Electromagnetic tensors

2015

Recently, several discussions on the possible observability of 4-vector fields have been published in literature. Furthermore, several authors recently claimed existence of the helicity=0 fundamental field. We re-examine the theory of antisymmetric tensor fields and 4-vector potentials. We study the massless limits. In fact, theoretical motivation for this venture is the old papers of Ogievetskiȋ and Polubarinov, Hayashi, and Kalb and Ramond. They proposed the concept of the notoph, whose helicity properties are complementary to those of the photon. We analyze the quantum field theory with taking into account mass dimensions of the notoph and the photon. We also proceed to derive equations for the symmetric tensor of the second rank on the basis of the Bargmann-Wigner formalism. They are consistent with the general relativity. Particular attention has been paid to the correct definitions of the energy-momentum tensor and other Nöther currents. We estimate possible interactions, fermion-notoph, graviton-notoph, photon-notoph.

2008

Symmetry, 2019

A fully relativistically covariant and manifestly gauge-invariant formulation of classical Maxwell electrodynamics is presented, purely in terms of gauge-invariant potentials without entailing any gauge fixing. We show that the inhomogeneous equations satisfied by the physical scalar and vector potentials (originally discovered by Maxwell) have the same symmetry as the isometry of Minkowski spacetime, thereby reproducing Einstein’s incipient approach leading to his discovery of special relativity as a spacetime symmetry. To arrive at this conclusion, we show how the Maxwell equations for the potentials follow from stationary electromagnetism by replacing the Laplacian operator with the d’Alembertian operator, while making all variables dependent on space and time. We also establish consistency of these equations by deriving them from the standard Maxwell equations for the field strengths, showing that there is a unique projection operator which projects onto the physical potentials....

1998

Quantum electrodynamics is a well-accepted theory. But, we believe it useful to look at formalisms which provide alternative ways to describe light, because the development of quantum ®eld theories based primarily on the gauge principle have, in recent years, met with considerable diculties. There are numerous generalized theories and, mainly, they are characterized by introducing some additional parameters and/or longitudinal modes of electromagnetism. The Majorana±Oppenheimer form of electrodynamics, the Sachs' theory of Elementary Matter, the analysis of the action-at-adistance concept, presented recently by Chubykalo and Smirnov-Rueda, and the analysis of the claimed`longitudinality' of the anti±symmetric tensor ®eld after quantization are examined in this article. We list also recent advances in the Weinberg 2(2J+1) formalism (which is built on ®rst principles) and in the Majorana theory for neutral particles. They can serve as starting points for constructing the quantum theory of light. Ã Here and below in this historical essay we try to keep the notation and the metric of the original papers.

Archive for Rational Mechanics and Analysis, 2004

We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in studies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials. Our detailed treatment aims to encourage the use of spacetime algebra as a readily available and mature extension to existing vector calculus and tensor methods that can greatly simplify the analysis of fundamentally relativistic objects like the electromagnetic field.

Journal of Physics A: Mathematical and General, 2000

The gauge freedom in the electromagnetic potentials indicates an underdeterminacy in Maxwell's theory. This underdeterminacy will be found in Maxwell's (1864) original set of equations by means of Helmholtz's (1858) decomposition theorem. Since it concerns only the longitudinal electric field, it is intimately related to charge conservation, on the one hand, and to the transversality of free electromagnetic waves, on the other hand (as will be discussed in Pt. II). Exploiting the concept of Newtonian and Laplacian vector fields, the role of the static longitudinal component of the vector potential being not determined by Maxwell's equations, but important in quantum mechanics (Aharonov-Bohm effect) is elucidated. These results will be exploited in Pt.III for formulating a manifest gauge invariant canonical formulation of Maxwell's theory as input for developing Dirac's (1949) approach to special-relativistic dynamics.

2017

The covariant derivative of the 4-components electromagnetic potential in a flat Minkowski space-time is split into its antisymmetric and symmetric parts. While the former is well known to describe the electromagnetic field, we show that the latter describes the associated particles. When symmetry principles are applied to the invariants in operations of the Poincaré group, one finds equations which describe the structure of the particles. Both parts of the tensor unify the concept of matter-wave duality. Charge and mass are shown to be associated to the potential.La dérivée covariante du potentiel électromagnétique à 4 composantes dans l'espace plat de Minkowski est séparée en ses deux parties antisymmétrique et symmétrique. Alors que la première représente le champ électromagnétique, nous montrons que la seconde décrit les particules associées. Quand les principes de symétrie sont appliqués aux invariants dans une opération du groupe de Poincaré, on trouve les équations permet...

