belief, the classical Maxwell electrodynamics of a point-like particle may be formulated as an infinite-dimensional Hamiltonian system. We derive well defined quasi-Hamiltonian which possesses direct physical interpretation being equal to the total energy of the composed (field + particle) system. The phase space of this system is endowed with an interesting symplectic structure. We prove that this structure is strongly non-degenerated and, therefore, enables one to define consistent Poisson bracket for particle's and field degrees of freedom. We stress that this formulation is perfectly gauge-invariant.

A new approach for solving electromagnetic wave propagation problems is currently being developed at the Naval Command, Control and Ocean Surveillance Center (NCCOSC), RDT and E Division (NRaD). This new approach is based upon an 8 by 8 matrix representation of the Maxwell field equations. In addition, a computer software package based on this matrix representation of electromagnetic theory is also being written and tested at NRaD to handle a variety of scenarios involving electromagnetic wave propagation through matter. This software package is referred to as the MATURE Program. MATURE is the acronym for Matrix Approach To Understanding Relativistic Electrodynamics. The MATURE Program is written in MATLAB code for use on a Sun 4 SPARCstation 2 workstation. Under Independent Research (IR) FY 92 funding, this matrix approach was successfully employed in solving problems dealing with electromagnetic wave propagation through dielectric, crystalline, linear electro-optic, and magneto- optic materials of infinite extent. Under the Office of Naval Research (ONR) FY 93 funding, this matrix formulation was extended to handle problems involving wave propagation through multilayer dielectric media with planar boundaries. Presented in this technical document is the underlying theory of this matrix approach. Several numerical examples, based on the use of the MATURE Program, are also included to illustrate the use of the matrix approach in solving electromagnetic wave propagation problems.

The usual action integral of classical electrodynamics is derived starting from Lanczos's electrodynamics -- a pure field theory in which charged particles are identified with singularities of the homogeneous Maxwell's equations interpreted as a generalization of the Cauchy-Riemann regularity conditions from complex to biquaternion functions of four complex variables. It is shown that contrary to the usual theory based on the inhomogeneous Maxwell's equations, in which charged particles are identified with the sources, there is no divergence in the self-interaction so that the mass is finite, and that the only approximation made in the derivation are the usual conditions required for the internal consistency of classical electrodynamics. Moreover, it is found that the radius of the boundary surface enclosing a singularity interpreted as an electron is on the same order as that of the hypothetical "bag" confining the quarks in a hadron, so that Lanczos's electrodynamics is engaging the reconsideration of many fundamental concepts related to the nature of elementary particles.

Numerical values of the Fresnel intensity reflection coefficients in tables and graphs are presented. Reflection coefficients are given for normal and oblique incidence for approximately 2500 indices of refraction. Graphs illustrate the solutions of the Fresnel equa tions. This volume 1 of the 2 volume report contains all the narrative, discussion, and graphs.

We propose a new relativistic Lorentz-invariant spin-noncommutative algebra. Using the Weyl ordering of noncommutative position operators, we build an analogue of the Moyal-Groenewald product for the proposed algebra. The Lagrange function of an electromagnetic field in the space with spin noncommutativity is constructed. In such a space electromagnetic field becomes non-abelian. A gauge transformation law of this field is also obtained. Exact nonlinear field equations of noncommutative electromagnetic field are derived from the least action principle. Within the perturbative approach we consider field of a point charge in a constant magnetic field and interaction of two plane waves. An exact solution of a plane wave propagation in a constant magnetic and electric fields is found.

Expectation values of the electromagnetic field and the electric current are introduced at space-time resolution which belongs to the quantum domain. These allow us to approach some key features of classical electrodynamics from the underlying QED. One is the emergence of the radiation field in the retarded solution of the Maxwell equation, derived from an action principle. Another question discussed is the systematic derivation of the polarizability of a charge system. Furthermore, the decoherence and the consistency of the photon field is established by a perturbative calculation of the reduced density matrix for the electromagnetic field within the Closed Time Path formalism.

A non-existence theorem of classical electrodynamics in odd-dimensional spacetimes is shown to be invalid. The source of the error is pointed out, and is then demonstrated during the derivation of the fields generated by a uniformly moving point source.

Classical Electrodynamics Schwinger, Deraad, Milton, Tsai

It is proved that, according to Classical Mechanics and Electrodynamics, the trajectory of the center of mass of a neutral system of electrical charges can be deflected by an inhomogeneous magnetic field, even if its internal angular momentum is zero. This challenges the common view about the function of the Stern-Gerlach apparatus, as resolving the eigen-states of an intrinsic angular momentum. Doubts are cast also on the supposed failure of Schrodinger's theory to explain the properties of atoms in presence of magnetic fields without introducing spin variables.

