# Commutator between covariant derivative, field strength

1. Jul 29, 2013

### oliveriandrea

Hello,
i try to prove that
μFμ$\nu$ + ig[Aμ, Fμ$\nu$] = [Dμ,Fμ$\nu$]
with the Dμ = ∂μ + igAμ

but i have a problem with the term Fμ$\nu$μ ...
i try to demonstrate that is nil, but i don't know if it's right...

Fμ$\nu$μ $\Psi$ = $\int$ (∂$\nu$Fμ$\nu$) (∂μ$\Psi$) + $\int$ Fμ$\nu$μ$\nu$ $\Psi$ = (∂$\nu$Fμ$\nu$) [$\Psi$ ] - $\int$$\Psi$∂μ$\nu$Fμ$\nu$ = 0

with $\Psi$ a smooth function, nil at infinity

if it's wrong please do you post the right answers? and why it is wrong...
thank you

2. Jul 30, 2013

### vanhees71

This is right. Of course, if you need it mathematically accurate, you have to think about, how fast you test function has to go to 0 at infinity, but if it's a physics question, what you did should be sufficient.

3. Jul 30, 2013

### oliveriandrea

Thank you (yes it's physics question)