Hermitian adjoint of operator

[student111]

1. The problem statement, all variables and given/known data

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!
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

[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, \tau_j[/tex] are the pauli spin matrices, [itex]\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?"