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Lie Algebra of Poincaré Group

  1. Jun 3, 2013 #1
    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):

    BCaSwpm.png

    2. Relevant equations

    The definitions of [itex]L_{\mu \nu}[/itex] and [itex]P_{\rho}[/itex] 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:

    [itex]-i\hbar(\partial_{\rho}x_{\mu}-x_{\mu}\partial_{\rho})P_{\nu}+i\hbar(\partial_{\rho}x_{\nu}-x_{\nu}\partial_{\rho})P_{\mu}[/itex]

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

    Calculating the derivatives of the coordinates and rearranging I get:

    [itex]-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})[/itex]

    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. jcsd
  3. Jun 3, 2013 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

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

    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}##.
     
  4. Jun 3, 2013 #3
    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!
     
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