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

Question

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.

Answer 1

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?