# Motivation behind action when deriving ''Strings as Harmonic oscilators" in Zwiebach's book on String theory

+ 1 like - 0 dislike
319 views

Page 248 gives us this action and he simply says that we will assume it correct.

$$S=\int d \tau d \sigma \mathcal{L}=\frac{1}{4 \pi \alpha^{\prime}} \int d \tau \int_{0}^{\pi} d \sigma\left(\dot{X}^{I} \dot{X}^{I}-X^{I^{\prime}} X^{I^{\prime}}\right)$$

Besides giving us the right answer at the end, what is the motivation for this action, how was it thought up? It seems like a modified Nambu-Goto action.

+ 0 like - 0 dislike

What I was thinking was instead of going from the Nambu-Goto action, doing it from the Polyakov.

https://en.wikipedia.org/wiki/Polyakov_action

Using diffeomorphisms and Weyl transformation, with a Minkowskian target space, one can make a physically insignificant transformation thus writing the action in the conformal gauge which is basically what we have here.

Then put light cone coordinates and do it in natural units.

Another way I thought of doing it is directly doing this substitution.

$$\delta S=\int_{\tau_{i}}^{\tau_{f}} d \tau\left[\delta X^{\mu} \mathcal{P}_{\mu}^{\sigma}\right]_{0}^{\sigma_{1}}-\int_{\tau_{i}}^{\tau_{f}} d \tau \int_{0}^{\sigma_{1}} d \sigma \delta X^{\mu}\left(\frac{\partial \mathcal{P}_{\mu}^{\tau}}{\partial \tau}+\frac{\partial \mathcal{P}_{\mu}^{\sigma}}{\partial \sigma}\right)$$

$$\frac{\partial \mathcal{L}}{\partial \dot{X}^{I}}=\frac{1}{2 \pi \alpha^{\prime}} \dot{X}^{I}=\mathcal{P}^{\tau I}$$

That's it I think?

answered Jun 10, 2019 by (5 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysic$\varnothing$OverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.