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:

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

H is an isospiner, \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.<br /> <br /> I want to calculate (D_\mu H) ^{\dagger} (D^\mu H) but keep getting the wrong answer. So I&#039;ve begun to doubt wether i do (D_\mu H) ^{\dagger} correct. Is it:<br /> <br /> &lt;br /&gt; (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 ?&lt;br /&gt;<br /> <br /> or will the first term be: H^{\dagger} \partial_\mu ?<br /> <br /> Any help would be much appreciated!<br />

<br /> What does it look like in the momentum basis? Whenever you have derivatives, you should ask yourself, &quot;can I understand this better, or calculate this more easily, in the momentum basis?&quot;