Deccan Education Society's Willingdon College,Sangli, 2022

To describe successfully the properties and dynamics of the particle, we need field theory. The dynamics of gauge bosons are described by Gauge Field Theory, a type of quantum field theory. In this project, we work in four-dimensional Minkowski space. Gauge field theory is characterized by the presence of a vector field, we defined the Lagrangian for the gauge field (photon field) and showed that it is Lorentz invariant and gauge invariant. Any physical theory must be Lorentz invariant. We derived the equation of motion for the gauge field and derived all Maxwell's equations in terms of the electric and magnetic fields. We check the invariance of the action under global and local symmetry transformations using Noether's theorem. We derived the stress-energy tensor that gives us energy density, and the momentum of the field and determined the conserved charges associated with the gauge field. We then quantize the gauge field by treating each field as a harmonic oscillator. We derived the expression for vector field in terms of annihilation and creation operators. The gauge invariance implies that in four dimensions the photon has only two physical degrees of freedom

General Relativity and Gravitation, 2005

When sources are added at their right-hand sides, and g (ik) is a priori assumed to be the metric, the equations of Einstein’s Hermitian theory of relativity were shown to allow for an exact solution that describes the general electrostatic field of n point charges. Moreover, the injunction of spherical symmetry of g (ik) in the infinitesimal neighbourhood of each of the charges was proved to yield the equilibrium conditions of the n charges in keeping with ordinary electrostatics. The tensor g (ik), however, cannot be the metric of the theory, since it enters neither the eikonal equation nor the equation of motion of uncharged test particles. A physically correct metric that rules both the behaviour of wave fronts and of uncharged matter is the one indicated by Hély. In the present paper it is shown how the electrostatic solution predicts the structure of the n charged particles and their mutual positions of electrostatic equilibrium when Hély’s physically correct metric is adopted.

Journal of Physics A: Mathematical and General, 1989

IJPSR, 2023

EM theory started from electricity and its current, as the carriers or objects, mediated by the fields and potentials. In the opposite sense, the fields are formal features of the potentials, limited by the carriers. Apart from the central Coulomb's law, similar Ampere's law is here generalized. The radial-static and transverse-kinetic, are thus supplemented by longitudinal-dynamic forces. The fields are introduced in the three ways: as the evident forces, via the object densities and by analogy of the potentials with fluid mechanics. As the simplest basic set, the two algebraic relations of J. J. Thomson operate by the two moving fields. Instead of the parallel or hierarchical processes, they form a causal loop with the constitutive field relations. The spatial derivatives of the algebraic pair give the four differential forms, wider from Maxwell's equations. The elimination of excessive, and explanation of remaining terms, convincingly relate the two sets. Maxwell's equations are finally presented in Einstein's tensor form, concerning 4D space.

We present new mathematical foundations of classical Maxwell–Lorentz electrodynamic models and related charged particles interaction-radiation problems, and analyze the fundamental least action principles via canonical Lagrangian and Hamiltonian formalisms. The corresponding electrodynamic vacuum field theory aspects of the classical Maxwell–Lorentz theory are analyzed in detail. Electrodynamic models of charged point particle dynamics based on a Maxwell type vacuum field medium description are described, and new field theory concepts related to the mass particle paradigms are discussed. We also revisit and reanalyze the mathematical structure of the classical Lorentz force expression with respect to arbitrary inertial reference frames and present new interpretations of some classical special relativity theory relationships.

International Journal of Modern Physics A, 1998

After a summary of a recently proposed new type of instant form of dynamics (the Wigner-covariant rest-frame instant form), the reduced Hamilton equations in the covariant rest-frame Coulomb gauge for the isolated system of N scalar particles with pseudoclassical Grassmann-valued electric charges plus the electromagnetic field are studied. The Lienard-Wiechert potentials of the particles are evaluated and it is shown how the causality problems of the Abraham-Lorentz-Dirac equation are solved at the pseudoclassical level. Then, the covariant rest-frame description of scalar electrodynamics is given. Applying to it the Feshbach-Villars formalism, the connection with the particle plus electromagnetic field system is found.