In 1985, Harmuth wrote that Maxwell's equations are incompatible with causality, and overcame the problem by adding a term for magnetic dipole currents, and as a consequence the problem of infinite zero-point energy and renormalization disappears. At least according to Harmuth's book:

The foreword is readable at Calculus of finite differences in quantum electrodynamics, by Henning F. Harmuth, Beate Meffert. The modified Maxwell equations read (page 3):

$$ \begin{aligned} \mathrm{curl}\,\boldsymbol H&=\frac{\partial\boldsymbol D}{\partial t}+\boldsymbol g_e\\ -\mathrm{curl}\,\boldsymbol E&=\frac{\partial\boldsymbol B}{\partial t}+\boldsymbol g_m\\ \mathrm{div}\,\boldsymbol D&=\rho_e\\ \mathrm{div}\,\boldsymbol B&=0\quad\text{or}\quad\mathrm{div}\,\boldsymbol B=\rho_m \end{aligned} $$

Are Harmuth's modifications generally accepted by the physics community as a more accurate description of reality than the unmodified equations?


For what it's worth, throughout my career as a professor of theoretical physics specializing in high energy physics and the foundations of Lorentz covariant quantum theory, I never heard of Harmuth or his modification of Maxwell's equations until reading this question and the indicated foreword to his book. Now it's hard to keep track of everything going on in science, even within ones specialty. But I think it's safe to say that Harmuth's theory is not accepted by the physics community as an improvement on Maxwell's equations. There have been proposed modifications that have received extensive attention and study, such as Born-Infeld electrodynamics, and ever since the development of the Electro-Weak unified theory our understanding of the status of the EM field has been altered. But no proposals for modifying Maxwell's equations, per se, have been accepted as established improvements.


I have done two projects when I was a graduate student on magnetic monopoles. It seems to me that Harmuth's equations are just the same as those for electromagnetism with magnetic monopoles, which has been studied by Dirac before Harmuth. But I may be missing something here.

In any case, while magnetic monopoles are not part of mainstream physics in the sense that they have never been detected and that they don't show up in the Standard Model, they are an important part in many of the theories that try to go beyond the Standard Model.

  • $\begingroup$ Yes, the equations are completely standard if you include magnetic monoples. Like here:en.wikipedia.org/wiki/Magnetic_monopole . I read a little further in his text (next page, got bored then), and he claims the magnetic monopoles are optional, but then he goes on with a constitutive relation g_m=s H, and this is inconsistent then as the usual argument you get that div g_m has to vanish for the M.E. to be consistent. But we know that div H is not equal to zero (and getting worse with a function s =0 outside of such a situation) $\endgroup$
    – lalala
    Jul 3 at 14:14

Generalized Maxwell's Equations or Symmetric Maxwell's Equations

A. I. Arbab, Complex Maxwell's Equations, Chinese Physics B, Volume 22, Number 3 (2013), [equation 45-46] <click here> \begin{equation*}\left.\begin{split}\nabla\cdot\mathbf{E}&=\frac1{\epsilon_0}\rho_e\cr\nabla\cdot\mathbf{B}&=\rho_m\cr\nabla\times\mathbf{E}&=-\left(\mathbf{J}_m+\frac{\partial\mathbf{B}}{\partial t}\right)\cr\nabla\times\mathbf{B}&=\mu_0\mathbf{J}_e+\frac1{c^2}\frac{\partial\mathbf{E}}{\partial t}\end{split}\right.\end{equation*}

  • $\rho_e$ is electric charge density
  • $\rho_m$ magnetic charge density
  • $\mathbf{J}_e$ electric current density
  • $\mathbf{J}_m$ magnetic current density
  • $c=\frac1{\sqrt{\mu_0\epsilon_0}}$
  • I. BIALYNICKI-BIRULA and Z. BIALYNICKA-BIRULA, Magnetic Monopoles in the Hydrodynamic Formulation of Quantum Mechanics, Physical Review D, Volume 3, Number 10, 15 May 1971, [equation 20-23] <click here> \begin{align*}\nabla\cdot\mathbf{E}&=\rho_e\cr\nabla\cdot\mathbf{B}&=\rho_m\cr\nabla\times\mathbf{E}&=-\frac1{c}\left(\mathbf{J}_m+\frac{\partial\mathbf{B}}{\partial t}\right)\cr\nabla\times\mathbf{B}&=\frac1{c}\left(\mathbf{J}_e+\frac{\partial\mathbf{E}}{\partial t}\right)\end{align*}Valeri V Dvoeglazov, Generalized Maxwell equations from the Einstein postulate, Journal of Physics A: Mathematical and General, Volume 33, Number 28 [equation 4-7] <click here>\begin{align*}\nabla\cdot\mathbf{E}&=-\frac1{c}\frac{\partial}{\partial t}\text{Re}\chi\cr\nabla\cdot\mathbf{B}&=\frac1{c}\frac{\partial}{\partial t}\text{Im}\chi\cr\nabla\times\mathbf{E}&=-\frac1{c}\frac{\partial\mathbf{B}}{\partial t}+\nabla\text{Im}\chi\cr\nabla\times\mathbf{B}&=\frac1{c}\frac{\partial\mathbf{E}}{\partial t}+\nabla\text{Re}\chi\end{align*}


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