The inadequacy of Liénard-Wiechert potentials is demonstrated as one of the examples related to the inconsistency of the conventional classical electrodynamics. The insufficiency of the Faraday-Maxwell concept to describe the whole electromagnetic phenomena and the incompleteness of a set of solutions of Maxwell equations are discussed and mathematically proved. Reasons of the introduction of the so-called "electrodynamics dualism concept" (simultaneous coexistence of instantaneous Newton long-range and Faraday-Maxwell short-range interactions) have been displayed. It is strictly shown that the new concept presents itself as the direct consequence of the complete set of Maxwell equations and makes it possible to consider classical electrodynamics as a self-consistent and complete theory, devoid of inward contradictions. In the framework of the new approach, all main concepts of classical electrodynamics are reconsidered. In particular, a limited class of motion is revealed when accelerated charges do not radiate electromagnetic field.

We start with the fact that the actual consensus on the empirical verification of the standard retardation condition does not take into account the complex structure of the whole electromagnetic field in the near zone. The most rigorous and methodologically consistent way to the empirical test of the causal behavior of the whole EM field should be based on testing causal properties for each component—velocity dependent (bound) and acceleration dependent (radiation)—taken separately. Preliminary discussions on the relative contribution of bound and radiation terms for an idealized magnetic dipole provided us with a methodological approach to causal characteristics of bound fields. In view of practical implementation of this method, we made an analysis of the finite size loop antennas. The use of multisection loop antennas was fully justified by a substantial rise of the ratio of bound-to-radiation field strength. Finally, we effected numerical calculations, taking into account particular experimental settings. Comparison with the experimentally obtained data showed considerable discrepancy with the predictions of the standard electromagnetic theory. A possible interpretation in terms of nonlocal properties of bound fields in near zone is proposed.

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 twoparameter 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 exclusively instantaneously.

Some mathematical inconsistencies in the conventional form of Maxwell's equations extended by Lorentz for a single charge system are discussed. To surmount these in framework of Maxwellian theory, a novel convection displacement current is considered as additional and complementary to the famous Maxwell displacement current. It is shown that this form of the Maxwell-Lorentz equations is similar to that proposed by Hertz for electrodynamics of bodies in motion. Original Maxwell's equations can be considered as a valid approximation for a continuous and closed (or going to infinity) conduction current. It is also proved that our novel form of the Maxwell-Lorentz equations is relativistically invariant. In particular, a relativistically invariant gauge for quasistatic fields has been found to replace the non-invariant Coulomb gauge. The new gauge condition contains the famous relationship between electric and magnetic potentials for one uniformly moving charge that is usually attributed to the Lorentz transformations. Thus, for the first time, using the convection displacement current, a physical interpretation is given to the relationship between the components of the four-vector of quasistatic potentials. A rigorous application of the new gauge transformation with the Lorentz gauge transforms the basic field equations into an independent pair of differential equations responsible for longitudinal and transverse fields, respectively. The longitudinal components can be interpreted exclusively from the standpoint of the instantaneous "action at a distance" concept and leads to necessary conceptual revision of the conventional Faraday-Maxwell field. The concept of electrodynamic dualism is proposed for self-consistent classical electrodynamics. It implies simultaneous coexistence of instantaneous long-range (longitudinal) and Faraday-Maxwell short-range (transverse) interactions that resembles in this aspect the basic idea of Helmholtz's electrodynamics.

We analyzed the triple Higgs boson self-coupling at future e + e − colliders energies, with the reactions e + e − → bbHH, ttHH. We evaluate the total cross-sections for both bbHH and ttHH, and calculate the total number of events considering the complete set of Feynman diagrams at tree-level. We vary the triple coupling κλ 3H within the range κ = −1 and +2. The numerical computation is done for the energies expected to be available at a possible Future Linear e + e − Collider with a center-of-mass energy 800, 1000, 1500 GeV and a luminosity 1000 f b −1 . Our analysis is also extended to a center-of-mass energy 3 T eV and luminosities of 1000 f b −1 and 5000 f b −1 . We found that for the process e + e − → bbHH, the complete calculation differs only by 3% from the approximate calculation e + e − → ZHH(Z → bb), while for the process e + e − → ttHH, the expected number of events, considering the decay products of both t and H, is not enough to obtain an accurate determination of the triple Higgs boson self-coupling.