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
<br /> [$J J^{\mu \nu}, J^{\rho \sigma}] = i (g^{\nu \rho} J^{\mu \sigma} - g^{\mu<br /> \rho} J^{\nu \sigma} - g^{\nu \sigma} J^{\mu \rho} + g^{\mu \sigma} J^{\nu<br /> \rho})$<br />
Where the generators are defined as

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

 

[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.