Cornelius Lanczos’s Derivation of the Usual Action Integral of 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 approximations 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.     

On “Gauge Renormalization” in Classical Electrodynamics

In this paper we pay attention to the inconsistency in the derivation of the symmetric electromagnetic energy–momentum tensor for a system of charged particles from its canonical form, when the homogeneous Maxwell’s equations are applied to the symmetrizing gauge transformation, while the non-homogeneous Maxwell’s equations are used to obtain the motional equation. Applying the appropriate non-homogeneous Maxwell’s equations to both operations, we obtained an additional symmetric term in the tensor, named as “compensating term”. Analyzing the structure of this “compensating term”, we suggested a method of “gauge renormalization”, which allows transforming the divergent terms of classical electrodynamics (infinite self-force, self-energy and self-momentum) to converging integrals. The motional equation obtained for a non-radiating charged particle does not contain its self-force, and the mass parameter includes the sum of mechanical and electromagnetic masses. The motional equation for a radiating particle also contains the sum of mechanical and electromagnetic masses, and does not yield any “runaway solutions”. It has been shown that the energy flux in a free electromagnetic field is guided by the Poynting vector, whereas the energy flux in a bound EM field is described by the generalized Umov’s vector, defined in the paper. The problem of electromagnetic momentum is also examined.     

Comparison between Weber’s electrodynamics and classical electrodynamics

We present the main aspects of Weber’s electrodynamics and of Maxwell’s equations. We discuss Maxwell’s point of view related to Weber’s electrodynamics. We compare Weber’s force with Lorentz’s force. We analyse the relation between Weber’s law and Maxwell’s equations. Finally, we discuss some experiments performed and proposed with which we can distinguish Weber’s force from Lorentz’s one.     

Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions

We study the Maxwell–Dirac equations in a manifestly gauge invariant presentation using only the spinor bilinear scalar and pseudoscalar densities, and the vector and pseudovector currents, together with their quadratic Fierz relations. The internally produced vector potential is expressed via algebraic manipulation of the Dirac equation, as a rational function of the Fierz bilinears and first derivatives (valid on the support of the scalar density), which allows a gauge invariant vector potential to be defined. This leads to a Fierz bilinear formulation of the Maxwell tensor and of the Maxwell–Dirac equations, without any reference to gauge dependent quantities. We show how demanding invariance of tensor fields under the action of a fixed (but arbitrary) Lie subgroup of the Poincaré group leads to symmetry reduced equations. The procedure is illustrated, and the reduced equations worked out explicitly for standard spherical and cylindrical cases, which are coupled third order nonlinear PDEs. Spherical symmetry necessitates the existence of magnetic monopoles, which do not affect the coupled Maxwell–Dirac system due to magnetic terms cancelling. In this paper we do not take up numerical computations. As a demonstration of the power of our approach, we also work out the symmetry reduced equations for two distinct classes of dimension 4 one-parameter families of Poincaré subgroups, one splitting and one non-splitting. The splitting class yields no solutions, whereas for the non-splitting class we find a family of formal exact solutions in closed form.     

Are Words the Quanta of Human Language? Extending the Domain of Quantum Cognition

In previous research, we showed that ‘texts that tell a story’ exhibit a statistical structure that is not Maxwell–Boltzmann but Bose–Einstein. Our explanation is that this is due to the presence of ‘indistinguishability’ in human language as a result of the same words in different parts of the story being indistinguishable from one another, in much the same way that ’indistinguishability’ occurs in quantum mechanics, also there leading to the presence of Bose–Einstein rather than Maxwell–Boltzmann as a statistical structure. In the current article, we set out to provide an explanation for this Bose–Einstein statistics in human language. We show that it is the presence of ‘meaning’ in ‘texts that tell a story’ that gives rise to the lack of independence characteristic of Bose–Einstein, and provides conclusive evidence that ‘words can be considered the quanta of human language’, structurally similar to how ‘photons are the quanta of electromagnetic radiation’. Using several studies on entanglement from our Brussels research group, we also show, by introducing the von Neumann entropy for human language, that it is also the presence of ‘meaning’ in texts that makes the entropy of a total text smaller relative to the entropy of the words composing it. We explain how the new insights in this article fit in with the research domain called ‘quantum cognition’, where quantum probability models and quantum vector spaces are used in human cognition, and are also relevant to the use of quantum structures in information retrieval and natural language processing, and how they introduce ‘quantization’ and ‘Bose–Einstein statistics’ as relevant quantum effects there. Inspired by the conceptuality interpretation of quantum mechanics, and relying on the new insights, we put forward hypotheses about the nature of physical reality. In doing so, we note how this new type of decrease in entropy, and its explanation, may be important for the development of quantum thermodynamics. We likewise note how it can also give rise to an original explanatory picture of the nature of physical reality on the surface of planet Earth, in which human culture emerges as a reinforcing continuation of life.     

