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Modified by a logarithmic term, the non-linear electrodynamics (NED) model of the Born-Infeld (BI) action is reconsidered. Unlike the standard BI action, this choice provides interesting integrals of the Einstein-NED equations. It is found that the spherical matching process for a regular black hole entails indispensable surface stresses that vanish only for a specific value of the BI parameter. This solution represents a classical model of an elementary particle whose radius coincides with the horizon. In flat space time, a charged particle becomes a conducting shell with a radius proportional to the BI parameter.

We are doubtlessly familiar with some edition of Jackson's tome on electrodynamics, and Schwinger's calculation of the anomalous magnetic moment of the electron in QED. From the perspective of strong interactions, however, electromagnetic effects usually amount to negligible contributions. Despite this fact, electromagnetic probes have always been a fundamental source for our knowledge of QCD experimentally. Elastic scattering of electrons off nucleons provides us a window to their distributions of charge and magnetism. To account for the spectrum of QCD at the percent level, moreover, we need isospin breaking introduced from both quark masses and electric charges. This overview concerns some of the prospects and progress of studying electromagnetic effects in QCD. Our focus is divided between classical and quantum effects. In classical electromagnetic fields, the dynamical response of QCD to external conditions can be investigated. The vacuum and hadrons alike should be viewed as media which respond to external fields: both magnetize and polarize in magnetic fields, for example. At the quantum level, electromagnetism and QCD renormalize each other. In the era of high precision lattice computations, both strong and electromagnetic contributions must be accounted for to make predictions at the percent level.

We analyze wave propagation in the vacuum of Podolsky regularized electrodynamics. Two kinds of waves were found in the theory: the traditional non-dispersive waves of Maxwell electrodynamics, and a dispersive wave reminiscent of wave propagation in a collisionless plasma. Charged particle concentration was determined, and found to be huge in this vacuum. We interpret the results in terms of vacuum polarization effects induced in an otherwise classical theory.

In the present work foundations of the law of the energy conservation and the introduction of particles in the classical electrodynamics are discussed. We pay attention to a logic error which takes place at an interpretation of the Poynting's theorem as the law of conservation of energy. It was shown that the laws of conservation of energy and momentum of the system of electromagnetic fields and charged particles does not follow from the equations of electrodynamics and the violation of these laws is displayed at the energy change of particles. Particular examples are considered which make it possible to restrict a possible kind of fields of a non-electromagnetic origin. We hope that this work will permit to produce a more comprehensive analysis and to stimulate the next step to the development of the foundations of the classical and quantum electrodynamics.

It is shown how point charges and point dipoles with finite self-energies can be accomodated into classical electrodynamics. The key idea is the introduction of constitutive relations for the electromagnetic vacuum, which actually mirrors the physical reality of vacuum polarization. Our results reduce to conventional electrodynamics for scales large compared to the classical electron radius $r_0\approx 2.8\times10^{-13}$ cm. A classical simulation for a structureless electron is proposed, with the appropriate values of mass, spin and magnetic moment.

Theory and experiment on the London moment is reviewed. A simple mathematical model is motivated and then used to study the responses of a spherical superconductor to an external field and to rotation. It reveals a connection between perfect diamagnetism (Meissner effect) and the London moment. In the model neither of these are exact but the deviation from B=0 internal field in the former and from $B=(2mc/e)\Omega$ in the latter case is described by the same dimensionless parameter. Apart from its pedagogical values the model might throw some light on the controversy surrounding the correction to the London moment.

In a $U(1)_{\star}$-noncommutative (NC) gauge field theory we extend the Seiberg-Witten (SW) map to include the (gauge-invariance-violating) external current and formulate - to the first order in the NC parameter - gauge-covariant classical field equations. We find solutions to these equations in the vacuum and in an external magnetic field, when the 4-current is a static electric charge of a finite size $a$, restricted from below by the elementary length. We impose extra boundary conditions, which we use to rule out all singularities, $1/r$ included, from the solutions. The static charge proves to be a magnetic dipole, with its magnetic moment being inversely proportional to its size $a$. The external magnetic field modifies the long-range Coulomb field and some electromagnetic form-factors. We also analyze the ambiguity in the SW map and show that at least to the order studied here it is equivalent to the ambiguity of adding a homogeneous solution to the current-conservation equation.

