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- Homework Statement
- Trying to derive the open string Hamiltonian given as ##H=\frac{1}{2}\sum_{n\in\mathbb{Z}}\alpha_{-n}\cdot\alpha_{n}(2.72)## (Becker Becker Schwartz; string theory) using the solution for the open string

##X^{\mu}(\tau,\sigma)=x^{\mu}+l^{2}_{s}p^{\mu}\tau+il_{s}\sum_{m\neq0}\frac{1}{m}\alpha^{\mu}_{m}e^{-im\tau}\cos(m\sigma)##

- Relevant Equations
- My Hamiltonian

##H=\frac{T}{2}\int_{0}^{\pi}(\dot{X}^{2}+X^{'2}). \tag{2.69} ##

And my open string solution as

##X^{\mu}(\tau,\sigma)=x^{\mu}+l^{2}_{s}p^{\mu}\tau+il_{s}\sum_{m\neq0}\frac{1}{m}\alpha^{\mu}_{m}e^{-im\tau}\cos(m\sigma)##

Where

##\dot{X}=l_{s}\sum_{m\in\mathbb{Z}}\alpha^{\mu}_{m}e^{-im\tau}\cos(m\sigma)## Is derivative with respect to ##\tau## and

##{X}^{'}=-il_{s}\sum_{m\neq0}\alpha^{\mu}_{m}e^{-im\tau}\sin(m\sigma)## with respect to ##\sigma##

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

where we can calculate our terms $$\dot{X}=l_{s}\sum_{m\in\mathbb{Z}}\alpha^{\mu}_{m}e^{-im\tau}\cos(m\sigma)$$

and

$${X}^{'}=-il_{s}\sum_{m\neq0}\alpha^{\mu}_{m}e^{-im\tau}\sin(m\sigma)$$

remembering that $$\alpha^{\mu}_{0}=l_{s}p^{\mu}$$

If I am correct, plugging our expressions into our Hamiltonian gives us

$$H=\frac{T}{2}l^{2}_{s}\sum_{m,n\in\mathbb{Z}}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\int_{0}^{\pi}\cos(m\sigma)\cos(p\sigma)d\sigma-\frac{T}{2}l^{s}_{2}\sum_{m,p\neq 0}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\int_{0}^{\pi}\sin(m\sigma)\sin(p\sigma)d\sigma$$

Evaluating our integrals gives us

$$H=\frac{T}{2}l^{2}_{s}\sum_{m,n\in\mathbb{Z}}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\frac{\pi}{2}-\frac{T}{2}l^{s}_{2}\sum_{m,p\neq 0}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\frac{\pi}{2}$$

By ***equation 2.72*** I know that I should get

$$H=\frac{1}{2}\sum_{n\in\mathbb{Z}}\alpha_{-n}\cdot\alpha_{n}(2.62)$$

The issue that I am stuck on is based on my equation that I found

$$H=\frac{T}{2}l^{2}_{s}\sum_{m,n\in\mathbb{Z}}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\frac{\pi}{2}-\frac{T}{2}l^{s}_{2}\sum_{m,p\neq 0}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\frac{\pi}{2}$$

I can use ##m=-p ##I think to get

$$ H=\frac{T}{2}l^{2}_{s}\sum_{p\in\mathbb{Z}}\alpha_{-p}\cdot\alpha_{p}\frac{\pi}{2}-\frac{T}{2}l^{s}_{2}\sum_{p\neq 0}\alpha_{-p}\cdot\alpha_{p}\frac{\pi}{2}$$

but I am not sure how to get ***equation 2.72*** from here. In addition if I write out my sums the only term that survives is the m=0 terms I am not sure what went wrong here whether it was my mistake in doing m=-p or evaluating my integrals incorrect which I don't think is it the case.

Also the preview section is not working so I’m not sure how my equations looked or not unfortunately. I’ve tried different browsers and my phone to try and use the preview function but it didn’t work

Considering the open string we have

**$$X^{\mu}(\tau,\sigma)=x^{\mu}+l^{2}_{s}p^{\mu}\tau+il_{s}\sum_{m\neq0}\frac{1}{m}\alpha^{\mu}_{m}e^{-im\tau}\cos(m\sigma)$$**where we can calculate our terms $$\dot{X}=l_{s}\sum_{m\in\mathbb{Z}}\alpha^{\mu}_{m}e^{-im\tau}\cos(m\sigma)$$

and

$${X}^{'}=-il_{s}\sum_{m\neq0}\alpha^{\mu}_{m}e^{-im\tau}\sin(m\sigma)$$

remembering that $$\alpha^{\mu}_{0}=l_{s}p^{\mu}$$

If I am correct, plugging our expressions into our Hamiltonian gives us

$$H=\frac{T}{2}l^{2}_{s}\sum_{m,n\in\mathbb{Z}}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\int_{0}^{\pi}\cos(m\sigma)\cos(p\sigma)d\sigma-\frac{T}{2}l^{s}_{2}\sum_{m,p\neq 0}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\int_{0}^{\pi}\sin(m\sigma)\sin(p\sigma)d\sigma$$

Evaluating our integrals gives us

$$H=\frac{T}{2}l^{2}_{s}\sum_{m,n\in\mathbb{Z}}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\frac{\pi}{2}-\frac{T}{2}l^{s}_{2}\sum_{m,p\neq 0}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\frac{\pi}{2}$$

By ***equation 2.72*** I know that I should get

$$H=\frac{1}{2}\sum_{n\in\mathbb{Z}}\alpha_{-n}\cdot\alpha_{n}(2.62)$$

The issue that I am stuck on is based on my equation that I found

$$H=\frac{T}{2}l^{2}_{s}\sum_{m,n\in\mathbb{Z}}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\frac{\pi}{2}-\frac{T}{2}l^{s}_{2}\sum_{m,p\neq 0}\alpha_{m}\cdot\alpha_{p}e^{-i(m+p)\tau}\frac{\pi}{2}$$

I can use ##m=-p ##I think to get

$$ H=\frac{T}{2}l^{2}_{s}\sum_{p\in\mathbb{Z}}\alpha_{-p}\cdot\alpha_{p}\frac{\pi}{2}-\frac{T}{2}l^{s}_{2}\sum_{p\neq 0}\alpha_{-p}\cdot\alpha_{p}\frac{\pi}{2}$$

but I am not sure how to get ***equation 2.72*** from here. In addition if I write out my sums the only term that survives is the m=0 terms I am not sure what went wrong here whether it was my mistake in doing m=-p or evaluating my integrals incorrect which I don't think is it the case.

Also the preview section is not working so I’m not sure how my equations looked or not unfortunately. I’ve tried different browsers and my phone to try and use the preview function but it didn’t work

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