- #1
- 457
- 40
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 :
<br /> F_{\mu\nu}F^{\mu\nu}<br />
Proca (massive vector):
<br /> A_\mu A^\mu<br />
QCD :
<br /> G^{\alpha}_{\mu\nu} G^{\mu\nu}_{\alpha}<br />
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, isn't it?
REEDIT: Ah its still a scalar though, just not NECESSARILY invariant.
Well then what about
<br /> B^{\mu}B_{\nu}<br />
so that you still get some 16 term scalar, but its not a similar-indice contraction.
QED :
<br /> F_{\mu\nu}F^{\mu\nu}<br />
Proca (massive vector):
<br /> A_\mu A^\mu<br />
QCD :
<br /> G^{\alpha}_{\mu\nu} G^{\mu\nu}_{\alpha}<br />
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, isn't it?
REEDIT: Ah its still a scalar though, just not NECESSARILY invariant.
Well then what about
<br /> B^{\mu}B_{\nu}<br />
so that you still get some 16 term scalar, but its not a similar-indice contraction.
