# Lie Algebra of Poincaré Group

1. Jun 3, 2013

### malaspina

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

The problem statement is to prove the following identity (the following is the solution provided on the worksheet):

2. Relevant equations

The definitions of $L_{\mu \nu}$ and $P_{\rho}$ are apparent from the first line of the solution.

3. The attempt at a solution

I get to the second line and calculate the commutators explicitly:

$-i\hbar(\partial_{\rho}x_{\mu}-x_{\mu}\partial_{\rho})P_{\nu}+i\hbar(\partial_{\rho}x_{\nu}-x_{\nu}\partial_{\rho})P_{\mu}$

The derivatives of the coordinates give the metric tensor e.g. $\partial_{\rho}x_{\mu}=g_{\rho \mu}$

Calculating the derivatives of the coordinates and rearranging I get:

$-i\hbar g_{\rho \mu}P_{\nu}+i \hbar g_{\rho \nu}P_{\mu} +i\hbar(x_{\mu}\partial_{\rho}P_{\nu}-x_{\nu}\partial_{\rho}P_{\mu})$

The first two terms are the solution I'm looking for, so I'd deduce the last term should be equal to zero.

Is this correct? and if it is, how do I prove that the last term is in fact equal to zero?

2. Jun 3, 2013

### TSny

The last term should not be there. You need to be careful when evaluating $-i\hbar(\partial_{\rho}x_{\mu}-x_{\mu}\partial_{\rho})P_{\nu}+i\hbar(\partial_{\rho}x_{\nu}-x_{\nu}\partial_{\rho})P_{\mu}$

For example, if you bring in the $P_{\nu}$ in the first term you get $-i\hbar(\partial_{\rho}x_{\mu}P_{\nu}-x_{\mu}\partial_{\rho}P_{\nu})$

The first term in the parentheses should be interpreted as $\partial_{\rho}(x_{\mu}P_{\nu})$ where the derivative acts on the product of $x_\mu$ and $P_{\nu}$.

3. Jun 3, 2013

### malaspina

Jesus, I was completely dumbfounded and it was so obvious. Now that I've re-done it I don't even know how I was missing it. Thank you a lot!