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We know the Yang-Mills theory Equation of Motion (eom) without source

$$ * D * F = * (d (* F ) + [A, (* F )])= 0. $$

My question is that what are the most simple form we can boil down this eom to its minimal?

$$ * (d (* (d A + A \wedge A) ) + [A, (* (d A + A \wedge A) )]) $$

$$=* d * d A + * d * (A \wedge A) +A( * (d a + A \wedge A) ) - (* (dA + A \wedge A) ) A=0 $$

This is what I get. How can we massage it further in order to make it as simple as possible but similar to the Maxwell's ---

$$ * d * d A + ... =0? $$ What is the simplest form of $ ...$ term?

p.s. What I got so far is that $ ...$ term is $$C=* d * (A \wedge A) +A( * (d a + A \wedge A) ) - (* (dA + A \wedge A) ) A.$$ Do we have better way to simplify this complicated $C$ in terms of $A$?

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  • $\begingroup$ $\star d \star dA$ is not gauge invariant, so why do you want to isolate it? $\endgroup$ – Ryan Thorngren May 20 at 21:07
  • $\begingroup$ $F$ is also not gauge invariant, but only covariant. I want to see $C just for comparing the higher order terms. $\endgroup$ – annie heart May 20 at 21:17
  • $\begingroup$ $F$ transforms homogeneously so it's much easier to understand than $\star d \star A$. $\endgroup$ – Ryan Thorngren May 20 at 21:33

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