Electromagnetic tensors

Overview & Synopsis

The electromagnetic potential A i is a quadrivector in Minkowski space-time x k and its gradient a i k is a tensor of rank two whose elements are the sixteen partial derivatives ∂A i /∂x k. We study in this article the properties of a family of tensors resulting from [a i k ]. We first introduce the covariant tensor [a ki ]. Four initial tensors are obtained by separating [a i k ] on the one hand, and [a ki ] on the other hand into their symmetric and antisymmetric parts. These are (s i k , [f i k ], [S ki ], [F ki ]). As the lowering-raising index operations and symmetrization-antisymmetrization operations do not commute, these four tensors are different. We associate a Lagrangian density L to the determinant of [a i k ] which is invariant in an operation of symmetry of the Poincaré group. In the first part of the article, we show that there is a particular coordinate system where the scalar potential obeys the Hemholtz equation. The solutions allow to describe the "electromagnetic particles", characterized by three quantum numbers n, and m. We give the tensors corresponding to the first five solutions. They describe energy and electric charge distributions. The condition of existence of these particles is related to a property of the electron described in the Wheeler-Feynman's absorber theory. In a second part, we first check that [F ki ] is the usual electromagnetic tensor whose components are the electric and magnetic fields. We prove that Maxwell's equations are obtained by applying the principle of least action to the 4-potential endowed with L. The source terms (ρ and − → j) are expressed in terms of the components of [S ki ]. The results obtained are covariant. The formulation of these tensors being independant of scale, they unify the human and the electron scales, giving a new way to understand elementary particles.