We know the classical Maxwell equation of motion (eom) with both electric and magnetic source can be written as:

(1) Explicit form

enter image description here or more schematically as:

(2) Differential form $$ d * F = * J_e $$ $$ dF =* J_m $$

My question is that do we have such classical Yang-Mills equation of motion with both electric and magnetic source in both

(1) Explicit form?

(2) Differential form? Naively, we may write $$ D * F = * J_e $$ $$ D F =* J_m $$ where $F= dA + A \wedge A$ and $D=d + [A, ]$ as the covariant derivative version of exterior derivative $d$.

But: To be aware that for example, the $SU(2)$ Yang-Mills and $SO(3)$ Yang-Mills theory may have distinct constraint on the magnetic monopole (or the t Hooft loop). It does not seem to me that $J_e$ or $J_m$ contain such information?

  • $\begingroup$ Please clarify your question. Are you asking why the same equations can be written in such different forms? $\endgroup$ – G. Smith May 19 at 23:59
  • $\begingroup$ No. But I am interested in knowing both (1) Explicit form and (2) Differential form for YM with e and m source $\endgroup$ – annie heart May 20 at 0:02
  • $\begingroup$ OK, that makes it clearer. Thank you. Do you really want the Yang-Mills equations written in terms of $\mathbf{E}$ and $\mathbf{B}$? They ate usually written in terms of $F^{\mu\nu}$. $\endgroup$ – G. Smith May 20 at 0:03
  • $\begingroup$ written as $E^a$ and $B^a$ are preferred. But you can convert to $F$ first $\endgroup$ – annie heart May 20 at 1:16

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