20 pages, LaTeX, Internal Report FIAN 1975, the Preprint FIAN No 196, 1980. This paper is the Preprint FIAN No 35, 1997 enlarged by Appendix, some Comments and foot-notes. Presented to Uspekhi Phyzicheskich Nauk
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.
The axiomatic structure of the electromagnetic theory is outlined. We will base classical electrodynamics on (1) electric charge conservation, (2) the Lorentz force, (3) magnetic flux conservation, and (4) on the Maxwell-Lorentz spacetime relations. This yields the Maxwell equations. The consequences will be drawn, inter alia, for the interpretation and the dimension of the electric and magnetic fields.
Stochastic electrodynamics is the classical electrodynamic theory of interacting point charges which includes random classical radiation with a Lorentz-invariant spectrum whose scale is set by Planck's constant. Here we give a cursory overview of the basic ideas of stochastic electrodynamics, of the successes of the theory, and of its connections to quantum theory.
In this paper we propose an approach to the problem of two body motion in classical electrodynamics that takes into account the electromagnetic radiation and the radiation reaction forces. The resulting differential equations are solved numerically.
In nonrelativistic approximation one-dimensional motion of Sommerfeld sphere in the case of potential barrier is numerically investigated. The effect of classical tunneling is confirmed once more - Sommerfeld sphere overcomes the barrier and finds itself in the forbidden, from classical point of view, area.
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.
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.
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.
In this paper we express the retarded fields of Maxwell's theory in terms of the instantaneous fields of a Galilei-invariant electromagnetic and we find the vector function whose spatial and temporal derivatives transform the instantaneous fields into the retarded ones. We conclude that the instantaneous fields can formally be introduced as unphysical objects into classical electrodynamics which can be used to find the physical retarded fields.
The concept of gauge invariance in classical electrodynamics assumes tacitly that Maxwell's equations have unique solutions. By calculating the electromagnetic field of a moving particle both in Lorenz and in Coulomb gauge and directly from the field equations we obtain, however, contradicting solutions. We conclude that the tacit assumption of uniqueness is not justified. The reason for this failure is traced back to the inhomogeneous wave equations which connect the propagating fields and their sources at the same time.
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 th…
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.
It is shown that the well-known disparity in classical electrodynamics between the power radiated in electromagnetic fields and the power-loss, as calculated from the radiation reaction on a charge undergoing a non-uniform motion, is successfully resolved when a proper distinction is made between quantities expressed in terms of a "real time" and those expressed in terms of a retarded time. It is shown that the expression for the real-time radiative power loss from a charged particle is somewhat different from the familiar Larmor's formula, or in a relativistic case, from Liénard's formula.
It is shown that the total energy of the static "field + particle" system, defined in the framework of classical, renormalized electrodynamics of particles and fields, depends in an unstable way upon the field boundary data. It is argued that this phenomenon may be also an origin of the unstable dynamical behaviour of the system (i.e. existence of "runaway solutions"). It is proved that a suitable polarization mechanism of the particle restores the stability, at least on the level of statics. Whether or not it restores also the full, dynamical stability of the theory is still an open question.
The force due to electromagnetic induction on a test charge is calculated in different reference frames. The Faraday-Lenz Law and different formulae for the fields of a uniformly moving charge are used. The classical Heaviside formula for the electric field of a moving charge predicts that, for the particular spatial configuration considered, the inductive force vanishes in the frame in which the magnet is in motion and the test charge at rest. In contrast, consistent results, in different frames, are given by the recently derived formulae of relativistic classical electrodynamics.
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 t…
Flaws and ambiguities are pointed out upon examining the comment attempting to solve a problem as raised recently --- the currently accepted formulation of electromagnetic radiation of an accelerated charge violates the principle of conservation of energy. This problem is not solved by the comment, due to a misunderstanding in the meaning of the total radiated power crossing a sphere. An experiment is suggested to determine whether on not the currently accepted formulation is valid.
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…
Convergence of the full energy (mass) of point charged particle by means of direct calculation is proved. The consideration is based on the strict solutions of nonlinear equations system of electrostatics and gravistatics in the classical field approach. Analytical calculation in the case of point charge mass of Markov's "Friedmon" M=e/(G^0.5) has confirmed. It is shown that mass defect caused by gravitational interaction leads to degradation of bare (phenomenological) mass and to its full vanishing from a total mass of system in the limit of point particle.
