# Poincare Algebra -- Quick Question

• binbagsss
binbagsss

## Homework Statement

Does ##x_p\partial_v\partial_u-x_v\partial_p\partial_u=0##

## Homework Equations

I need this to be true to show a poincare algebra commutator.

We have just shown that ##[P_u, P_v] =0 ##, i.e. simply because partial derivatives commute.

Where ##P_u=\partial_u##

## The Attempt at a Solution

[/B]
I'm guess it's using that partial derivatives commute and renaming the ##v## and ##p## for each other, however I'm a bit confused with this, because, I know this isn't really the context here, but these are free indicies not dummy indices, i..e not an index summing over, so I'm a bit unclear how you could simply rename.

Many thanks

Staff Emeritus

## Homework Statement

Does ##x_p\partial_v\partial_u-x_v\partial_p\partial_u=0##

## Homework Equations

I need this to be true to show a poincare algebra commutator.

We have just shown that ##[P_u, P_v] =0 ##, i.e. simply because partial derivatives commute.

Where ##P_u=\partial_u##

## The Attempt at a Solution

[/B]
I'm guess it's using that partial derivatives commute and renaming the ##v## and ##p## for each other, however I'm a bit confused with this, because, I know this isn't really the context here, but these are free indicies not dummy indices, i..e not an index summing over, so I'm a bit unclear how you could simply rename.

Many thanks

Free indices means that it must be true for every possible choice of $u,p,v$, and for all possible values of $x_p$ and $x_v$. So try the particular case where $u,p,v$ are all different indices, and $x_p = 0$, and $x_v \neq 0$.