- #1

- 21

- 0

## Homework Statement

Given a Poincaré transformation, Lorentz+translation, I have to find the Poincaré generators in the scalar field representation and then prove that the commutation relations.

I've done the first part but I can't prove the commutation relations.

## Homework Equations

[tex]P_{\mu}=i\partial_{\mu}[/tex]

[tex]M_{\mu\nu}=i\left(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}\right)[/tex]

## The Attempt at a Solution

For example for the mixed commutator after doing some straight-forward algebra

[tex]\left[M_{\mu\nu},P_{\rho}\right]=i^{2}\left[\left(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}\right),\partial_{\rho}\right]=\left[\partial_{\rho},\left(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}\right)\right]=\partial_{\rho}x_{\mu}\partial_{\nu}-\partial_{\rho}x_{\nu}\partial_{\mu}[/tex]

Now if we recall the definition of the generator of the translations [tex]

P_{\mu}=i\partial_{\mu}\implies\partial_{\mu}=\frac{P_{\mu}}{i}=-iP_{\mu}

[/tex]

[tex]\left[M_{\mu\nu},P_{\rho}\right]=\partial_{\rho}x_{\mu}\partial_{\nu}-\partial_{\rho}x_{\nu}\partial_{\mu}=\partial_{\rho}x_{\mu}\left(-iP_{\nu}\right)-\partial_{\rho}x_{\nu}\left(-iP_{\mu}\right)=i\left(\partial_{\rho}x_{\nu}P_{\mu}-\partial_{\rho}x_{\mu}P_{\nu}\right)[/tex]

I know the results of the commutators from the Poincaré algebra so [tex]

\partial_{\rho}x_{\mu}=g_{\rho\mu}[/tex] but I don't understand it. I thought that

[tex]

\partial_{\rho}x_{\mu}=\delta_{\rho\mu}[/tex]

Any help in order to prove the penultimate relation ? Because I don't know how to go from

[tex]

\partial_{\rho}x_{\mu}=\delta_{\rho\mu}[/tex] to

[tex]

\partial_{\rho}x_{\mu}=g_{\rho\mu}[/tex]

Thanks