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QFT : Why do tensors in lagrangian densities contract? 
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#1
Jul1709, 06:21 PM

PF Gold
P: 472

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 nonreal 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 similarindice contraction. 


#2
Jul1709, 06:36 PM

Emeritus
Sci Advisor
PF Gold
P: 16,464

That's not a "16term 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". 


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