Recent content by Overtoad123

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    How can negative integers be used in deriving the Hamiltonian for open strings?

    I need negative integers in order to derive my Hamiltonian for the open string which is given as $$H=\frac{1}{2}\sum_{n\in\mathbb{Z}}\alpha_{-n}\cdot\alpha_{n}(2.62)$$ Altough TSny pointed out that since my sums in...
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    How can negative integers be used in deriving the Hamiltonian for open strings?

    So in other words I only need $$\int_{0}^{\pi}\cos(m\sigma)\cos(p\sigma)d\sigma =\frac{\pi}{2} \delta_{m,p}$$ since I have a real Fourier series ?? But if i don’t incorporate negative integers through integral then how would I go about introducing negative integers in my sum ?? Sorry if I’m a...
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    How can negative integers be used in deriving the Hamiltonian for open strings?

    So I get the following now...
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    How can negative integers be used in deriving the Hamiltonian for open strings?

    This is really insightful as I was not aware you can express these orthogonality relationships in such way. I have normally been taught that $$\int_{0}^{\pi}\cos(m\sigma)\cos(p\sigma)d\sigma =\frac{\pi}{2} \delta_{m,p}$$ but considering that cosine is even then it’s also fair to to include the...
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    How can negative integers be used in deriving the Hamiltonian for open strings?

    thank you ! :) I’ve updated my post and it looks readable now !
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    How can negative integers be used in deriving the Hamiltonian for open strings?

    On ***page 38*** of Becker Becker Schwarz, we're given ***equation 2.69*** which is the Hamiltonian for a string given as $$H=\frac{T}{2}\int_{0}^{\pi}(\dot{X}^{2}+X^{'2})$$ Considering the open string we have...