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I am reading GSW volume 1 and it says in chapter 2 that the Hamiltonian for closed strings turn out to be $$ H = \frac{1}{2} \sum_{n=-\infty}^{\infty} (\alpha_{-n}.\alpha_{n} + \tilde{\alpha}_{-n}\cot \tilde{\alpha}_{n})\;.$$ I tried to calculate it and I got twice of this i.e. $$ H = \sum_{n=-\infty}^{\infty} (\alpha_{-n}.\alpha_{n} + \tilde{\alpha}_{-n}.\tilde{\alpha}_{n})\;.$$ I also calculated it in terms of virasoro modes and got $H = 2 (L_0 + \tilde{L_0}) $ as opposed to $H = (L_0 + \tilde{L_0}) $ mentioned in GSW and here I just used the definitions of $ H $, $L_0$ and $\tilde{L_0} $ without using $\alpha$ anywhere. I have the images of my calculation here. Could somebody please see it and tell me what mistake I might be doing?

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closed as off-topic by user36790, ACuriousMind, Sebastian Riese, Daniel Griscom, FraSchelle Feb 4 '16 at 17:51

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    $\begingroup$ Hi, Himanshu! Welcome to Physics Stack Exchange!! Please note that this is not a homework-helping site. For more guidance, please check our meta site. $\endgroup$ – user36790 Feb 2 '16 at 11:43
  • $\begingroup$ Hi Himanshu Khanchandani. Linking to private clouds, dropbox, etc, is for various reasons not acceptable on SE, cf. this meta post. $\endgroup$ – Qmechanic Oct 6 at 10:44