Excerpts are presented from a graduate course on Classical Electrodynamics held during the spring semester of 2000 at the Institute of Physics, Guanajuato State University, Mexico

We show that there exists a choice of gauge in which the electromagnetic 4-potential may be written as the difference of two 4-velocity vector fields describing the motion of a two-component space-filling relativistic fluid. Maxwell's equations are satisfied immediately, while the Lorentz force equation follows from the interactions of sources and sinks. The usual electromagnetic quantities then admit new interpretations as functions of the local 4-velocities. Electromagnetic waves are found to be described by oscillations of the underlying medium which can therefore be identified with the `luminiferous aether'. The formulation of electrodynamics in terms of 4-velocities is more general than that of the standard 4-potential in that it also allows for a classical description of a large class of vacuum energy configurations. Treated as a self-gravitating fluid, the model can be explicitly identified with Nelson's stochastic formulation of quantum mechanics, making it a promising candidate as the classical field theory unifying gravitation, electromagnetism and quantum theory which Einstein had sought.

We show that there exists a choice of gauge in which the electromagnetic 4-potential may be written as the difference of two 4-velocity vector fields describing the motion of a two-component space-filling relativistic fluid. Maxwell's equations are satisfied immediately, while the Lorentz force equation follows from the interactions of sources and sinks. The usual electromagnetic quantities then admit new interpretations as functions of the local 4-velocities. Electromagnetic waves are found to be described by oscillations of the underlying medium which can therefore be identified with the `luminiferous aether'. The formulation of electrodynamics in terms of 4-velocities is more general than that of the standard 4-potential in that it also allows for a classical description of a large class of vacuum energy configurations. Treated as a self-gravitating fluid, the model can be explicitly identified with Nelson's stochastic formulation of quantum mechanics, making it a promising candidate as the classical field theory unifying gravitation, electromagnetism and quantum theory which Einstein had sought.

We study the stability of the circular orbits of the electromagnetic two-body problem of classical electrodynamics. We introduce the concept of resonant dissipation, i.e. a motion that radiates the center-of-mass energy while the interparticle distance performs bounded oscillations about a metastable orbit. The stability mechanism is established by the existence of a quartic resonant constant generated by the stiff eigenvalues of the linear stability problem. This constant bounds the particles together during the radiative recoil. The condition of resonant dissipation predicts angular momenta for the metastable orbits in reasonable agreement with the Bohr atom. The principal result is that the emission lines agree with the predictions of quantum electrodynamics (QED) with 1 percent average error even up to the $40^{th}$ line. Our angular momenta depend logarithmically on the mass of the heavy body, such that the deuterium and the muonium atoms have essentially the same angular momenta, in agreement with QED. Analogously to QED, our stability analysis naturally uses the eigenvalues of an infinite-dimensional linear operator; the infinite-dimensionality is brought in by the delay of the electromagnetic interaction.

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.

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.

These digital-whiteboard video lectures were recorded by Professor Kirill Belashchenko for a graduate course on electrodynamics that was taught at the University of Nebraska-Lincoln in the flipped-classroom format. These lectures do not cover the entire material in the course and are intended to be used in combination with tutorials, which are available at Prof. Belashchenko's website. Special relativity 1. Interval 2. Lorentz transformation 3. Minkowski diagram 4. Twin paradox 5. 4-vectors and covariance 6. Relativistic mechanics Covariant electromagnetism 7. Electromagnetic field, part 1 8. Electromagnetic field, part 2 9. Electromagnetic action and Maxwell equations Conservation laws 10. Energy and momentum Electromagnetic field in matter 11. Field equations in a medium 12. Dissipation in a medium 13. Effective energy of the field in a medium 14. Reflection and refraction 15. Optical multilayers Waveguides and resonant cavities 16. Fields near the surface of a conductor 17. Cylindrical waveguide Radiation 18. Retarded potentials

This is the second part of the  Essential Graduate Physics  series of lecture notes and problems, which also includes 3 other parts (Classical Mechanics, Quantum Mechanics, and Statistical Mechanics),

This is the second part of the  Essential Graduate Physics  series of lecture notes and problems, which also includes 3 other parts (Classical Mechanics, Quantum Mechanics, and Statistical Mechanics),

One model for the motion of n charged particles on the x-axis leads to a system of delay differential equations with delays dependent on the unknown trajectories. If appropriate past histories of the trajectories are given, say on alpha, zero, then for sufficiently small equal to or greater than zero one has a system of n squared ordinary differential equations of the form y' = f(t,y) with y(0) = y sub zero are given. The function f, which involves the known past histories of the trajectories, is continuous; so existence of solutions is assured. However, f does not satisfy the Lipschitz condition usually used for proving uniqueness.

The problem of self forces and radiation reaction is solved by conservation of energy methods. The longstanding problem of constant acceleration is solved, and it is shown that the self force does indeed affect the particle's motion, as expected on physical grounds. The relativistic generalization is also presented.

