# QFT : Why do tensors in lagrangian densities contract?

by Hepth
Tags: contract, densities, lagrangian, tensors
 PF Gold P: 434 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 : $$F_{\mu\nu}F^{\mu\nu}$$ Proca (massive vector): $$A_\mu A^\mu$$ QCD : $$G^{\alpha}_{\mu\nu} G^{\mu\nu}_{\alpha}$$ Like could I imagine some non-real lagrangian that is $$B^{\mu\nu}B^{\mu}_{\nu}$$ 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 $$B^{\mu}B_{\nu}$$ so that you still get some 16 term scalar, but its not a similar-indice contraction.
 Mentor P: 15,370 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".
 PF Gold P: 434 Ok, just making sure.

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