This question relates to mathematical physics, so it's debatable whether it belongs here or in Maths.SE. I chose to post here because I feel the physics context is strong. Content from, though not directly related to this Phys.SE question are present in this question:
The Lagrangian for the $\mathrm{U}(N)$ Higgs model is $$\mathcal{L}=-\frac12\mathrm{Tr}F_{\mu\nu} F^{\mu\nu}+\left(D_\mu\phi\right)^\dagger D^\mu\phi-m^2\phi^\dagger\phi-\frac12\lambda\left(\phi^\dagger\phi\right)^2$$ where $\phi$ is a complex $N$-component Lorentz scalar field, $D_\mu=\partial_\mu+igA_\mu$ and $F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu+ig\left[A_\mu, A_\nu\right]$, and $A_\mu=A_\mu^at^a$ is the gauge field. The equation of motion for the scalar field $\phi$ is $$D_\mu D^\mu\phi+m^2\phi + \lambda\left(\phi^\dagger\phi\right)\phi=0\tag{A}$$ Using the equation of motion, $(\mathrm{A})$, show explicitly that $j^\mu$ satisfies $$\mathrm{D}^{ab}_\mu j^{\mu b}=0\tag{B}$$ where $$j^{\mu b}= ig\left(\phi^\dagger t^b (D^\mu\phi)-(D^\mu \phi)^\dagger t^b\phi\right)\tag{1}$$ and the covariant derivative, $\mathrm{D}^{ab}_\mu$ in the adjoint representation is $$\mathrm{D}^{ab}_\mu=\left(\delta^{ab}\partial_\mu+gA_{\mu}^cf^{abc}\right)\tag{2}$$
and $f^{abc}$ are the structure constants which obey $T^a_{bc}=-if^{abc}$ in the adjoint representation.
This is how I would tackle the problem, first, by direct substitution of $(1)$ and $(2)$ into $(\mathrm{B})$ yields:
$$D_\mu^{ab}j^{\mu b}=ig\left[\delta^{ab}\partial_\mu+gA_\mu^cf^{abc}\right]\left[\phi^\dagger t^b\left(D^\mu\phi\right)-\left(D^\mu\phi\right)^\dagger t^b\phi\right]$$ $$=ig\left[\delta^{ab}\partial_\mu\{\phi^\dagger t^b (D^\mu \phi)\}-\delta^{ab}\partial_\mu \{(D^\mu\phi)^\dagger t^b\phi\}+g A_\mu^c f^{abc}\phi^\dagger t^b(D^\mu \phi)-g A_\mu^cf^{abc}(D^\mu)^\dagger t^b\phi\right]$$ $$=ig\bigg[(\partial_\mu \phi^\dagger)t^a(D^\mu\phi)+\phi^\dagger t^a\partial_\mu(D^\mu\phi)+gA_\mu^c f^{abc}\phi^\dagger t^b(D^\mu\phi)$$ $$-\partial_\mu(D^\mu \phi)^\dagger t^a \phi-(D_\mu\phi)^\dagger t^a\partial_\mu\phi-gA_\mu^cf^{abc}(D^\mu\phi)^\dagger t^b\phi\bigg]\tag{3}$$
Since the covariant derivative is defined as $D_\mu=\partial_\mu+igA_\mu$ $$\implies \partial_\mu=D_\mu-igA_\mu\tag{C}$$ and therefore, $$\partial_\mu\phi=D_\mu\phi-igA_\mu\phi\tag{D}$$ and $$\partial_\mu\phi^\dagger=(D_\mu-igA_\mu)\phi^\dagger=D_\mu\phi^\dagger-ig\phi^\dagger A_\mu\tag{E}$$
Substituting $(\mathrm{C})$, $(\mathrm{D})$ and $(\mathrm{E})$ into $(3)$ (the previous expression for $D_\mu^{ab}j^{\mu b}$), gives $$D_\mu^{ab}j^{\mu b}=ig\bigg[\{D_\mu\phi^\dagger-ig\phi^\dagger A_\mu\}t^aD^\mu\phi+\phi^\dagger t^a\{D_\mu-igA_\mu\}D^\mu\phi+gA_\mu^c f^{abc}\phi^\dagger t^bD^\mu\phi$$ $$-\{D_\mu-igA_\mu\}(D^\mu\phi)^\dagger t^a\phi-D^\mu\phi^\dagger t^a\{D_\mu\phi-igA_\mu\phi\}-gA_\mu^cf^{abc}D^\mu\phi^\dagger t^b\phi\bigg]$$
$$=ig\bigg[\color{blue}{D_\mu\phi^\dagger t^a D^\mu\phi}-ig\phi^\dagger A_\mu t^aD^\mu\phi+\phi^\dagger t^a D_\mu (D^\mu\phi)-ig\phi^\dagger t^a A_\mu D^\mu\phi+g A_\mu^cf^{abc}\phi^\dagger t^b D^\mu\phi$$
$$-D_\mu(D^\mu\phi)^\dagger t^a\phi+igA_\mu D^\mu\phi^\dagger t^a\phi-\color{blue}{D^\mu\phi^\dagger t^a D_\mu \phi}+igD^\mu\phi^\dagger t^a A_\mu\phi-gA_\mu^cf^{abc}D^\mu\phi^\dagger t^b \phi\bigg]$$
This is as far as I can get, I am unable to simplify the equation anymore except by noting that the blue terms are similar and may cancel iff $\color{blue}{(D^\mu\phi)^\dagger t^a(D_\mu \phi)}=g_{\mu\alpha}(D^\alpha \phi)^\dagger t^a g^{\mu\beta}(D_\beta \phi)\stackrel{\color{red}{?}}{=}(D_\mu\phi)^\dagger t^a (D^\mu\phi)$. However, I am doubtful that this type of manipulation is valid due to the presence of the Hermitian adjoint, hence the red question mark over the equality.
