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Generalisation of classical electrodynamics to admit a scalar field and longitudinal waves
Abstract
The theory of electrodynamics can be cast into biquaternion form. Usually Maxwells’ equations are invariant with respect to a gauge transformation of the potentials and one can choose freely a gauge condition. For instance, the Lorentz gauge condition yields the potential Lorenz inhomogeneous wave equations. It is possible to introduce a scalar field in the Maxwell equations such that the generalized Maxwell theory, expressed in terms of the potentials, automatically satisfy the Lorentz inhomogeneous wave equations, without any gauge condition. This theory of electrodynamics is no longer gauge invariant with respect to a transformation of the potentials: it is electrodynamics with broken gauge symmetry. The appearence of the extra scalar field terms can be described as a conditional current regauge that does not violate the conservation of charge, and it has several consequences: the prediction of a longitudinal electroscalar wave (LES wave) in vacuum; superluminal wave solutions, and possibly classical theory about photon tunneling; a generalized Lorentz force expression that contains an extra scalar term; generalized energy and momentum theorems, with an extra power flow term associated with LES waves. A charge density wave that only induces a scalar field is possible in this theory.
... Actually, many papers deal with the application of geometric algebra to Maxwell's equation (see [4][5][6][7][8][9] and many others), but few of them deal with the concept of scalar field. Among the most interesting works we can find a paper by Bettini [5], two papers written by van Vlaenderen [10,11] and two papers of Hively [12,13]. ...
... where S = S 1 + S 2 + S 3 + S 4 is a scalar field, whose meaning will be clarified later. It is to be noted that equating (13) to zero, i.e. S = 0, gives an expression that takes the form of the "Lorenz gauge" condition if A t = −ϕ/c, where ϕ is the scalar potential of the electric field [4,8,10,17]. Equation (13) can be rewritten as ...
... is the generalized Poynting vector [10,11]. Beside the usual term E × B here a new energy term appears, namely SE, which is associated to a longitudinal scalar wave [11] and that is not further investigated in the present work. ...
In this paper a straightforward application of Occam’s razor principle to Maxwell’s equation shows that only one entity, the electro- magnetic four-potential, is at the origin of a plurality of concepts and entities in physics. The application of the so called “Lorenz gauge” in Maxwell’s equations denies the status of real physical entity to a scalar field that has a gradient in space-time with clear physical meaning: the four-current density field. The mathematical formalism of space-time Clifford algebra is introduced and then used to encode Maxwell’s equations starting only from the electromagnetic four-potential. This approach suggests a particular Zitterbewegung (ZBW) model for charged elementary particles.
... One aspect of the Proca equations is, a zero photon mass results in the original Maxwell equations of classical electrodynamics. In our interest to determine the relation between S and µ γ , we will turn to the biquaternion form of electromagnetics in order to determine the form of ∇A, in which A is the biquaternion containing the scalar and vector potentials [8]. The biquaternion defines a four dimensional Minkowski space as ...
... By applying the relations (4, 3) we find the scalar portion of this biquaternion is analogous to (8), and the vector portion is analogous to (7) by applying (1) and (2), in other words ...
We show the scalar field defined in the vectoral form of Maxwell's equations is found to break the Lorentz Gauge in the case of massive photons. Using the biquaternion form of Maxwell's equations, the differential forms of the scalar field is shown to have relation to photon mass. We show both the scalar field gradient and photon mass are related to the displacement current using the Proca equations, and propose a means to determine the spatial geometric structure of the massive photon based on its intrinsic capacitance.
... A suitable extension of the Maxwell equations is needed, and such an extension actually exists, called Aharonov-Bohm electrodynamics. This has been studied by several authors ( [11,12] and refs.) and in [13] a compact covariant version has been found, in which the degree of freedom S = ∂ µ A µ is explicitly eliminated. Here (Sect. ...
... In the low frequency and near field limits, the electric field will be essentially determined by these two dipole moments, since the magnetic contributions are smaller by many orders of magnitude. (This can be checked by including the appropriate transport current in (12), such that the continuity equation holds, and then solving the standard Maxwell equations.) The dipole moment d of ρ is readily found: d = qa cos(ωt). ...
