I would like to calculate $\Box\phi$
whereby $\phi = exp(ip_{\mu}x^{\mu})$ and $\Box = \partial_{\mu}\partial^{\mu}$ and whereby $\partial_{\mu} = \frac{\partial}{\partial x^{\mu}} $ and $\partial^{\mu} = \frac{\partial}{\partial x_{\mu}}$.
However, I am stuck on how to do this since I was always told that we can never have more than two (a covariant and contravariant component) of the same indices in the same expression but the expressions we were given will mean I will get multiple of the same index in one term.
How would I go about solving this problem and avoiding that. I tried doing it so that I define $\Box = \partial_{\nu}\partial^{\nu}$ but it jut does not give me the correct answer, if someone could walk me through this I would greatly appreciate it.
As far as I understand you operate in four-dimensional flat Minkowski space. In this case all indices lifting is done by means of the metric tensor $g^{ik} =diag (1 -1 -1 -1)$. For example, $\partial^i=g^{ik}\partial_k$. Descent of indices is done via $g_{ik} =diag (1 -1 -1 -1)$; $g_{ik}g^{kl}=\delta_i^l =diag (1 1 1 1)$. Everywhere the summation over repeated indices is supposed.
Now you can easily take the derivatives:
$\partial_{k}x^i=\delta_k^i$
$\partial_{k}\exp(ip_{m}x^{m})=\exp(ip_{m}x^{m})ip_l\delta_k^l=$$\exp(ip_{m}x^{m})ip_k$
$\Box=g^{ik}\partial_{i}\partial_{k}\exp(ip_{m}x^{m})=g^{ik}\partial_{i}(\exp(ip_{m}x^{m})ip_k)=(ip_k)(ip_i)g^{ik}\exp(ip_{m}x^{m})$$=-p_kp^k\exp(ip_{m}x^{m})$, etc.
$p_kp^k =p_0p^0+p_1p^1+p_2p^2+p_3p^3=g^{ik}p_ip_k=p_0p_0-p_1p_1-p_2p_2-p_3p_3$
$g^{ik}\partial_{i}\partial_{k}=\partial_0\partial_0-\partial_1\partial_1-\partial_2\partial_2-\partial_3\partial_3$