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QFT : Why do tensors in lagrangian densities contract?

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Hepth
#1
Jul17-09, 06:21 PM
PF Gold
Hepth's Avatar
P: 471
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.
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Vanadium 50
#2
Jul17-09, 06:36 PM
Mentor
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P: 16,385
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".
Hepth
#3
Jul17-09, 06:54 PM
PF Gold
Hepth's Avatar
P: 471
Ok, just making sure.


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