# QFT : Why do tensors in lagrangian densities contract?

1. Jul 17, 2009

### Hepth

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.

2. Jul 17, 2009