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

[Onamor]

Not a particularly direct question, just something I don't mathematically understand and would very much appreciate help with.

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

From what I understand its basically [itex]\Sigma_{\mu}\left(\frac{\partial}{\partial x^{\mu}}\phi \right)^{2}[/itex] 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 [itex]\partial^{\mu} \phi^{*} \partial^{\mu} \phi[/itex]? Is it because I wouldn't then be allowed to sum over the [itex]\mu[/itex] 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.

 

[vela]

[itex]\partial_\mu \phi^* \partial^\mu \phi[/itex] is the same as [itex]\eta^{\mu\nu}\partial_\mu \phi^* \partial_\nu \phi[/itex] where
[tex]\partial_\mu = \frac{\partial}{\partial x^\mu}[/tex]and repeated indices imply summation, so you have
[tex]\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[/tex]
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.

 

[CompuChip]

Also, just to ask, why wouldn't I write this [itex]\partial^{\mu} \phi^{*} \partial^{\mu} \phi[/itex]? Is it because I wouldn't then be allowed to sum over the [itex]\mu[/itex] 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.

 

[Onamor]

Thank you both, very helpful as always.
Much appreciated.

 

[paranormal]

The expression \partial_{\mu} \phi^{*}\partial^{\mu} \phi can be understood as the action of the derivative operator \partial_{\mu} on the complex conjugate of the scalar field \phi^{*}, followed by the action of the derivative operator \partial^{\mu} on the original scalar field \phi. In mathematical terms, this can be written as \partial_{\mu} \phi^{*} = \frac{\partial \phi^{*}}{\partial x^{\mu}} and \partial^{\mu} \phi = \frac{\partial \phi}{\partial x_{\mu}}. Therefore, the expression \partial_{\mu} \phi^{*}\partial^{\mu} \phi can be rewritten as \frac{\partial \phi^{*}}{\partial x^{\mu}} \frac{\partial \phi}{\partial x_{\mu}}. This is a scalar quantity, as it does not depend on any specific coordinate system.

The reason for writing it as \partial_{\mu} \phi^{*}\partial^{\mu} \phi instead of \partial^{\mu} \phi^{*} \partial^{\mu} \phi is because of the Einstein summation convention. In this convention, repeated indices in a product are automatically summed over, unless specified otherwise. In this case, the index \mu appears twice, so it is automatically summed over. This is a common notation used in tensor calculus.

The contraction of indices in this expression is indeed Lorentz invariant, meaning it remains the same under Lorentz transformations. This is because the derivative operators \partial_{\mu} and \partial^{\mu} are covariant and contravariant vectors, respectively, and their product results in a scalar quantity.

I hope this explanation helps clarify the notation and the meaning behind it. If you have any further questions, please don't hesitate to ask.