Revisiting Quaternion Dual Electrodynamics

Dual electrodynamics and corresponding Maxwell’s equations (in the presence of monopole only) are revisited from dual symmetry and accordingly the quaternionic reformulation of field equations and equation of motion is developed in simple, compact and consistent manner.     

Complex symmetries of electrodynamics

We prove that a set of nonsingular free solutions of Maxwell's equations forms a representation of the group obtained by analytic continuation of the Poincaré group to complex values of the group parameters, and that a set of singular solutions forms a representation of the group obtained by analytic continuation of the conformal group to complex values of the group parameters. These results are obtained by constructing a theory governing 2 × 2 complex matrix fields defined for complex values of position and time; the equations of this theory are invarient with respect to complex Poincaré transformations and complex conformal transformations, but the set of nonsingular solutions is in one-to-one correspondence with a set of nonsingular solutions of Maxwell's equations, and a similar correspondence exists for the singular solutions. Certain collections of solutions of Maxwell's equations for the field of a current form representations of these complex groups if both magnetic and electric currents are permitted, in which case complex transformations provide a natural connection between electric and magnetic charge. A class of complex transformations also yield natural relations between sources moving slower than light and sources moving faster than light.     

Two Mathematically Equivalent Versions of Maxwell’s Equations

This paper is a review of the canonical proper-time approach to relativistic mechanics and classical electrodynamics. The purpose is to provide a physically complete classical background for a new approach to relativistic quantum theory. Here, we first show that there are two versions of Maxwell’s equations. The new version fixes the clock of the field source for all inertial observers. However now, the (natural definition of the effective) speed of light is no longer an invariant for all observers, but depends on the motion of the source. This approach allows us to account for radiation reaction without the Lorentz-Dirac equation, self-energy (divergence), advanced potentials or any assumptions about the structure of the source. The theory provides a new invariance group which, in general, is a nonlinear and nonlocal representation of the Lorentz group. This approach also provides a natural (and unique) definition of simultaneity for all observers.     

Photon spin and Bose-Einstein statistics

It is shown that Maxwell’s equations for the electric and magnetic fields free of sources can be inferred from Dirac’s pair of first-order equations for a zero-mass, zero-charge particle. This result is interpreted as a Lorentz invariant form of the transverse nature of photonic propagation in which only two components of the spin-1 field exist in nature.Canonical quantization of Dirac’s equations leads to a time average of the electromagnetic energy in agreement with the standard result of quantum electrodynamics. It is shown that the spin-statistics theorem is not violated for canonical quantization of the Dirac field provided the mass of the particle is zero.     

Superposition de deux ondes d'indices d'heterogeneite differents

Heterogeneous plane and uniform electromagnetic waves are particular solutions of Maxwell's equations. The sum of two of these solutions of the same frequency is generally not a wave of the same kind. A new type of interference appears for two heterogeneous plane and uniform components of different heterogeneity. The experimental proof of this phenomenon would show the dependence of the phase velocity on the heterogeneity. In the general case, the equal-amplitude surfaces of the various fields components are different; the same holds for the equal-phase surfaces. It is not always possible to define an eikonal and a wave surface for such a solution of Maxwell's equations. These difficulties disappear for waves of equal or opposite heterogeneity index.     

Perturbation theory for studying the effect of the τ∈ term in lens-like media

In this paper we derive solutions of Maxwell's equations for parabolic-index media, using first-order perturbation theory and taking into account the τ∈ term that is most often neglected as being negligibly small. The theory presented here should find applications in the propagation of electromagnetic waves through Selfoc fibers and p-n junctions.     

On transitions to stationary states in Hamiltonian nonlinear wave equations

We consider the convergence to stationary states of all finite energy solutions to nonlinear wave equations without dissipation in the long-time limits t → ±∞. The investigation is inspired by Bohr's postulate on the transitions between stationary states, by de Broglie's wave-particle duality, and by radiative damping in classical electrodynamics.     

Electromagnetic waves refraction on the interface of transparent with absorptive right or left-handed media

The expressions for the time-averaged Pointing’s vector in absorptive left-handed medium based on the classical vector Maxwell’s equations for a continuum have been obtained. It was found that in the case of inclined incidence with mixed polarization the additional plane component of Poynting’s vector perpendicular to incidence plane has appeared for negative effective refraction. This addition of the in-plane component leads to non-coincidence of refraction and incidence plane.     

