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QFT - rewriting a conserved quantity

  1. Aug 27, 2012 #1
    Hey! I'm trying to learn QFT now and I'm currently reading David Tong's online lectures;


    At page 17 finds the conserved current

    [tex] j^\mu = - \omega^\rho_{\ \nu} T^{\mu}_{\ \rho} x^\nu[/tex]

    where i have understood T to be the energy momentum tensor. He further states that it can be rewritten as

    [tex](J^\mu)^{\sigma \rho} = x^\rho T^{\mu \sigma} - x^\sigma T^{\mu \rho}.[/tex]

    I am not that good manipulating tensors yet and my question is how one goes about showing this, step by step.
  2. jcsd
  3. Aug 27, 2012 #2


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    Since ωμν is antisymmetric, ωμν = ½(ωμv - ω). We can therefore write (1.54) as
    jμ = - ½(ωρv - ω)Tμρxν.
    In the second term, switch the index labels ρ and v:
    jμ = ½ωρv(Tμvxρ - Tμρxv)
    He then defines (Jμ)ρv to be the quantity in parentheses.
  4. Aug 27, 2012 #3
    Thanks! And there has also been one applied lowering operator and one raising operator?
  5. Aug 27, 2012 #4
    What is the motivation behind doing this manipulation btw? To make it look kind of like a cross product?
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