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.(adsbygoogle = window.adsbygoogle || []).push({});

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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# QFT : Why do tensors in lagrangian densities contract?

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