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Nauj Onerom

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**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".**

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

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? ?

**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? ?

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