The principle of equivalence in gravitational physics and its mathematical base are reviewed. It is demonstrated how this principle can be realized in classical electrodynamis. In general, it is valid at any given single point or along a path without selfintersections unless the field considered satisfies some conditions.

In the present article, we discuss a modification of classical electrodynamics in which ``ordinary'' point charges are absent. The modified equations contain additional terms describing the induced charges and currents. The densities of the induced charges and currents depend on the vector k and the vectors of the electromagnetic field E and B. It is shown that the vectors E and B can be defined in terms of two 4-potentials and the components of k are the components of the 4-tensor of the third rank. The Lagrangian of modified electrodynamics is defined. The conditions are derived at which only one 4-potential determines the behavior of the electromagnetic field. It is also shown that static modified electrodynamics can describe the electromagnetic field in the inner region of the electric monopole. In the outer region of the electric monopole the electric field is governed by the Maxwell equations. It follows from boundary conditions at the interface between the inner and outer regions of the monopole that the vector k has a discrete spectrum. The electric and magnetic fields, energy and angular momentum of the monopole are found for different eigenvalues of k.

Presented in this document is the development of a new matrix description of electromagnetic wave propagation in optical media of infinite extent. This material will interest individuals desiring a description of electromagnetic wave propagation that deviates from the traditional vector calculus approach. Our starting point will be with the fundamental equations of classical electrodynamics, namely the Maxwell field equations. From the vector form of Maxwell's equations, and 8-by-8 differential matrix operator formulation of Maxwell's equations will be developed. The matrix form of the Maxwell field equations allows for simple and direct derivation of matrix representations of the electromagnetic wave and charge continuity equations, the Lorentz conditions and definition of the electromagnetic potentials, the electromagnetic potential wave equations, and Poynting's conservation of energy theorem. The matrix form of the Maxwell field equations and the electromagnetic wave and continuity equations will be used to solve a variety of wave-propagation problems dealing with linear, homogeneous, anisotropic optical media of infinite extent in the presence of monochromatic plane-wave electromagnetic fields. The indices of refraction as well as corresponding states of polarization, associated with wave propagation in crystalline, optically active, and electrooptical media, will be determined by using these matrix representations.

The general theory for electric current multipoles appearing at the motion of magnetic dipoles and change in these values or orientation has been suggested. Static multipoles, including an anapole, have been studied in detail.

The problems of Classical Electrodynamics with the electron equation of motion and with non-integrable singularity of its self-field stress tensor are well known. They are consequences, we show, of neglecting terms that are null off the charge world line but that gives a non null contribution on its world line. The self-field stress tensor of a point classical electron is integrable, there is no causality violation and no conflict with energy conservation in its equation of motion, and there is no need of any kind of renormalization nor of any change in the Maxwell's theory for this. (This is part of the paper hep-th/9510160, stripped , for simplicity, of its non-Minkowskian geometrization of causality and of its discussion about the physical meaning of the Maxwell-Faraday concept of field).

Classical electrodynamics is reformulated in terms of wave functions in the classical phase space of electrodynamics, following the Koopman-von Neumann-Sudarshan prescription for classical mechanics on Hilbert spaces {\em sans} the superselection rule which prohibits interference effects in classical mechanics. This is accomplished by transforming from a set of commuting observables in one Hilbert space to another set of commuting observables in a larger Hilbert space. This is necessary to clarify the theoretical basis of much recent work on quantum-like features exhibited by classical optics. Furthermore, following Bondar et al ({\em Phys.Rev. A} {\bf 88}, 052108, (2013)), it is pointed out that quantum processes that preserve the positivity or nonpositivity of the Wigner function can be implemented by classical optics. This may be useful in interpreting quantum information processing in terms of classical optics.

It is shown that the well-known procedure for proving the equivalence of the expressions for the electric field calculated using the Lorentz and Coulomb gauges is incorrect. The difference between the two gauges is due to the difference in the speed of propagation of a disturbance of the scalar potential. As an auxiliary result, it is proven that the solution for the electric field cannot be obtained directly from the Maxwell equations, i.e. without introducing the scalar and vector potentials.

We present the Fortran code SIMLA, which is designed for the study of charged particle dynamics in laser and other background fields. This can be done classically via the Landau-Lifshitz equation, or alternatively, via the simulation of photon emission events determined by strong-field quantum-electrodynamics amplitudes and implemented using Monte-Carlo type routines. Multiple laser fields can be included in the simulation and the propagation direction, beam shape (plane wave, focussed paraxial, constant crossed, or constant magnetic), and time envelope of each can be independently specified.

We present the Fortran code SIMLA, which is designed for the study of charged particle dynamics in laser and other background fields. This can be done classically via the Landau-Lifshitz equation, or alternatively, via the simulation of photon emission events determined by strong-field quantum-electrodynamics amplitudes and implemented using Monte-Carlo type routines. Multiple laser fields can be included in the simulation and the propagation direction, beam shape (plane wave, focussed paraxial, constant crossed, or constant magnetic), and time envelope of each can be independently specified.

