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I am currently studying David Tong's notes on String theory and there’s a step taken in writing out the worldsheet constraint in lightcone coordinates $\sigma^{\pm}$ for the closed string that I’m not sure about. We have the constraint eq 1.38 written out on page 26 as $$(\partial_{-}X)^{2}=\frac{\alpha^{‘}}{2}\sum_{m,p}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\sigma^{-}}=\\\frac{\alpha^{‘}}{2}\sum_{m,n}\alpha_{m}\cdot\alpha_{n-m}e^{-in\sigma^{-}}.$$

It looks like my $p$ index was changed to $p=n-m$ but I’m unsure how this action is valid considering I have an exponent hanging around. Also Wouldn’t this change in $p$ change my summation? How am I able to have a summation for $n$ after this change in $p$. I’m not sure if I’m overthinking this change but I can’t seem to convince myself why this change in $p$ would be valid.

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It is just a dummy variable change, from $p$ to $n:=m+p$. Since $m$ and $p$ run through all integers, $n$ also runs through all integers. $\newcommand{\ex}[1]{\mathrm{e}^{#1}}$ Stripping off the physics we have $$ \sum_{m=-\infty}^\infty\sum_{p=-\infty}^\infty a_m\ b_p \ c_{m+p} = \sum_{m=-\infty}^\infty\sum_{n-m=-\infty}^\infty a_m\ b_{n-m} \ c_{n} = \sum_{m=-\infty}^\infty\sum_{n=-\infty}^\infty a_m\ b_{n-m} \ c_{n}.$$

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  • $\begingroup$ Thank you for clarifying this!!! This opened my eyes. I kept ignoring the simple fact both p and m run through all the integers therefore n does as well. Thank you again ! $\endgroup$ Commented Jul 22, 2022 at 8:19

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