Telegraphic Transport Processes and Their Fractional Generalization: A Review and Some Extensions

We address the problem of telegraphic transport in several dimensions. We review the derivation of two and three dimensional telegrapher’s equations—as well as their fractional generalizations—from microscopic random walk models for transport (normal and anomalous). We also present new results on solutions of the higher dimensional fractional equations.     

The Kirchhoff gauge

We discuss the Kirchhoff gauge in classical electrodynamics. In this gauge, the scalar potential satisfies an elliptical equation and the vector potential satisfies a wave equation with a nonlocal source. We find the solutions of both equations and show that, despite of the unphysical character of the scalar potential, the electric and magnetic fields obtained from the scalar and vector potentials are given by their well-known retarded expressions. We note that the Kirchhoff gauge pertains to the class of gauges known as the velocity gauge.     

Hidden Dissipation and Irreversibility in Maxwell’s Demon

Maxwell’s demon is an entity in a 150-year-old thought experiment that paradoxically appears to violate the second law of thermodynamics by reducing entropy without doing work. It has increasingly practical implications as advances in nanomachinery produce devices that push the thermodynamic limits imposed by the second law. A well-known explanation claiming that information erasure restores second law compliance fails to resolve the paradox because it assumes the second law a priori, and does not predict irreversibility. Instead, a purely mechanical resolution that does not require information theory is presented. The transport fluxes of mass, momentum, and energy involved in the demon’s operation are analyzed and show that they imply “hidden” external work and dissipation. Computing the dissipation leads to a new lower bound on entropy production by the demon. It is strictly positive in all nontrivial cases, providing a more stringent limit than the second law and implying intrinsic thermodynamic irreversibility. The thermodynamic irreversibility is linked with mechanical irreversibility resulting from the spatial asymmetry of the demon’s speed selection criteria, indicating one mechanism by which macroscopic irreversibility may emerge from microscopic dynamics.     

Ultra-short scalar and vector solitons in self-inductively transparent media

The analytical form of the one-soliton solutions to the Maxwell–Bloch equations is found without the slowly-varying envelope approximation with application to the ultra-short (few-cycle or sub-cycle) light pulses propagating in media of two-level atoms as well as to fluxons in the long Josephson junctions. Also, we discuss the dynamics of the ultra-short vector solitons propagating in specific three-level media and magnetic-flux transmission lines (of two long Josephson junctions sharing a common superconducting plate). Studies of the (ultra-short) pulse collisions lead to the prediction of pulse stability against the collisions. In particular, the collisions of the ultra-short vector solitons are investigated in detail. Their collision-induced polarization transform is found to be similar to the polarization transform of the vector (Manakov) solitons propagating in self-focusing media.     

Method of semiclassical representation in quantum electrodynamics

The operators of the classical amplitudes of an electromagnetic field are introduced and a method of transferring from quantum electrodynamics to the semiclassical approximation both in the case of a free field and in the case of the interaction of the field with a quantum system is given. The method considered enables one to set up solutions of quantum electrodynamics in the case of an intense field from the solutions of the semiclassical problem. An operator method of obtaining solutions of the equations of semiclassical electrodynamics is considered. The physical meaning of the quantum corrections to the semiclassical electrodynamics of an intense field is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 77–98, February, 1980.     

Interior penalty DG methods for Maxwell’s equations in dispersive media

In this paper, we develop a fully-discrete interior penalty discontinuous Galerkin method for solving the time-dependent Maxwell’s equations in dispersive media. The model is described by a vector integral–differential equation. Our scheme is proved to be unconditionally stable and achieve optimal error estimates in both L² norm and energy norm. The scheme is implemented and numerical results supporting our analysis are presented.     

Recent Advances in Conservation–Dissipation Formalism for Irreversible Processes

The main purpose of this review is to summarize the recent advances of the Conservation–Dissipation Formalism (CDF), a new way for constructing both thermodynamically compatible and mathematically stable and well-posed models for irreversible processes. The contents include but are not restricted to the CDF’s physical motivations, mathematical foundations, formulations of several classical models in mathematical physics from master equations and Fokker–Planck equations to Boltzmann equations and quasi-linear Maxwell equations, as well as novel applications in the fields of non-Fourier heat conduction, non-Newtonian viscoelastic fluids, wave propagation/transportation in geophysics and neural science, soft matter physics, etc. Connections with other popular theories in the field of non-equilibrium thermodynamics are examined too.