A detailed study is made of the space-time transformation properties of intercharge forces and the associated electric and magnetic force fields, both in classical electrodynamics and in a recently developed relativistic classical electrodynamical theory. Important differences are found and serious errors are found in the traditional treatment of special-relativistic effects in classical electromagnetism. Fields associated with radiation processes are also considered and classical and quantum mechanical predictions are compared. An important, though trivial, mathematical error in the derivation of the retarded Li\'{e}nard-Wiechert retarded potentials is pointed out. It is also shown why electric and magnetic fields produced by moving source charges or current distributions are not `classical fields', and that they do not, in general, constitute an antisymmetic second-rank tensor, as is usually assumed.

Mainly on nano-scale, but maybe not exclusively, it can be imagined a spontaneous charge disjunction inside certain media due to the fluctuations, collisions, wall effects, radiation or/and other presently unknown interactions. The repulsive Casimir-Lifshitz force may be also a good candidate as a source of this kind of phenomenon. To cover this assumption mathematically an additional term should appear in the Maxwell equations. As a consequence of this term, similar Klein-Gordon equations with a negative mass term will be obtained for the electric and magnetic fields. These equations may have tachyonic solutions depending on the parameter of the charge disjunction process. Finally, the propagator is formulated, the electric field and the charge density are calculated during the charge disjunction.

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.

In the present work, we investigate classical solutions of the Maxwell-Carroll-Field-Jackiw-Proca (MCFJP) electrodynamics for the cases a purely timelike and spacelike Lorentz-violating (LV) background. Starting from the MCFJP Lagrangian and the associated wave equations written for the potential four-vector, the tensor form of the Green function is achieved. In the timelike case, the components of the stationary Green function are explicitly written. The classical solutions for the electric and magnetic field strengths are then evaluated, being observed that the electric sector is not modified by the LV background, keeping the Maxwell-Proca behavior. The magnetic field associated with a charge in uniform motion presents an oscillating behavior that also provides an oscillating MCFJ solution (in the limit of a vanishing Proca mass), but does not recover the Maxwell-Proca solution in the limit of vanishing background. In the spacelike case, the stationary Green function is written and also explicitly carried out in the regime of a small background. The electric and magnetic fields reveal to possess an exponentially decaying behavior, that recover the Maxwell-Proca solutions.

Fundamental errors in the Chubykalo et al paper [Found. of Phys. Lett, 19, 37-46 (2006)] are highlighted. Contrary to their claim that "... the irrotational component of the electric field has a physical meaning and can propagate exclusively instantaneously," it is shown that this instantaneous component is physically irrelevant because it is always canceled by a term contained into the solenoidal component. This result follows directly from the solution of the wave equation that satisfies the solenoidal component. Therefore the subsequent inference of these authors that there are two mechanisms of transmission of energy and momentum in classical electrodynamics, one retarded and the other one instantaneous, has no basis. The example given by these authors in which the full electric field of an oscillating charge equals its instantaneous irrotational component on the axis of oscillations is proved to be false.

The symmetry studies of Maxwell equations gave new insight on the nature of electromagnetic (EM) field. It has in general case quaternion single structure, consisting of four independent field constituents, which differ with each other by the parities under space inversion and time reversal. Generalized Maxwell equations for quaternion four-component EM-field are obtained. Invariants for EM-field, consisting of dually symmetric parts are found. It is shown, that there exists physical conserving quantity, which is simultaneously invariant under both Rainich dual and additional hyperbolic dual symmetry transformation of Maxwell equations. It is spin in general case and spirality in the geometry, when electrical and magnetic vectors $\vec{E}$, $\vec{H}$ are directed along coordinate axes in ($\vec{E}$, $\vec{H}$) functional space. It is additional proof for quaternion four component structure of EM-field to be a single whole. Canonical Dirac quantization method is developed in two aspects. The first aspect is its application the only to observable quantities. The second aspect is the realization along with well known time-local quantization of space-local quantization and space-time-local quantization. It is also shown, that Coulomb field can be quantized in 1D and 2D systems. New model of photons is proposed. The photons in quantized EM-field are main excitations in oscillator structure of EM-field, which is equivalent to spin S = 1 "boson-atomic" structure, like matematically to well known spin S = 1 boson matter structure - carbon atomic backbone chain structure in many conjugated polymers. They have two kind nature. The photons of the first kind and the second kind represent themselves respectively neutral chargeless EM-solitons and charged spinless EM-solitons of Su-Schrieffer-Heeger family.

