Commutator of 4-momentum and position

[kilokhan]

Is there a commutation relation between $x^{\mu}$ and $\partial^{\nu}$ if you treat them as operators? I think I will need that to prove this
$[J J^{\mu \nu}, J^{\rho \sigma}] = i (g^{\nu \rho} J^{\mu \sigma} - g^{\mu<br /> \rho} J^{\nu \sigma} - g^{\nu \sigma} J^{\mu \rho} + g^{\mu \sigma} J^{\nu<br /> \rho})$
Where the generators are defined as

$<br /> J J^{\mu \nu} = i (x^{\mu} \partial^{\nu} - x^{\nu} \partial^{\mu})$

 

[kilokhan]

Never mind, I found the appropriate relation, $\partial_{\mu}x^{\nu}=g^{\mu \nu}$

But I'm not entirely sure why this is true. If someone could explain that would be great.