Maxwell's equations generalize very nicely if we add in magnetic monopoles: we get $$\begin{align*} \partial_\mu F^{\mu \nu} &= J^\nu \\ \partial_\mu \tilde{F}^{\mu \nu} &= \tilde{J}^\nu, \end{align*}$$ where $F_{\mu \nu}$ is the electromagnetic tensor, $\tilde{F}_{\mu \nu} := \epsilon_{\mu \nu \rho \sigma} F^{\rho \sigma}$ is the dual tensor, and $J^\mu$ and $\tilde{J}^\mu$ are the electric and magnetic current densities respectively. We can use differential forms to make these equations even more compact: $$\begin{align*} \mathrm{d} (\star F) &= J,\\ \mathrm{d} F &= \tilde{J}. \end{align*}\tag1$$

It is known that QED can be generalized to incorporate point magnetic monopoles, which leads to all kind of interesting phenomena like the Dirac quantization condition. (This is done by essentially just excising the monopoles from spacetime completely and only defining the gauge field $A^\mu$ away from the monopoles, so that they act as topological defects that convert the trivial principle $\mathrm{U}(1)$ fiber bundle on which the gauge field lives to a nontrivial bundle. This is straightforward when the monopole trajectories are specified by hand, but I think there are also reasonably tractable ways to give the monopoles their own dynamics.)

But what about the case of continuous distributions of magnetic monopoles? (To avoid cheating, let's say that the $\tilde{J}$ field's support is the entire region of spacetime whose dynamics we are considering, although still bounded so that the fields are well-defined.) In this case the equation $\mathrm{d}F \neq 0$ prevents us from introducing a gauge field $A$ such that $F = \mathrm{d}A$ in the first place. It's probably not at all clear how to quantize such a theory, so let's just stick to the classical case. To make things even simpler, we can ignore the Lorentz force law and treat the source currents as background rather than dynamical fields, so that we only need to worry about the EM field's time evolution. In this case the equations (1) form a perfectly well-posed system of coupled differential equations. Is there any known/possible Lagrangian or Hamiltonian whose equations of motion are given by (1)?


1 Answer 1


Let \begin{equation} S[\chi,\tilde\chi,F,\tilde F]\overset{\mathrm{def}}=\int\mathrm dx\ \chi_\nu(\partial_\mu F^{\mu \nu} - J^\nu)+ \tilde{\chi}_\nu(\partial_\mu \tilde{F}^{\mu \nu} - \tilde{J}^\nu) \end{equation}

Variations with respect to $\chi$ and $\tilde \chi$ give you the Maxwell equations. Variation with respect to $F$ and $\tilde F$ give you \begin{equation} \partial_{[\mu}\chi_{\nu]}=\partial_{[\mu}\tilde\chi_{\nu]}=0 \end{equation}

  • $\begingroup$ Haha okay fair, but I was looking for a somewhat less trivial example, which doesn't involve introducing any new dummy pure gauge fields which don't couple to anything physically observable. (I.e. the only equations of motion should be Maxwell's equations. Since your dummy $\chi$ don't have any kinetic terms, they're really just Lagrange multipliers that happen to satisfy some differential equation, rather than true dynamical fields.) But I can't deny that this does technically answer my question. $\endgroup$
    – tparker
    Commented Jul 5, 2017 at 17:42

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