I would like to calculate the following expression: $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ where $D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$ and $A_\mu^a$ are the components of a real $SU(2)$ gauge Field and $\Phi$ is a complex two component spinor.

Unfortunately I have not yet come to a result. Here is what I have tried, $$(\tau^a)^\dagger = \tau^a .$$

Inserting the definition of $D_\mu$ yields
$\partial_\mu\Phi^\dagger \partial^\mu \Phi - \partial^\mu(\Phi^\dagger)\frac{ig}{2}\tau^a A_\mu^a\Phi+\frac{ig}{2} \Phi^\dagger \tau^aA^{a}_{\mu}\partial^\mu \Phi+\frac{g^2}{4}\Phi^\dagger\tau^aA^a_\mu \tau^bA^{b\mu}\Phi.$

can I simplify this expression even further?

I would like to calculate the following expression: $$(D_\mu\Phi)^\dagger(D^\mu\Phi)$$ where $$D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$$ and $A_\mu^a$ are the components of a real $SU(2)$ gauge Field and $\Phi$ is a complex two component spinor.

Unfortunately I have not yet come to a result. Here is what I have tried, $$(\tau^a)^\dagger = \tau^a .$$

Inserting the definition of $D_\mu$ yields
$$\partial_\mu\Phi^\dagger \partial^\mu \Phi - \partial^\mu(\Phi^\dagger)\frac{ig}{2}\tau^a A_\mu^a\Phi+\frac{ig}{2} \Phi^\dagger \tau^aA^{a}_{\mu}\partial^\mu \Phi+\frac{g^2}{4}\Phi^\dagger\tau^aA^a_\mu \tau^bA^{b\mu}\Phi.$$

can I simplify this expression even further?

I would like to calculate the following expression: $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ where $D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$ and $A_\mu^a$ are the components of a real $SU(2)$ gauge Field and $\Phi$ is a complex two component spinor.

Unfortunately I have not yet come to a result. Here is what I have tried:

$(\tau^a)^\dagger = \tau \\$

Inserting the definition of $D_\mu$ yields to:

$\partial_\mu\Phi^\dagger \partial^\mu \Phi - \partial_\mu(\Phi^\dagger)\frac{ig}{2}\tau^a A_\mu^a\Phi\frac{ig}{2}\tau^aA^{a\mu} \Phi^\dagger \partial^\mu (\Phi)+\frac{g^2}{4}\tau^aA^a_\mu \Phi^\dagger\tau^aA^{a\mu}\Phi$

can i simplify this expression even further?

I would like to calculate the following expression: $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ where $D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$ and $A_\mu^a$ are the components of a real $SU(2)$ gauge Field and $\Phi$ is a complex two component spinor.

Unfortunately I have not yet come to a result. Here is what I have tried, $$(\tau^a)^\dagger = \tau^a .$$

Inserting the definition of $D_\mu$ yields
$\partial_\mu\Phi^\dagger \partial^\mu \Phi - \partial^\mu(\Phi^\dagger)\frac{ig}{2}\tau^a A_\mu^a\Phi+\frac{ig}{2} \Phi^\dagger \tau^aA^{a}_{\mu}\partial^\mu \Phi+\frac{g^2}{4}\Phi^\dagger\tau^aA^a_\mu \tau^bA^{b\mu}\Phi.$

can I simplify this expression even further?