The dynamic $U (1)$ symmetry of classical electrodynamics without charges, similar to the known phase symmetry of quantum mechanics, is analyzed. Using this symmetry, alternative Lagrangians of classical electrodynamics, similar to quantum-mechanical ones, were found, and their symmetry properties were investigated.
We will display the fundamental structure of classical electrodynamics. Starting from the axioms of (1) electric charge conservation, (2) the existence of a Lorentz force density, and (3) magnetic flux conservation, we will derive Maxwell's equations. They are expressed in terms of the field strengths $(E,{\cal B})$, the excitations $({\cal D},H)$, and the sources $(ρ,j)$. This fundamental set of four microphysical equations has to be supplemented by somewhat less general constitutive assumptions in order to make it a fully determined system with a well-posed initial value problem. It is only …
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 onl…
We define and compute the renormalized four-momentum of the composed physical system: classical Maxwell field interacting with charged point particles. As a `reference' configuration for the field surrounding the particle, we take the Born solution. Unlike in the previous approach [Gen. Relat. Grav. 26 (1994) 167; Acta Phys. Pol. A 85 (1994) 771; Commun. Math. Phys. 198 (1998), 711], based on the Coulomb `reference', a dependence of the four-momentum of the particle (`dressed' with the Born solution) upon its acceleration arises in a natural way. This will change the resulting equations of mot…
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…
One-dimensional motion of Sommerfeld sphere in the case of potential barrier is numerically investigated. The effect of classical tunneling is found out - Sommerfeld sphere overcomes the barrier and finds itself in the forbidden, from classical point of view, area.
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.
It is demonstrated how all the mechanical equations of classical electrodynamics (CEM) may be derived from only Coulomb's inverse square force law, special relativity and Hamilton's Principle. The instantaneous nature of the Coulomb force in the centre-of-mass frame of two interacting charged objects, mediated by the exchange of space-like virtual photons, is predicted by QED. The interaction Lagrangian of QED is shown to be identical, in the appropriate limit, to the potential energy term in the Lorentz-invariant Lagrangian of CEM. A comparison is made with the Feynman-Wheeler action-at-a-dis…
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 following is the very first set of the series in 'Problems and Solutions in a Graduate Course in Classical Electrodynamics'. In each of the sets of the problems we intend to follow a theme, which not only makes it unique but also deals with the investigation of a certain class of problems in a thorough and rigorous manner. Figures are not included.
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.
On simple example is shown that a model of mechanically extended source of radiation in classical electrodynamics indeed has some problems with radiation reaction force. So one cannot, in general case, state, that the present situation with classical dynamics of mechanically extended sources of radiation is "very satisfactory".
We look at the foundations of electromagnetism in this 1st LeCosPA Symposium. For doing this, after some review (constraints on photon mass etc.), we use two approaches. The first one is to formulate a Parametrized Post-Maxwellian (PPM) framework to include QED corrections and a pseudoscalar photon interaction. PPM framework includes lowest corrections to unified electromagnetism-gravity theories based on connection approach. It may also overlap with corrections implemented from generalized uncertainty principle (GUP) when electromagnetism-gravity coupling is considered. We discuss various vac…
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.
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.
In this note we show that Jackson's criticism (hep-ph/0203076) of our work "Necessity of simultaneous co-existence of..." is based on an inexact understanding of the basic assumptions and conclusions of our work.
I show that the main statement of [Phys. Rev. Lett. 105, 094802 (2010), arXiv:1005.2617] is wrong. Other inconsistences of this Letter are also discussed.
Equation of motion of Sommerfeld sphere in the field of Coulomb center is numerically investigated. It is shown that contrary to Lorentz-Dirac equation in the attractive case there are physical solutions. In the repulsive case sphere gains less energy then that should be according to relativistic equation of motion of point charge without radiation force.
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 electromagneti…
If the conventional Maxwell--Lorentz formulation of classical electrodynamics is adopted in a flat spacetime of arbitrary odd dimension, then the retarded vector potential $A^μ$ generated by a point charge turns out to be pure gauge, $A^μ=\partial^μχ$. By Gauss' law, the charge shows up as zero. The classical electromagnetic coupling is thus missing from odd-dimensional worlds. If the action is augmented by the addition of the Chern--Simons term, then the classical interaction picture in the three-dimensional world becomes nontrivial.