In previous paper derivations of the Green function have been given for 5D off-shell electrodynamics in the framework of the manifestly covariant relativistic dynamics of Stueckelberg (with invariant evolution parameter $\tau$). In this paper, we reconcile these derivations resulting in different explicit forms, and relate our results to the conventional fundamental solutions of linear 5D wave equations published in the mathematical literature. We give physical arguments for the choice of the Green function retarded in the fifth variable $\tau$.

Offshell electrodynamics based on a manifestly covariant off-shell relativistic dynamics of Stueckelberg, Horwitz and Piron, is five-dimensional. In this paper, we study the problem of radiation reaction of a particle in motion in this framework. In particular, the case of above-mass-shell is studied in detail, where the renormalization of the Lorentz force leads to a system of non-linear differential equations for 3 Lorentz scalars. The system is then solved numerically, where it is shown that the mass-shell deviation scalar $\ve$ either smoothly falls down to 0 (this result provides a mechanism for the mass stability of the off-shell theory), or strongly diverges under more extreme conditions. In both cases, no runaway motion is observed. Stability analysis indicates that the system seems to have chaotic behavior in the divergent case. It is also shown that, although a motion under which the mass-shell deviation $\ve$ is constant but not-zero, is indeed possible, but, it is unstable, and eventually it either decays to 0 or diverges.

classical electrodynamics

classical electrodynamics

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.

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.

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.

In the classical theory of electromagnetism, the permittivity and the permeability of free space are constants whose magnitudes do not seem to possess any deeper physical meaning. By replacing the free space of classical physics with the quantum notion of the vacuum, we speculate that the values of the aforementioned constants could arise from the polarization and magnetization of virtual pairs in vacuum. A classical dispersion model with parameters determined by quantum and particle physics is employed to estimate their values. We find the correct orders of magnitude. Additionally, our simple assumptions yield an independent estimate for the number of charged elementary particles based on the known values of the permittivity and the permeability, and for the volume of a virtual pair. Such interpretation would provide an intriguing connection between the celebrated theory of classical electromagnetism and the quantum theory in the weak field limit.

We introduce an effective field theory approach that describes the motion of finite size objects under the influence of electromagnetic fields. We prove that leading order effects due to the finite radius $R$ of a spherically symmetric charge is order $R^2$ rather than order $R$ in any physical model, as widely claimed in the literature. This scaling arises as a consequence of Poincar\'e and gauge symmetries, which can be shown to exclude linear corrections. We use the formalism to calculate the leading order finite size correction to the Abraham-Lorentz-Dirac force.

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.

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.

In this article we will discuss some general aspects of mathematical physics. In Section:II we will show that a line is not just a collection of points and we will have to introduce one-dimensional line-intervals as fundamental geometric elements. Similar discussions are valid for area and volume-elements. In Section:III we will discuss a few mathematical aspects of Classical Electrodynamics with particular reference to the Robertson-Walker-Friedmann cosmological model with closed spatial geometry. We will first give an alternate derivation of the Gauss' law for the electric field of a point charge in the flat space-time not using the Gauss' divergence theorem. We will thereafter derive an exact expression for the electrostatic energy. The expression for the electrostatic energy obtained in this article agrees with the standard expression when the sources are not point particles. However we will show that the electrostatic self-energy of a point charge may be zero. We will also consider the consistency of the Maxwell's equations for the electromagnetic potentials in the Robertson-Walker-Friedmann universe with closed spatial geometry. We find that the contravariant electromagnetic fields are invariant under the gauge transformations and so also are the Maxwell's equations when they are expressed in terms of the contravariant electromagnetic fields. However the Maxwell's equations are not invariant under the gauge transformations when they are expressed in terms of the electromagnetic potentials. This may also remain valid in an arbitrary curved space-time.

A classical field theory is proposed for the electric current and the electromagnetic field interpolating between microscopic and macroscopic domains. It represents a generalization of the density functional for the dynamics of the current and the electromagnetic field in the quantum side of the crossover and reproduces standard classical electrodynamics on the other side. The effective action derived in the closed time path formalism and the equations of motion follow from the variational principle. The polarization of the Dirac-see can be taken into account in the quadratic approximation of the action by the introduction of the deplacement field strengths as in conventional classical electrodynamics. Decoherence appears naturally as a simple one-loop effect in this formalism. It is argued that the radiation time arrow is generated from the quantum boundary conditions in time by decoherence at the quantum-classical crossover and the Abraham-Lorentz force arises from the accelerating charge or from other charges in the macroscopic or the microscopic side, respectively. The functional form of quantum renormalization group, the generalization of the renormalization group method for the density matrix, is proposed to follow the scale dependence through the quantum-classical crossover in a systematical manner.

