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Show that [J_a,G_a] = 0, commutation relationships

  1. Oct 28, 2015 #1
    1. The problem statement, all variables and given/known data
    Using the given equations prove that

    upload_2015-10-28_12-22-32.png


    2. Relevant equations


    upload_2015-10-28_12-22-47.png ,

    upload_2015-10-28_12-23-42.png ,


    upload_2015-10-28_12-21-41.png + upload_2015-10-28_12-21-50.png

    (it won't render together in Maple for whatever reason)

    3. The attempt at a solution

    So I started with expanding the Jacobi Identity (the third relevant equation) and through tedious algebra arrived at proving it unnecessarily and then finding that

    upload_2015-10-28_12-27-32.png

    then by applying the second equation this goes to

    upload_2015-10-28_12-27-49.png

    I am unsure how to proceed from here however.

    The question also gives the hint that the trick to solve it is to take the appropriate particular values of the labels a,b,y in the Jacobi identity but I am unsure how to use this.

    Any help would be greatly appreciated.
     
  2. jcsd
  3. Oct 28, 2015 #2

    fzero

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    You're not actually given ##[G_a,G_b]##, so it's a bit of a hassle to use that Jacobi identity. Try to find a way to relate ##[J_a,G_b]## to the expression ## [ J_a, [J_b,J_c]]## and use the Jacobi identity for that.
     
  4. Oct 28, 2015 #3
    Hmm okay, tbh I'm not sure how to proceed with that
     
  5. Oct 29, 2015 #4

    fzero

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    To push you along, consider
    $$ {\epsilon_a}^{bc} [J_b,J_c] = i {\epsilon_a}^{bc} {\epsilon^d}_{bc} G_d.$$
    Using an identity for the Levi-Civita symbol will give you an expression for ##G_a## in terms of the commutator of ##J##s.
     
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