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Quick question, lets say I have the following term

\begin{equation} a_{\mu}b^{\mu}c_{\nu}d^{\nu}\tag{1} \end{equation}

I have repeated indices over $a$ and $b$ so their components are summed over, and I have repeated indices over $c$ and $d$ so their components are summed over. However, since the indices are repeated, I may relabel them:

\begin{equation} a_{\mu}b^{\mu}c_{\mu}d^{\mu} \tag{2} \end{equation}

This suggests that I can now sum the components of $b$ and $c$ and likewise with $a$ and $d$, etc. however I don't think this is valid and I just wanted to confirm. More specifically, as a general rule you should keep repeated indices unique to other repeated indices so confusions such as this don't occur. Can anyone confirm?

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  • $\begingroup$ Think about this analogy: in the double integral $\iint xy\ dx\ dy$, $x$ and $y$ are dummy variables, so you may relabel them. Does this mean you can change $y$ to $x$? $\endgroup$
    – Javier
    Feb 13, 2017 at 8:48

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Yes, your conclusion is correct. The initial expression evaluates to just $(a \cdot b) (c \cdot d)$ while the rewriting of it to $a_{\mu}b^{\mu} c_{\mu} d^{\mu} $ is then meaningless (In Einstein summation convention you have an upper and lower repeated index and indices to be summed over come in pairs).

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  • $\begingroup$ Hi thank you very much! I just edited my question slightly so take note to edit your answer accordingly. (I changed the final four vector c to d to make the statement more general) $\endgroup$
    – Decebalus
    Feb 13, 2017 at 7:43

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