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Poincare Algebra -- Quick Question

  1. Apr 25, 2017 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant 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##

    3. The attempt at a solution

    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
     
  2. jcsd
  3. Apr 28, 2017 #2

    stevendaryl

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    Staff Emeritus
    Science Advisor

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