Covariant derivatives commutator - field strength tensor

[caimzzz]

Homework Statement

So I've been trying to derive field strength tensor. What to do with the last 2 parts ? They obviously don't cancel (or do they?)

Homework Equations

The Attempt at a Solution

$$[D_{\mu},D_{\nu}] = (\partial_{\mu} + A_{\mu})(\partial_{\nu} + A_{\nu}) - (\mu <-> \nu) = [\partial_{\mu},\partial_{\nu}] + [\partial_{\mu},A_{\nu}]+[A_{\mu},\partial_{\nu}] +[A_{\mu}, A_{\nu}] =$$
$$\partial_{\mu}A_{\nu} - \partial_{\nu} A_{\mu} + [A_{\mu},A_{\nu}] -A_{\nu} \partial_{\mu} + A_{\mu} \partial_{\nu}$$

 

[Orodruin]

They are canceled by the first two terms when the derivative acts on whatever function the entire operator is acting on. What remains is just the derivative acting on the A fields.

 

[caimzzz]

Thank you