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Levi Civita and commutators

  1. Apr 21, 2010 #1
    Sorry for spamming the forums, but one last question for today!

    If

    [tex]\Sigma^k=\frac{i}{2} \epsilon_{kij} [\gamma^i , \gamma^j][/tex]

    where [A,B]=AB-BA

    Why does [tex]{\Sigma^1=2i \gamma^2\gamma^3[/tex] (that's what my notes say, anyway)

    I think it should equal:

    [tex]\Sigma^1=\frac{i}{2}\epsilon_{123}[\gamma^2,\gamma^3] + \frac{i}{2}\epsilon_{132}[\gamma^3,\gamma^2] [/tex]
    [tex] = \frac{i}{2}(+1)}[\gamma^2,\gamma^3] + \frac{i}{2}(-1)(-[\gamma^2,\gamma^3])[/tex]
    [tex] =i[\gamma^2,\gamma^3] [/tex]
     
  2. jcsd
  3. Apr 21, 2010 #2
    Disregard that question - I've just worked it out. For anyone who's interested: it's because

    [tex]
    =\gamma^\mu\gamma^\nu = -\gamma^\nu\gamma^\mu
    [/tex]
     
  4. Apr 24, 2010 #3

    that is not true, it is only true for the "spatial" components, for the time component: gamma^0 gamma^0 = gamma^0 gamma^0 = 1
     
    Last edited: Apr 24, 2010
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