Derivation of the Polyakov Action

Quantum systems are essentially defined by their symmetries. For example, in QFT's you expect all terms not forbidden by the symmetries of the problem to appear in the Lagrangian, with irrelevant operators suppressed by large scales, etc.

So I think your first step in this approach would be to write down the most general 2D QFT respecting the 2D Diff and internal Poincare symmetries. The Diff symmetry motivates you to introduce a dynamical metric, since you've already done a similar thing for the point particle. This doesn't quite get you to the Polyakov action, since the Polyakov action has a Weyl symmetry that the NG action doesn't. You've introduced a fake degree of freedom on the worldsheet that wasn't present in the NG action, so you need some local symmetry principle to remove the redundant degrees of freedom. I don't know of a particular way to reason that this symmetry has to be Weyl invariance, maybe someone else does.

But once you believe the theory should have a local lagrangian with Diff, Poincare and Weyl symmetries, you are basically stuck with the Polyakov action. The Polyakov action (with the Euler characteristic term) is the most general 2D action with the Diff, Poincare and Weyl symmetries and the associated field content (Polchinski p 15 ).

So the guiding principle should be the symmetries of the NG action.

I) The closest cosmetic resemblance between the Nambu-Goto action and the Polyakov action is achieved if we write them as

$$\tag{1} S_{NG}~=~ -\frac{T_0}{c} \int d^2{\rm vol} ~\det(M)^{\frac{1}{2}} , $$

and

$$\tag{2} S_{P}~=~ -\frac{T_0}{c}\int d^2{\rm vol}~ \frac{{\rm tr}(M)}{2} , $$

respectively. Here $h_{ab}$ is an auxiliary world-sheet (WS) metric of Lorentzian signature $(-,+)$, i.e. minus in the temporal WS direction;

$$\tag{3} d^2{\rm vol}~:=~\sqrt{-h}~d\tau \wedge d\sigma$$

is a diffeomorphism-invariant WS volume-form (an area actually);

$$\tag{4} M^{a}{}_{c}~:=~(h^{-1})^{ab}\gamma_{bc} $$

is a mixed tensor; and

$$\tag{5} \gamma_{ab}~:=~(X^{\ast}G)_{ab}~:=~\partial_a X^{\mu} ~\partial_b X^{\nu}~ G_{\mu\nu}(X) $$

is the induced WS metric via pull-back of the target space (TS) metric $G_{\mu\nu}$ with Lorentzian signature $(-,+, \ldots, +)$.

Note that the Nambu-Goto action (1) does actually not depend on the auxiliary WS metric $h_{ab}$ at all, while the Polyakov action (2) does.

II) As is well-known, varying the Polyakov action (2) wrt. the WS metric $h_{ab}$ leads to that the $2\times 2$ matrix

$$\tag{6} M^{a}{}_{b}~\approx~\frac{{\rm tr}(M)}{2} \delta^a_b~\propto~\delta^a_b $$

must be proportional to the $2\times 2$ unit matrix on-shell. This implies that

$$\tag{7} \det(M)^{\frac{1}{2}} ~\approx~ \frac{{\rm tr}(M)}{2},$$

so that the two actions (1) and (2) coincide on-shell, see e.g. the Wikipedia page. (Here the $\approx$ symbol means equality modulo eom.)

III) Now, let us imagine that we only know the Nambu-Goto action (1) and not the Polyakov action (2). The only diffeomorphism-invariant combinations of the matrix $M^{a}{}_{b}$ are the determinant $\det(M)$, the trace ${\rm tr}(M)$, and functions thereof.

If furthermore the TS metric $G_{\mu\nu}$ is dimensionful, and we demand that the action is linear in that dimension, this leads us to consider action terms of the form

$$\tag{8} S~=~ -\frac{T_0}{c}\int d^2{\rm vol}~ \det(M)^{\frac{p}{2}} \left(\frac{{\rm tr}(M)}{2}\right)^{1-p} , $$

where $p\in \mathbb{R}$ is a real power. Alternatively, Weyl invariance leads us to consider the action (8). Obviously, the Polyakov action (2) (corresponding to $p=0$) is not far away if we would like simple integer powers in our action.