# Covariant derivatives commutator - field strength tensor

1. Jun 20, 2015

### caimzzz

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

2. Relevant equations

3. The attempt at a solution
$$[D_{\mu},D_{\nu}] = (\partial_{\mu} + A_{\mu})(\partial_{\nu} + A_{\nu}) - (\mu <-> \nu) = [\partial_{\mu},\partial_{\nu}] + [\partial_{\mu},A_{\nu}]+[A_{\mu},\partial_{\nu}] +[A_{\mu}, A_{\nu}] =$$
$$\partial_{\mu}A_{\nu} - \partial_{\nu} A_{\mu} + [A_{\mu},A_{\nu}] -A_{\nu} \partial_{\mu} + A_{\mu} \partial_{\nu}$$

2. Jun 20, 2015

### Orodruin

Staff Emeritus
They are cancelled 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.

3. Jun 20, 2015

Thank you