Why Is the Middle Term in the QED Field Strength Tensor Commutator Non-Zero?

[malawi_glenn]

Hi, I was reading earlier today in Peskin about QED field strenght tensor:

equation 15.15 and 15.16
$$[D_\mu, D_\nu ] \psi= [\partial _\mu , \partial _\nu ] \psi + ([\partial _\mu, A_\nu] - [\partial _\nu, A_\mu]) \psi + [A_\mu,A_\nu] \psi$$


Where A is the gauge field...

That part, I have control over.

Now I know that the first and last commutator is zero (abelian gauge theory and partial derivatives commute), but the middle one is really bothering me!

I obtain:

$$(\partial _\mu A_\nu - \partial _{\nu} A_\mu) \psi + (-A_\nu\partial _\mu + A_\mu \partial _\nu) \psi$$

And that last $(-A_\nu\partial _\mu + A_\mu \partial _\nu) \psi$ should be ZERO, so that:

$$F_{\mu \nu} = [D_\mu, D_\nu ] = \partial _\mu A_\nu - \partial _{\nu} A_\mu$$

BUT I don't know why ...

Any help or insight would, I would be very thankful of :shy:

 

[malawi_glenn]

I found an older thread "Yang Mills field stress tensor" where George Jones gave an excellent answer!

:-)