Proving Poincare Algebra Using Differential Expression of Generator

[crime9894]

Homework Statement

Using differential expressions for the generator to verify the commutator expression in Poincare group

Relevant Equations

Definition for the diffrential expressions of the generators are given below

Using differential expressions for the generator, verify the commutator expression for $[J_{\mu\nu},P_{\rho}]=i(\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu})$ in Poincare group

Generator of translation: $P_{\rho}=-i\partial_{\rho}$
Generator of rotation: $J_{\mu\nu}=i(x_{\mu}\partial_{\nu}-x_{\nu}\partial{_\mu})$

Here is my working, I operate the commutator on a vector $x^j$:
$[J_{\mu\nu},P_{\rho}]x^j$
$=(J_{\mu\nu}P_{\rho}-P_{\rho}J_{\mu\nu})x^j$
$=0-iP_{\rho}(x_{\mu}\partial_{\nu}-x_{\nu}\partial{_\mu})x^j$
$=-[(\partial_{\rho}x_{\mu})(\partial_{\nu}x^j)-(\partial_{\rho}x_{\nu})(\partial_{\mu}x^j)]$
$=-[\eta_{\rho\mu}(\partial_{\nu}x^j)-\eta_{\rho\nu}(\partial_{\mu}x^j)]$
$=-i(\eta_{\rho\mu}P_{\nu}x^j-\eta_{\rho\nu}P_{\mu}x^j)$
$=-i(\eta_{\rho\mu}P_{\nu}-\eta_{\rho\nu}P_{\mu})x^j$

My answer had one extra negative sign on it. Where did it go wrong?

 

[TSny]

My answer had one extra negative sign on it. Where did it go wrong?

Your work looks correct to me. One has to be careful with sign conventions for $P_\rho$ and $J_{\mu \nu}$. For example, https://www.physik.uni-bielefeld.de/~borghini/Teaching/Symmetries/02_02.pdf they get the answer that you were asked to get, but they define $J_{\mu \nu}$ with the opposite sign.

 

[vanhees71]

And it should have the opposite sign, because $J_{\mu \nu}=x_{\mu} p_{\nu}-x_{\nu} p_{\mu}$ and then you just make the symbols operators.