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

$$ S=\int d \tau d \sigma ~\mathcal{L}=\frac{1}{4 \pi \alpha^{\prime}} \int d \tau \int_{0}^{\pi} d \sigma\left(\dot{X}^{I} \dot{X}^{I}-X^{I^{\prime}} X^{I^{\prime}}\right),\tag{12.81} $$ where $X^I$ denote transverse target-space coordinates.

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.

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It's the simplest action of a field $X^I(\sigma,\tau)$ defined on a 2d worldsheet which is manifestly Lorentz invariant in target space. Your action is a special case of the Polyakov action where both the worldsheet metric $h^{ab}(\sigma,\tau)$ and the target space metric $g_{\mu \nu}(X)$ are taken to be flat.

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  1. FWIW, it is in principle possible to systematically derive the light-cone action (12.81) from (the Hamiltonian formulation of) the Nambu-Goto (NG) string, see e.g. this related Phys.SE post.

  2. The NG action is in principle equivalent to the Polyakov action, cf. e.g. this Phys.SE post.

  3. In both cases, it takes a bit of work to consistently reduce to the transverse degrees of freedom of the light-cone formulation (12.81).

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

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

Then working in reverse to find the action. I'm not sure if it's cyclical or not but I don't like going the Hamiltonian way for the exact same reason.

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