Although standard quantum mechanics has some non-local features, the probability current of the Schr\"odinger equation is locally conserved, and this allows minimal electromagnetic coupling. For some important extensions of the Schr\"odinger equation, however, the probability current is not locally conserved. We show that in these cases the correct electromagnetic coupling requires a relatively simple extension of Maxwell theory which has been known for some time and recently improved by covariant integration of a scalar degree of freedom. We discuss some general properties of the solutions and examine in particular the case of an oscillating dipolar source. Remarkable mathematical and physical differences emerge with respect to Maxwell theory, as a consequence of additional current terms present in the equations for and . Several possible applications are mentioned.
... A theoretical method for quantizing the radiation field using transverse, longitudinal and scalar photons was introduced in [8]. An attempt to consider the existence of longitudinal electric waves using general framework of classical electrodynamics was tried in [26]. Possible existence and physical relevance of longitudinal electromagnetic waves from quantum electrodynamic point of view was studied in [15]. ...
... (62) Each of the three integral terms in the last expression were already evaluated in (24) - (26). If we substitute these results into (62), then we obtain ...
In this work by using the assumptions that wavelength is much smaller than charge separation distance of an electric dipole, which in turn is much smaller than a distance up to the point of observation, the new results for radiation of an electric dipole were obtained. These results generalize and extend the standard classical solution, and they indicate that under the above assumptions the electric dipole emits both long-range longitudinal electric and transverse electromagnetic waves. For a specific values of the dipole system parameters the longitudinal and transverse electric fields are displayed. Total power emitted by electric and electromagnetic waves are calculated and compared. It was shown that under the standard assumption of charge separation distance being much smaller than wavelength: a) classical solution correctly describes the transverse electromagnetic waves only; b) longitudinal electric waves are non-negligible; c) total radiated power is proportional to the fourth degree of frequency and to the second degree of the charge separation distance; d) transverse component of our solution reduces to classical solution. In case wavelength is much smaller than charge separation distance: a) the classical solution is not valid and it overestimates the total radiated power; b) longitudinal electric waves are dominant and transverse electromagnetic waves are negligible; c) total radiated power is proportional to the third degree of frequency and to the charge separation distance; d) most of the power is emitted in a narrow beam along the dipole axis, thus emission of waves is focused as with lasers.
... that we call "γ-models", in which the non-dimensional parameter γ is normally of very small magnitude, |γ| ≪ 1. These modified conservation laws are not compatible with the Maxwell equations but are fully compatible with the extended electrodynamics of Aharonov-Bohm [4][5][6][7][8][9][10][11]. We gave previous examples of simple γ-models in [12]. ...
In the framework of the quantum theory of many-particle systems, we study the compatibility of approximated non-equilibrium Green’s functions (NEGFs) and of approximated solutions of the Dyson equation with a modified continuity equation of the form ∂t〈ρ〉+(1−γ)∇·〈J〉=0. A continuity equation of this kind allows the e.m. coupling of the system in the extended Aharonov–Bohm electrodynamics, but not in Maxwell electrodynamics. Focusing on the case of molecular junctions simulated numerically with the Density Functional Theory (DFT), we further discuss the re-definition of local current density proposed by Wang et al., which also turns out to be compatible with the extended Aharonov–Bohm electrodynamics.
... Due to the difficulties of representing Maxwell's equations using quaternions, Heaviside rejected Hamilton's algebraic system and developed a system of vector notations using the scalar (dot) and cross products, which is the conventional system used today in engineering and physics. On the other hand, representation of Maxwell's equations in G 3 reveals the quaternionic symmetries in the vector derivative of the electromagnetic field [9,14], which are hidden in the conventional vector and tensor notations (e.g, Eq. (2.7)). ...
Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an extensive demonstration of the applications of Clifford algebras in electromagnetism using the geometric algebra G3 = Cl(3,0) as a computational model in the Maxima computer algebra system. We compare the geometric algebra-based approach with conventional symbolic tensor calculations supported by Maxima, based on the itensor package. The Clifford algebra functionality of Maxima is distributed as two new packages called clifford - for basic simplification of Clifford products, outer products, scalar products and inverses; and cliffordan - for applications of geometric calculus.
... The extended theory of electrodynamics by Aharonov and Bohm [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] is a generalization of the familiar Maxwell theory which allows to couple the e.m. field also to sources where charge is not locally conserved-a phenomenon that is expected to occur in certain physical system with meso-or macroscopic quantum effects. Just because of this more general setting, the theory is not gauge invariant, and the electric and magnetic potentials f, A become univocally defined. ...