A logical error in the usual derivation of the energy conservation law is analyzed, and a way to avoid the error is presented.

For the first time a method is devised for non-iterative modeling of motion of a radiating, electrified pointlike mass that has an internal structure. New, supplementary kinetic constants of accelerated charged particles are defined, that can be assessed by analysis of their trajectories in an accelerator.

A proof sheet for General Classical Electrodynamics

Various competing mathematical models are used in classical electrodynamics. To test, and possibly eliminate, some of these models one can apply them to the two body problem and see whether reasonable results are obtained. In one model it is assumed that each particle is influenced by both the past and future behavior of the other. The special case of two electrons moving symmetrically in one dimension was considered; and it was found that this curious model does make sense mathematically provided the two electrons never get too close together. Further studies under this grant led to a simple method for analyzing the asymptotic behavior of solutions problem of certain linear delay differential equations. This is useful in a one-body problem of electrodynamics with radiation reaction,in problems of control theory with time lags, in the telegraph equation, and other applications. Currently work in progress is aimed at understanding the simplest n-body problem of electrodynamics with interactions occurring only through retarded fields.

In [1, arXiv:1005.2617] effective field theory methods have been employed to compute the equations of motion of a spherically symmetric charged shell of radius R, taking into account the radiation reaction force exerted by the shell's own electromagnetic field up to O(R^2). The authors of Ref. [1] have stated that the known result for the self force of the shell as can be found from Eq. (16.28) of the textbook of Jackson [2] (see also Chap. 4 in the review of Pearle [3]) is incorrect, in that the term linear in R should be absent. We claim that this conclusion of Ref. [1] is incorrect, and that the textbook result, Eq. (1) does hold.

The work is devoted to studying some new classical electrodynamics models of interacting charged point particles and the aspects of the quantization via the Dirac procedure related to them. Based on the vacuum field theory no-geometry approach developed in [6,7,9], the Lagrangian and Hamiltonian reformulations of some alternative classical electrodynamics models are devised. The Dirac-type quantization procedure for the considered alternative electrodynamics models, based on the obtained canonical Hamiltonian formulations, is developed.

The region very close to an electron ($r < < r_0 = e^2/mc^2 \approx 2.8\times 10^{-13}$ cm) is, according to quantum electrodynamics, a seething maelstrom of virtual electron-positron pairs flashing in and out of existence. To take account of this well-established physical reality, a phenomenological representation for vacuum polarization is introduced into the framework of classical electrodynamics. Such a model enables a consistent picture of classical point charges with finite electromagnetic self-energy. It is further conjectured that the reaction of a point charge to its own electromagnetic field is tantamount to interaction with its vacuum polarization charge or "aura". This leads to a modification of the Lorentz-Dirac equation for the force on an accelerating electron, a new differential-difference equation which avoids the pathologies of preacceleration and runaway solutions.

We discuss, in the context of classical electrodynamics with a Lorentz invariant cut-off at short distances, the self-force acting on a point charged particle. It follows that the electromagnetic mass of the point charge occurs in the equation of motion in a form consistent with special relativity. We find that the exact equation of motion does not exhibit runaway solutions or non-causal behavior, when the cut-off is larger than half of the classical radius of the electron.

We present Dirac's method for using dual potentials to solve classical electrodynamics for an oppositely charged pair of particles, with a view to extending these techniques to non-Abelian gauge theories.

Minkowski spacetime is a convenient setting for the study of the relativistic dynamics of particles and fields in the vacuum. In order to study events that occur in a dielectric or other linear medium, we adopt the familiar continuum assumption of a linear, isotropic, homogeneous, transparent medium of refractive index n filling all space and seek the principle of relativity that applies in the filled spacetime. Applying the Einstein postulates with c/n as the speed of light, we show how the effective signal velocity results in a scaling of the proper time by the refractive index and examine the consequences for D'Alembert's principle, the Lagrange equations, and the canonical momentum field. The principles of dynamics in the filled spacetime are then applied to the electromagnetic Lagrangian and we derive equations of motion that are invariant with respect to a material Lorentz transformation. The new representation of the dynamics of macroscopic fields is shown to be consistent with the equal-time commutation relation for quantized macroscopic fields, quantum--classical correspondence, the principle of superposition, and electromagnetic boundary conditions.

The time evolution of a charged point particle is governed by a second-order integro-differential equation that exhibits advanced effects, in which the particle responds to an external force before the force is applied. In this paper we give a simple physical argument that clarifies the origin and physical meaning of these advanced effects, and we compare ordinary electrodynamics with a toy model of electrodynamics in which advanced effects do not occur.