Are there differences in notation for the d'Alembert operator?

Question

On Wikipedia the d'Alembert operator is defined as $$\square = \partial ^\alpha \partial_\alpha = \frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2 $$ However, my professor uses the notation: $$ \square = \partial _\alpha \partial^\alpha$$ with $\partial_\alpha = \frac{\partial}{\partial x^\alpha}$ and $\partial^\alpha = \frac{\partial}{\partial x_\alpha}$
Is there a difference or are both notations equivalent

Answer 1

As it was pointed out in the comments, both expressions are the same: Note that $$\partial^\alpha = \eta^{\alpha\beta}\, \partial_{\beta}$$ and hence

$$\square = \partial_\alpha\partial^\alpha = \partial_\alpha \eta^{\alpha\beta}\partial_\beta = \partial^\beta \partial_\beta \quad .$$

We have used the Minkowski space metric $(+1,-1,-1,-1)$.

Answer 2

If operators commute, namely if commutator : $$ [ \hat {x},\hat {y}] = \hat x \hat y - \hat y \hat x = 0$$, then there's no difference in order of operators applied, as in this case, because we know Clairaut's theorem : $$ {\frac {\partial }{\partial x_{i}}}\left({\frac {\partial f}{\partial x_{j}}}\right)\ =\ {\frac {\partial }{\partial x_{j}}}\left({\frac {\partial f}{\partial x_{i}}}\right) $$

So these two expressions : $$ \square = \partial _\alpha \partial^\alpha = \partial ^\alpha \partial_\alpha $$ are equivalent.