Aharonov-Bohm electrodynamics predicts the existence of traveling waves of pure potentials, with zero electromagnetic fields, denoted as gauge waves, or g-waves for short. In general, these waves cannot be shielded by matter since their lack of electromagnetic fields prevents the material from reacting to them. However, a not-locally-conserved electric current present in the material does interact with the potentials in the wave, giving the possibility of its detection. In [1] the basic theoretical description of a detecting circuit was presented, based on a phenomenological theory of materials that can sustain not-locally-conserved electric currents. In the present work we discuss how that circuit can be built in practice, and used for the effective detection of g-waves.
... Classical Maxwell's equations predict the existence of only transversal waves in a vacuum since, without material support, such as in a plasma medium where Langmuir waves are observed, there is no support for their propagation. Theoretical evidence for the existence of longitudinal waves in a vacuum has been documented from several sources, including [33][34][35], at the observational level [36][37][38], and in the proposal of two different apparatuses that are configured to transmit and/or receive scalar-longitudinal waves. More experimental evidence is needed to confirm their existence (see [39][40][41][42][43][44]). Attenuation is the gradual loss of energy that occurs as waves propagate through a medium. ...
This investigation introduces a new variational approach to refining Poisson’s equation, enabling the inclusion of a broader spectrum of physical phenomena, particularly in the emerging fields of spintronics and the analysis of resonant structures. The innovative formulation extends the traditional capabilities of Poisson’s equation, offering a nonlocal extension to classical theories of gravitation and opening new directions for energy conversion and enhanced communication technologies. By introducing a novel geometric structure, ω˜, into the equation, a deeper understanding of electrostatic potentials is achieved, and the intricate dynamics of the gravitational potential in systems characterized by radial vorticity fluctuations are illuminated. Furthermore, the research elucidates the generation of longitudinal electromagnetic waves and resonant phenomena within dusty plasma media, thereby contributing to the methodological advances in the study of nonequilibrium systems. These theoretical advances have the potential to transform the understanding of complex physical systems and open up opportunities for significant technological achievements across a range of scientific sectors.
... Although classical and quantum electrodynamics are among the most precise and successful theories in physics, there are reasons to look for extensions of Maxwell equations in various directions and physical contexts [1,2]. In particular, we are interested into the extension first proposed by Ohmura [3] and often called "Aharonov-Bohm electrodynamics" because of their major contribution ( [4]; see also [5][6][7][8][9][10][11][12][13]). The logical motivation for this extension, at least from a e-mail: minotti@df.uba.ar ...
The extended Aharonov–Bohm electrodynamics has a simple formal structure and allows to couple the e.m. field also to currents which are not locally conserved, like those resulting from certain non-local effective quantum models of condensed matter. As it often happens in physics and mathematics when one tries to extend the validity of some equations or operations, new perspectives emerge in comparison to Maxwell theory, and some “exotic” phenomena are predicted. For the Aharonov–Bohm theory the main new feature is that the potentials A μ become univocally defined and can be measured with probes in which the “extra-current” I = ∂ μ j μ is not zero at some points. As a consequence, it is possible in principle to detect pure gauge-waves with E = B = 0 , which would be regarded as non-physical in the Maxwell gauge-invariant theory with local current conservation. We discuss in detail the theoretical aspects of this phenomenon and propose an experimental realization of the detectors. A full treatment of wave propagation in anomalous media with extra-currents and of energy–momentum balance issues is also given.
Unidentified Anomalous Phenomena (UAP) challenge conventional physics with behaviours like instantaneous acceleration and apparent teleportation. By applying extended electrodynamics (EED) and extended gravitomagnetics (EGM), we can better grasp these phenomena as the manipulation of energy within a unified field. Maxwell’s extended electrodynamics proposes a scalar field behind these phenomena, enabling instant displacement and frictionless motion. Similarly, extended gravitomagnetic theories suggest alternative gravitational interactions to elucidate these dynamics. Exploring these concepts offers insights into advanced propulsion systems and unique electromagnetism-gravity interactions within a unified field. When designing and constructing trans medium superluminal crafts for interstellar transits, leveraging extended electrodynamics and gravitomagnetics allows for creative thinking while maintaining a foundation in Classical Electrodynamics (CED). The proposition arises that a unifying field may underpin the enigma of UAP, which due to erroneous mathematical constructs has been potentially overlooked until now.
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