The foundations of QM can find a consistent and exhaustive explanation in a new theoretical context. The Bridge Theory (BT) allows us to justify both classic and quantum electromagnetic phenomenology by using classical concepts. In this paper we review the bases of the BT originating from the role that the transverse component of the Poynting vector plays in localising energy in the neighbourhood of an electromagnetic source and we analyse the quantum implications.
Classical electrodynamics can be based on the conservation laws of electric charge and magnetic flux. Both laws are independent of the metric and the linear connection of spacetime. Within the framework of such a premetric electrodynamics -- provided a local and linear constitutive law of the vacuum is added -- the propagation of electromagnetic waves in the geometric-optics limit can be studied. The wave vectors of the wave fronts obey a quartic extended Fresnel equation. If one forbids birefringence in vacuum, the light cone emerges and Maxwell-Lorentz vacuum electrodynamics can be recovered…
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.
The energy-impulse flow of electromagnetic field, produced by the arbitrary moving charged particle, through the sphere of large radius is calculated. The result formulae are used in problem of radiation produced by hyperbolically moving particle and the problem of the synchrotron radiation energy loss. The energy-impulse radiated by a moving charged particle is shown to be not an invariant 4-tensor, for this reason calculation methods based on change of reference frame are not correct.
It is common in the literature on classical electrodynamics and relativity theory that the transformation rules for the basic electrodynamic quantities are derived from the pre-assumption that the equations of electrodynamics are covariant against these---unknown---transformation rules. There are several problems to be raised concerning these derivations. This is, however, not our main concern in this paper. Even if these derivations are regarded as unquestionable, they leave open the following fundamental question: Are the so-obtained transformation rules indeed identical with the true transf…
It is shown that the universal Newton time may be introduced in the classical electrodynamics. The statement results from an existence of the generalized symmetry of Maxwell equations with respect to Galilei transformations (physics/9701006). In the case of the extended Galilei transformations the postulate of invariance of the speed of light may be made compatible with the concept of the universal Newton time. Some physical consequences of the extended Galilei symmetry are considered.
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 give a concise axiomatic introduction into the fundamental structure of classical electrodynamics: It is based on electric charge conservation, the Lorentz force, magnetic flux conservation, and the existence of local and linear constitutive relations. The {\it inhomogeneous} Maxwell equations, expressed in terms of $D^i$ and $H_i$, turn out to be a consequence of electric charge conservation, whereas the {\it homogeneous} Maxwell equations, expressed in terms of $E_i$ and $B^i$, are derived from magnetic flux conservation and special relativity theory. The excitations $D^i$ and $H_i$, by m…
In this work we substantiate the applying of the Helmholtz vector decomposition theorem (H-theorem) to vector fields in classical electrodynamics. Using the H-theorem, within the framework of the two-parameter Lorentz-like gauge (so called v-gauge), we show that two kinds of magnetic vector potentials exist: one of them (solenoidal) can act exclusively with the velocity of light c and the other one (irrotational) with an arbitrary finite velocity $v$ (including a velocity more than c . We show also that the irrotational component of the electric field has a physical meaning and can propagate e…
Classical electrodynamics is a local theory describing local interactions between charges and electromagnetic fields and therefore one would not expect that this theory could predict nonlocal effects. But this perception implicitly assumes that the electromagnetic configurations lie in simply connected regions. In this paper we consider an electromagnetic configuration lying in a non-simply connected region, which consists of a charged particle encircling an infinitely-long solenoid enclosing a uniform magnetic flux, and show that the electromagnetic angular momentum of this configuration desc…
Invoking Maxwell's classical equations in conjunction with expressions for the electromagnetic (EM) energy, momentum, force, and torque, we use a few simple examples to demonstrate the nature of the EM angular momentum. The energy and the angular momentum of an EM field will be shown to have an intimate relationship; a source radiating EM angular momentum will, of necessity, pick up an equal but opposite amount of mechanical angular momentum; and the spin and orbital angular momenta of the EM field, when absorbed by a small particle, will be seen to elicit different responses from the particle…
It is shown that a newly derived "exact expression" for radiation of an accelerated charge in the recent literature is simply incorrect, having arisen because of a wrong relativistic transformation of the distance parameter. The ensuing claim that the newly derived expression alone that satisfies the energy conservation for the electromagnetic radiation, is based on a wrong reasoning where a proper distinction between the time during which the radiation is received and the time for emission (retarded time of the charge) was not maintained.
