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QFT : Why do tensors in lagrangian densities contract?

  1. Jul 17, 2009 #1

    Hepth

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    Gold Member

    What is the general rule behind why for any given lagrangian (QED/QCD show this) that any vectors or tensors contract indices? I know it must be something simple, but I just can't think of it offhand.
    QED :
    [tex]
    F_{\mu\nu}F^{\mu\nu}
    [/tex]
    Proca (massive vector):
    [tex]
    A_\mu A^\mu
    [/tex]
    QCD :
    [tex]
    G^{\alpha}_{\mu\nu} G^{\mu\nu}_{\alpha}
    [/tex]

    Like could I imagine some non-real lagrangian that is [tex]B^{\mu\nu}B^{\mu}_{\nu}[/tex]
    without worrying about gauge invariance?


    EDIT: its that the action has to be a scalar quantity, isnt it?
    REEDIT: Ah its still a scalar though, just not NECESSARILY invariant.

    Well then what about
    [tex]
    B^{\mu}B_{\nu}
    [/tex]
    so that you still get some 16 term scalar, but its not a similar-indice contraction.
     
  2. jcsd
  3. Jul 17, 2009 #2

    Vanadium 50

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    Staff Emeritus
    Science Advisor
    Education Advisor

    That's not a "16-term scalar" (which I don't think even makes sense). That's a tensor.

    You answered your own question with "the action has to be a scalar".
     
  4. Jul 17, 2009 #3

    Hepth

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    Gold Member

    Ok, just making sure.
     
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