Help with Vector Notation: \partial_{\mu} \phi^{*}\partial^{\mu} \phi

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The expression \partial_{\mu} \phi^{*}\partial^{\mu} \phi represents the sum of the squares of the derivatives of a scalar field, incorporating the complex conjugate. It is calculated using the summation convention, where repeated indices imply summation, leading to the form -\left\lvert\frac{\partial \phi}{\partial x^0}\right\rvert^2 + \sum_i \left\lvert\frac{\partial \phi}{\partial x^i}\right\rvert^2. The notation requires one index to be raised and one lowered to maintain proper Lorentz invariance and avoid malformed expressions. Writing \partial^{\mu} \phi^{*} \partial^{\mu} \phi would not allow for summation over the \mu index, violating this convention. Understanding these conventions is crucial for ensuring expressions behave correctly under Lorentz transformations.
Onamor
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Not a particularly direct question, just something I don't mathematically understand and would very much appreciate help with.

For some scalar field \phi, what would \partial_{\mu} \phi^{*}\partial^{\mu} \phi mean in mathematical terms. ie how would I calculate it?

From what I understand its basically \Sigma_{\mu}\left(\frac{\partial}{\partial x^{\mu}}\phi \right)^{2} because of the complex conjugate in the scalar field, and you sum over repeated indexes.

Also, just to ask, why wouldn't I write this \partial^{\mu} \phi^{*} \partial^{\mu} \phi? Is it because I wouldn't then be allowed to sum over the \mu index?
Or is it something to do with a contraction being Lorentz invariant?

Thanks for any help, let me know if I haven't been clear.
 
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\partial_\mu \phi^* \partial^\mu \phi is the same as \eta^{\mu\nu}\partial_\mu \phi^* \partial_\nu \phi where
\partial_\mu = \frac{\partial}{\partial x^\mu}and repeated indices imply summation, so you have
\partial_\mu \phi^* \partial^\mu \phi = - \left\lvert\frac{\partial \phi}{\partial x^0}\right\rvert^2+\sum_i \left\lvert\frac{\partial \phi}{\partial x^i}\right\rvert^2
In general, you shouldn't have a repeated index with both raised or both lowered. They should always come one up and one down, otherwise you have a malformed expression on your hands.
 
Onamor said:
Also, just to ask, why wouldn't I write this \partial^{\mu} \phi^{*} \partial^{\mu} \phi? Is it because I wouldn't then be allowed to sum over the \mu index?
Or is it something to do with a contraction being Lorentz invariant?

To complement vela's response: the answers are yes and yes.
Summation convention only applies to one upper and one lower index, and the whole idea is that doing this that given some objects behaving properly under Lorentz-transformations, the notation almost forces you into creating new objects behaving properly under Lorentz-transformations, rather than some arbitrary mathematical expression.
As vela shows, it means that if you use the simple trick of "one upper + one lower" index what you are actually doing is making sure you use the spacetime metric in precisely the places you need to get Lorentz-invariance right.
 
Thank you both, very helpful as always.
Much appreciated.
 
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