Helmholtz theorem for antisymmetric second-rank tensor fields and electromagnetism with magnetic monopoles

Abstract

A generalized Helmholtz’s theorem is proved, which states that an antisymmetric second-rank tensor field in 3+1 dimensional space-time, which vanishes at spatial infinity, is determined by its divergence and the divergence of its dual. When the divergence of the antisymmetric electromagnetic field strength tensor is equal to the electric charge-current density and the divergence of the dual of the electromagnetic field strength tensor is equal to the magnetic charge-current density, the equations of electromagnetism are obtained. As a convenience, in the solution of the equations of electromagnetism two different four-vector potentials can be used, one of which couples to the electric charge-current density and the other to the magnetic charge-current density.

The Helmholtz theorem and retarded fields

“…As may be seen, the sources of the field F are its divergence, curl and time derivative, all of them evaluated at the retarded time. In the Appendix A we derive (11). As expected, the causal Helmholtz theorem naturally applies to Maxwell's equations.…”

“…In this section we will review the extension of the Helmholtz theorem suggested by Hauser [13] and discussed by Heras [12] and Kobe [11]. We will see that this extension of the theorem also leads to the retarded fields.…”

“…A corollary of this theorem states that an antisymmetric tensor field vanishing sufficiently rapidly at spatial infinity is determined by specifying its divergence and the divergence of its dual. An explicit formula for this corollary involving the retarded Green function of the wave equation was introduced by Kobe [11] in 1986. This corollary can naturally be considered as another generalization of the Helmholtz theorem for antisymmetric tensor fields.…”

On the Helmholtz Theorem and Its Generalization for Multi-Layers

“…The theorem states that any vector field in space can be split into curl-free parts (irrotational or the gradient of a scalar function or potential) and divergence-free parts (solenoidal or the curl of a vector function or potential) (Morse & Feshbach, 1953;Plonsey & Collin, 1961;Collin, 1991;Arfken & Weber, 1995;Sprossig, 2010;Van Bladel, 1960, 1993a, p. 1005. Vector and scalar potential representations are valid and very important in almost every field theory, and Helmholtz theorem can be used to obtain the vector and scalar potentials (Morse & Feshbach, 1953;Kobe, 1984;Amrouche et al, 1998;Lindell & Dassios, 2001;Kurokawa, 2001Kurokawa, , 2008Chubykalo et al, 2006Chubykalo et al, , 2011Chew, 2014). In the literature (Papas, 1988;Dassios & Lindell, 2002;Zhou, 2007), it is mentioned that the theorem was previously presented by Stokes (1849), and it is also said in Zhou (2007) that Stokes did not introduce any scalar and vector potentials in his representations, resulting in an inconsistency.…”

“…The Helmholtz theorem is indispensable and very important in all of mathematical physics (Dassios & Lindell, 2002), and there are some generalizations of the theorem suggested for various applications previously and very recently (Kobe, 1984;Lindell & Dassios, 2000, 2001, 2003Sprossig, 2010;Ortigueira et al, 2015). Physical distributions can be correctly described in terms of mathematical distributions (Schwartz, 2008, p. 77;Van Bladel, 1995, p.18), and hence, the distribution theory is very convenient for the examination of physical problems.…”