Confused by raising and lowering indices

Question

I lack in understanding of some basic idea regarding 4-vectors and index raising and lowering. From what I understand that: $$ \partial^\mu = \eta^{\mu\nu}\partial_\mu$$ So then, is the following correct? $$ \partial_\mu\psi\partial^\mu\psi^* - \partial_\mu\psi^*\partial^\mu\psi =\partial_\mu\psi\partial^\mu\psi^* - \eta^{\mu\mu}\partial^\mu\psi^*\eta_{\mu\mu}\partial_\mu\psi = \partial_\mu\psi\partial^\mu\psi^* - \partial_\mu\psi\partial^\mu\psi^* = 0$$

EDIT: The line above is apparently wrong. What I think I meant was: $$ \partial_\mu\psi\partial^\mu\psi^* - \partial_\mu\psi^*\partial^\mu\psi =\partial_\mu\psi\partial^\mu\psi^* - \eta_{\mu\lambda}\partial^\mu\psi^*\eta^{\mu\lambda}\partial_\mu\psi = \partial_\mu\psi\partial^\mu\psi^* - \partial_\mu\psi\partial^\mu\psi^* = 0$$ where $*$ is just a complex conjugate.

And if this is correct, is this then true:

$$\eta^{\mu\lambda}\eta_{\mu\lambda} = \delta $$

Answer 1

There are some index issues with your expressions as mentioned by others. The correct way to raise/lower an index is by $$ \partial^\mu = \eta^{\mu\nu}\partial_\nu .$$

Notice how the free index $\mu$ appears in the up position exactly once on each side of the equal sign, and the dummy index $\nu$ is repeated in the up and down position on the same side. The dummy index $\nu$ is summed over, so the expression should be read as: $$ \partial^\mu = \sum_{\nu=0}^3 \eta^{\mu\nu}\partial_\nu. $$

The expression tells you how to get each of the 4 components of $\partial^\mu$.

When we apply that to the relevant part of your second expression we get:

$$ \left(\partial_\mu \psi^*\right) \, \left( \partial^\mu \psi \right) = \left( \eta_{\mu\nu} \partial^\nu\psi^*\right) \, \left( \eta^{\mu\lambda} \partial_\lambda \psi \right). $$

To get the indices correct we make sure that each term in parentheses maintains a single $\mu$ index in the correct position, and that in a given multiplicative statement no index is used more than once in a given up/down position. These means we need a different dummy index for each summation, otherwise we might not know which terms get summed with which.

The final statement is then: \begin{align} \partial_\mu \psi \partial^\mu \psi^* - \partial_\mu \psi^* \partial^\mu \psi &= \partial_\mu \psi \partial^\mu \psi^* - \eta_{\mu\nu} \partial^\nu\psi^* \eta^{\mu\lambda} \partial_\lambda \psi \\ &= \partial_\mu \psi \partial^\mu \psi^* - \left( \eta_{\mu\nu} \eta^{\mu\lambda} \right) \partial^\nu\psi^* \partial_\lambda \psi \\ &= \partial_\mu \psi \partial^\mu \psi^* - \partial_\nu \psi \partial^\nu \psi^* \\ &= \partial_\mu \psi \partial^\mu \psi^* - \partial_\mu \psi \partial^\mu \psi^* \\ &= 0 \end{align}

We used the fact that $$\eta_{\mu\nu} \eta^{\mu\lambda} = {\delta_\nu}^\lambda,$$ and that dummy indices can be renamed arbitrarily ($\nu\rightarrow\mu$).

Finally as a commenter stated, if you fully contract a metric with itself you get: $$\eta_{\mu\nu} \eta^{\mu\nu} = {\delta_\nu}^\nu = 4$$ in a 4-D space.

Answer 2

No. You cannot have more than one of the same index raised/lowered position. If you want to apply the metric to second set of terms, the second line should read \begin{equation} ....=\partial_\mu \psi \partial^\mu \psi^* - \eta_{\mu\nu} \partial^\nu \psi^* \eta^{\mu\lambda}\partial_\lambda\psi = .... \end{equation}