Why is $\partial_{\mu}\partial^{\mu}\phi=\partial^{\mu}\partial_{\mu}\phi$?

Question

I think I could probably find the answer if I knew what $\partial ^{\mu}$ actually was, but unfortunately our professor assumed we'd seen this before, and so didn't explain what it actually is. I know this $\partial_{\mu}\phi = \frac{\partial \phi}{\partial \mu}$, but I'd really appreciate it if someone could give a simple explanation as to what $\partial ^{\mu}$ is and why you can swap the order, or point me in the direction of an explanation.

Answer 1

While $\partial_{\mu} = \frac{\partial}{\partial x^{\mu}}$, $\partial^{\mu} = g^{\mu\nu}\partial_{\nu}$. So $$ \partial^{\mu}\partial_{\mu} \phi = g^{\mu\nu}\partial_{\nu}\partial_{\mu} \phi $$ But, $$ \partial_{\mu}\partial^{\mu} \phi = \partial_{\mu}(g^{\mu\nu}\partial_{\nu}\phi) = (\partial_{\mu}g^{\mu\nu}) \partial_{\nu}\phi + g^{\mu\nu}\partial_{\mu} \partial_{\nu}\phi \neq \partial^{\mu}\partial_{\mu} \phi $$ unless $g^{\mu\nu}$is a constant, for example $g^{\mu\nu}= \eta^{\mu\nu} = \text{diag}\,(1,1,1,-1)$ (so that $\partial_{\mu}g^{\mu\nu} = 0$). Probably this is the metric that used in your case (dispite the convention + and - here). I don't know if it has a name. Usually when $g^{\mu\nu}=\eta^{\mu\nu}$ its just denote by "square" $\square \, \phi = \partial^{\mu}\partial_{\mu}\phi$. This expression usually appear in scalar field theory. Look Ryder for example.