On differentiation and indices in field theory

[Nauj Onerom]

I'm self-studying field theory and trying to solidify my understanding of index manipulations. So I've been told that there is a general rule: " If the index is lowered on the 'denominator' then it's a raised index". My question is whether this is just a rule or something that can make sense mathematically. For example (taken from ch. 2 of P&S Intro to QFT), for a transformation $$\phi\rightarrow\phi + \alpha,$$ of the Lagrangian $$\mathcal{L} = \frac{1}{2}(\partial_\mu\phi)^2 ,$$ we get a conserved current

$$ j^{\mu} = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}.$$

If we differentiate normally we get $$\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)} = \partial_\mu\phi,$$ but this has a lowered index. How do I see that this is supposed to be raised? ?

 

[Orodruin]

What is meant by $(\partial_\mu \phi)^2$ is typically $(\partial_\mu\phi)(\partial^\mu\phi)$. Otherwise the term would not be Lorentz invariant.

 

[Nauj Onerom]

What is meant by $(\partial_\mu \phi)^2$ is typically $(\partial_\mu\phi)(\partial^\mu\phi)$. Otherwise the term would not be Lorentz invariant.

Very clear now, thanks !