How Is the Hermitian Adjoint of a Covariant Differential Operator Calculated?

[student111]

Homework Statement

Im am considering a covariant differential:

D_\mu H = ( partial_\mu + \frac{1}{2} i g \tau_j W_{j\mu} + ig B_\mu ) H

H is an isospiner, \tau_j are the pauli spin matrices, \partial_\mu is the four-gradient \frac{\partial}{\partial x^\mu} and W_{j \mu} and B_\mu are gauge fields.

I want to calculate (D_\mu H) ^{\dagger} (D^\mu H) but keep getting the wrong answer. So I've begun to doubt wether i do (D_\mu H) ^{\dagger} correct. Is it:

(D_\mu H)) ^{\dagger}= \partial_\mu H^{\dagger} - \frac{1}{2}ig H^{\dagger} \tau_j W_{j \mu} - H^{\dagger} i g B_\mu ?

or will the first term be: H^{\dagger} \partial_\mu ?

Any help would be much appreciated!

 

[turin]

Please use the [ tex ] ... [ / tex ] tags (without the spaces in the tags) for your equations. They are hard to read in plain text. I'll do this one for you:

Im am considering a covariant differential:

$$D_\mu H = ( \partial_\mu + \frac{1}{2} i g \tau_j W_{j\mu} + ig B_\mu ) H$$

H is an isospiner, [itex]\tau_j[/tex] are the pauli spin matrices, $\partial_\mu$ is the four-gradient $\frac{\partial}{\partial x^\mu}$ and $W_{j \mu}$ and $B_\mu$ are gauge fields.

I want to calculate $(D_\mu H) ^{\dagger} (D^\mu H)$ but keep getting the wrong answer. So I've begun to doubt wether i do $(D_\mu H) ^{\dagger}$ correct. Is it:

$$(D_\mu H) ^{\dagger}= \partial_\mu H^{\dagger} - \frac{1}{2}ig H^{\dagger} \tau_j W_{j \mu} - H^{\dagger} i g B_\mu ?$$

or will the first term be: $H^{\dagger} \partial_\mu$ ?

Any help would be much appreciated!

What does it look like in the momentum basis? Whenever you have derivatives, you should ask yourself, "can I understand this better, or calculate this more easily, in the momentum basis?"