Here is the author's solution which I am trying to understand:
$$D_\mu^{ab}j^{\mu b}=ig\left(\delta^{ab}\partial_\mu+gA_\mu^cf^{abc}\right)\left[\phi^\dagger t^b(D^\mu\phi)-(D^\mu\phi)^\dagger t^b \phi\right]$$ $$=ig\bigg[(\partial_\mu\phi^\dagger t^a (D^\mu\phi)+\phi^\dagger t^a \partial_\mu(D^\mu\phi)+gA_\mu^c f^{abc}\phi^\dagger t^b(D^\mu\phi)$$ $$-\color{red}{(\partial_\mu D^\mu\phi)^\dagger} t^a\phi-(D^\mu\phi)^\dagger t^a\partial_\mu\phi-gA_\mu^cf^{abc}(D^\mu\phi)^\dagger t^b\phi\bigg]\tag{F}$$
There is a problem here, I don't understand what allowed the author to bring the partial derivative, $\partial_\mu$ inside the Hermitian adjoint for the term marked red. Comparing this to the last line of eqn. $(3)$ I was cautious and did not commute the derivative operator into the adjoint as it seemed mathematically incorrect.
I could stop typing at this point as my doubts are already apparent, but I feel it's pertinent to typeset the rest of the solution given by the author to try to understand the logic that he/she employed:
Using $\partial_\mu=D_\mu-igA_\mu$, this becomes $$D_\mu^{ab}j^{\mu b}=ig\bigg[(D_\mu\phi)^\dagger t^a (D^\mu\phi)+ig\phi^\dagger A_\mu t^a(D^\mu\phi)+\phi^\dagger t^aD_\mu D^\mu\phi-ig\phi^\dagger t^a A_\mu(D^\mu\phi)$$ $$+gA_\mu^cf^{abc}\phi^\dagger t^b (D^\mu\phi)-(D_\mu D^\mu\phi)^\dagger t^a \phi-\color{#580}{ig(D^\mu\phi)^\dagger A_\mu t^a \phi}\tag{G}$$ $$-(D^\mu\phi)^\dagger t^a D_\mu\phi+ig(D^\mu\phi)^\dagger t^a A_\mu\phi-gA_\mu^cf^{abc}(D^\mu\phi)^\dagger t^b\phi\bigg]$$ $$=ig\bigg[(D_\mu\phi)^\dagger t^a (D^\mu\phi)+ig\phi^\dagger[A_\mu,\,t^a](D^\mu\phi)+\phi^\dagger t^a D_\mu D^\mu\phi+gA_\mu^cf^{abc}\phi^\dagger t^b(D^\mu\phi)$$ $$-(D_\mu D^\mu\phi)^\dagger t^a \phi-ig(D^\mu\phi)^\dagger [A_\mu,\,t^a]\phi - (D^\mu\phi)^\dagger t^a D_\mu\phi - gA_\mu^cf^{abc}(D^\mu\phi)^\dagger t^b\phi\bigg]$$
The first and seventh terms cancel. We can also write $[A_\mu,\,t^a]=A_\mu^c[t^c,t^a]=if^{abc}A_\mu^c t^b$, so we obtain $$D_\mu^{ab}j^{\mu b}=ig\bigg[-gA_\mu^cf^{abc}\phi^\dagger t^b(D^\mu\phi)+\phi^\dagger t^a D_\mu D^\mu\phi+gA_\mu^c f^{abc}\phi^\dagger t^b(D^\mu\phi)$$ $$-(D_\mu D^\mu \phi)^\dagger t^a\phi+gA_\mu^cf^{abc}(D^\mu\phi)^\dagger t^b \phi-gA_\mu^cf^{abc}(D^\mu\phi)^\dagger t^b\phi\bigg]$$
Finally, using the equation of motion $D_\mu D^\mu\phi=-m^2\phi-\lambda(\phi^\dagger\phi)\phi$, this becomes $$D_\mu^{ab}j^{\mu b}=ig\bigg[-m^2\phi^\dagger t^a\phi-\lambda(\phi^\dagger\phi)\phi^\dagger t^a\phi+m^2\phi^\dagger t^a\phi+\lambda(\phi^\dagger\phi)\phi^\dagger t^a \phi\bigg]=0$$ This confirms that the non-Abelian current indeed satisfies the covariant conservation law.
My main concerns in this question are:
- What allowed the author to bring the partial derivative, $\partial_\mu$ inside the Hermitian adjoint in eqn. $(\mathrm{F})$?
- Is the following equality $\color{blue}{(D^\mu\phi)^\dagger t^a(D_\mu \phi)}=g_{\mu\alpha}(D^\alpha \phi)^\dagger t^a g^{\mu\beta}(D_\beta \phi)\stackrel{\color{red}{?}}{=}(D_\mu\phi)^\dagger t^a (D^\mu\phi)$ true?