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 discuss a large class of phenomenological models incorporating quantum gravity motivated corrections to electrodynamics. The framework is that of electrodynamics in a birefringent and dispersive medium with non-local constitutive relations, which are considered up to second order in the inverse of the energy characterizing the quantum gravity scale. The energy-momentum tensor, Green functions and frequency dependent refraction indices are obtained, leading to departures from standard physics. The effective character of the theory is also emphasized by introducing a frequency cutoff. The analysis of its effects upon the standard notion of causality is performed, showing that in the radiation regime the expected corrections get further suppressed by highly oscillating terms, thus forbiding causality violations to show up in the corresponding observational effects.

In classical electrodynamics for rotating with variable angular velocity charged rigid sphere are found: the exact values of electromagnetic fields, the flux of radiating energy and the exact integral equation of rotation including the terms of self-rotation. The analysis of this eq. shows that on one hand, there is no "runaway" solutions, peculiar to the Lorentz-Dirac eq. of radiating point particle, on the other hand, there appear new problems, among them is the nonexistence of solution for some external moments of force.

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.

We present a pedagogical review of old inconsistencies of Classical Electrodynamics and of some new ideas that solve them. Problems with the electron equation of motion and with the non-integrable singularity of its self-field energy tensor are well known. They are consequences, we show, of neglecting terms that are null off the charge world-line but that give a non null contribution on its world-line. The electron self-field energy tensor is integrable without the use of any kind of renormalization; there is no causality violation and no conflict with energy conservation in the electron equation of motion, when its meaning is properly considered.

I will use the rationalised Lorentz-Heaviside system throughout (explained in Section 1). I will also assume, as prerequisite, that you know (i) the ordinary noncovariant form of Maxwell�s equations; (ii) the basic postulates of the Special Theory of Relativity, and (iii) how to get the Lagrangian and the Hamiltonian of a particle. In fact, for a certain section, I have to use the Lagrangian formulation for a �eld, but probably you have already encountered that in your Classical Mechanics course. Even if you have not, the formulation is analogous.

Contents: Maxwell�s electrodynamics; Electrostatics; Magnetostatics; Electromagnetic waves in matter; Electromagnetic radiation from slowly moving sources; Electrodynamics of point charges. Lecture Notes Collection FreeScience.info ID2225 Obtained from http://www.physics.uoguelph.ca/poisson/research/em.pdf http://www.freescience.info/go.php?pagename=books&id=2225

We show how a pair of superconducting qubits coupled to a microwave cavity mode can be used to engineer a single-atom laser that emits light into a non-classical state. Our scheme relies on the dressing of the qubit-field coupling by periodic modulations of the qubit energy. In the dressed basis, the radiative decay of the first qubit becomes an effective incoherent pumping mechanism that injects energy into the system, hence turning dissipation to our advantage. A second, auxiliary qubit is used to shape the decay within the cavity, in such a way that lasing occurs in a squeezed basis of the cavity mode. We characterize the system both by mean-field theory and exact calculations. Our work may find applications in the generation of squeezing and entanglement in circuit QED, as well as in the study of dissipative many-body phase transitions.

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.

We describe a seemingly unnoticed feature of the text-book Maxwell-Lorentz system of classical electrodynamics which challenges its formulation in terms of an initial value problem. For point-charges, even after appropriate renormalization, we demonstrate that most of the generic initial data evolves to develop singularities in the electromagnetic fields along the light cones of the initial charge positions. We provide explicit formulas for the corresponding fields, demonstrate how this phenomenon renders the initial value problem ill-posed, and show how such bad initial data can be ruled out by extra conditions in addition to the Maxwell constraints. These extra conditions, however, require knowledge of the history of the solution and, as we discuss, effectively turn the Maxwell-Lorentz system into a system of delay equations much like the Fokker-Schwarzschild-Tetrode equations. For extended charges such singular light fronts persist in a smoothened form and, as we argue, yield physically doubtful solutions. Our results also apply to some extent to expectation values of field operators in quantum field theory.

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).

There are known problems with the standard Lorentz-Dirac description of radiation reaction in classical electrodynamics. The model of extended in one dimension particle is proposed and is shown that for this model there is no total change in particle momentum due to radiation reaction

In Maxwell's classical theory of electrodynamics the fields are frequently expressed by potentials in order to facilitate the solution of the first order system of equations. This method obscures, however, that there exists an inconsistency between Faraday's law of induction and Maxwell's flux law. As a consequence of this internal contradiction there is neither gauge invariance, nor exist unique solutions in general. It is also demonstrated that inhomogeneous wave equations cannot be solved by retarded integrals.