Proving Commutation Relation in Poincaré Transformation

[Petraa]

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

$$P_{\mu}=i\partial_{\mu}$$

$$M_{\mu\nu}=i\left(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}\right)$$

The Attempt at a Solution

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

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

Now if we recall the definition of the generator of the translations $$P_{\mu}=i\partial_{\mu}\implies\partial_{\mu}=\frac{P_{\mu}}{i}=-iP_{\mu}$$
$$\left[M_{\mu\nu},P_{\rho}\right]=\partial_{\rho}x_{\mu}\partial_{\nu}-\partial_{\rho}x_{\nu}\partial_{\mu}=\partial_{\rho}x_{\mu}\left(-iP_{\nu}\right)-\partial_{\rho}x_{\nu}\left(-iP_{\mu}\right)=i\left(\partial_{\rho}x_{\nu}P_{\mu}-\partial_{\rho}x_{\mu}P_{\nu}\right)$$

I know the results of the commutators from the Poincaré algebra so $$\partial_{\rho}x_{\mu}=g_{\rho\mu}$$ but I don't understand it. I thought that
$$\partial_{\rho}x_{\mu}=\delta_{\rho\mu}$$

Any help in order to prove the penultimate relation ? Because I don't know how to go from
$$\partial_{\rho}x_{\mu}=\delta_{\rho\mu}$$ to
$$\partial_{\rho}x_{\mu}=g_{\rho\mu}$$

Thanks

 

[Orodruin]

I thought that
$$\partial_{\rho}x_{\mu}=\delta_{\rho\mu}$$

This is not true. What is true is $\partial_\rho x^\mu = \delta_\rho^\mu$, which is not the same thing.

 

[Petraa]

@Orodruin Can you develop it a little more pls ? Because I have problems with this little quibbling of index notation and such

 

[Orodruin]

The coordinate with covariant index does not fulfil the relation you quoted. It only holds for a contravariant index and when you lower it using the metric you get the relation you were looking for. Placing it on a more appropriate level is impossible if you do not specify exactly which part you are having problems with and what your current level of understanding is.

 

[Petraa]

So basically $$\partial_{\rho}x_{\nu}=\partial_{\rho}g_{v\alpha}x^{\alpha}=g_{v\alpha}\partial_{\rho}x^{\alpha}=g_{v\alpha}\delta_{\rho}^{\alpha}=g_{\nu\rho}=g_{\rho\nu}$$

Any mistake ?

 

[Orodruin]

Just beware that things will not work out like this once you start looking at GR and the metric becomes coordinate dependent ... The assumption here is that you are working with the Minkowski metric.

 

[Petraa]

Just beware that things will not work out like this once you start looking at GR and the metric becomes coordinate dependent ... The assumption here is that you are working with the Minkowski metric.

Yes, indeed I'm using the Minkowski metric. The part where I have problems, as I just realized right now, is the difference between $$\partial_{\alpha}x_{\nu}$$
$$\partial_{\rho}x^{\alpha}$$
$$\partial^{\rho}x_{\alpha}$$
$$\partial^{\rho}x^{\alpha}$$

If you know any book for dummies like me where this topic is covered I would appreciate that. Thank you for your time

 

[Orodruin]

The only thing to realize is essentially that $x^\mu$ is different from $x_\mu$ and that
$$
\partial_\mu x^\nu \equiv \frac{\partial x^\nu}{\partial x^\mu}$$
and there really is not much more to it. You can work it all out from there.