Polyakov action: difference induced metric and dynamical metric

Question

The Polyakov action is given by: $$ S_p ~=~ -\frac{T}{2}\int d^2\sigma \sqrt{-g}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu} ~=~ -\frac{T}{2}\int d^2\sigma \sqrt{-g}g^{\alpha\beta}\gamma_{\alpha\beta}, $$ where $\gamma_{\alpha\beta}$ is called the induced metric and $g_{\alpha\beta}$ the dynamical metric on the world sheet. I have difficulties understanding the differences between these two metrics. I know that the latter is introduced in order to be able to remove the square root in the Nambu-Goto action, but I don't know what it means. The space in which the string propagates has just the Minkowski metric $\eta_{\mu\nu}$, if I am not mistaken. Furthermore, I think that the induced metric is derived by demanding

$ds^2$(whole space) = $\eta_{\mu\nu}dx^{\mu}dx^{\nu}$ = $ds^2$(world sheet) = $\gamma_{\alpha\beta}d\sigma^{\alpha}d\sigma^{\beta}$

Is this correct? I am really confused by all these different metrics.

Answer 1

There are two manifolds that are involved in string propagation.

1. The spacetime in which the string propagates.

2. The worldsheet of the string itself.

The fields $X^\mu$ are embedding coordinates of the worldsheet in the spacetime manifold. This means that for each point $(\sigma^1, \sigma^1)$ on the worldsheet, $X^\mu(\sigma^1, \sigma^2)$ gives the coordinates of that point in the spacetime manifold.

In the case you are considering, the spacetime is taken to be Minkowski, so the metric is $\eta_{\mu\nu}$. Now we could ask

"Given that the worldsheet is a two dimensional embedded submanifold of Minkowski space, is there some way that this manifold inherits its metric from the metric on the ambient spacetime?"

This question is analogous to

"Given that the sphere $S^2$ is some two-dimensional embedded submanifold of Euclidean space $\mathbb R^3$, is there some natural sense in which it inherits its metric from $\mathbb R^3$?

The answer to both of these question is yes, and the metric on the submanifold that does this is precisely the induced metric. The formula expression the induced metric for a two-dimensional submanifold of some ambient manifold with metric $g_{\mu\nu}$ (not necessarily flat) in terms of embedding coordinates is $$ \gamma_{ab}(\sigma) = g_{\mu\nu}(X(\sigma))\partial_aX^\mu(\sigma)\partial_b X^\nu(\sigma), \qquad \sigma = (\sigma^2, \sigma^2) $$ You are right about the derivation of the induced metric, it comes from demanding that the distance measured between points on the embedded submanifold is calculated to be the same number whether you use the ambient metric, or the induced metric. To see that the above expression for the induced metric does this, simply note that the infinitesimal distance between any two points on the embedded submanifold can be written in terms of the ambient metric and the embedding coordinates as \begin{align} g_{\mu\nu}(X(\sigma))d(X^\mu(\sigma))d(X^\nu(\sigma)) &= g_{\mu\nu}(X(\sigma))\partial_a X^\mu(\sigma)\partial_bX^\nu(\sigma)d\sigma^ad\sigma^b \\ &= \gamma_{a b}(\sigma)d\sigma^ad\sigma^b \end{align} To get some intuition for all of this, recall that expression for embedding coordinates of $S^2$ in $\mathbb R^3$ is \begin{align} X(\theta, \phi) &= \sin\theta\cos\phi\\ Y(\theta, \phi) &= \sin\theta\sin\phi\\ Z(\theta, \phi) &= \cos\theta \end{align} and using these embeddings you should be able to show that the metric on the sphere is simply $$ \gamma_{ab}(\theta, \phi) = \mathrm{diag}(1, \sin^2\theta) $$

Let me know if that's unclear or if you need more detail!

Answer 2

I would like to add that the geometric picture and relationship between Nambu-Goto and Polyakov actions are only hints and heuristics. Specifically, string scattering amplitudes are computed in a Lorentzian space, but the worldsheets are Euclidean. One way to see it is that topology changes don't respect causality, so branching worldsheets are problematic for a Euclidean worldsheet. Would be great if a string theorist could elaborate.

Answer 3

That's right, there is the following equation:

$$X^{\mu}=Z^{\mu}(\sigma)) (dX^{\mu}=\partial_{a}Z^{\mu}(\sigma)d\sigma^{a})$$

for 2d surface embedded into, let say, a flat finite-dimensional target space inside the Polyakov and Nambu-Goto action. The main/only "reason" why people (mostly QFTs theorists who would like to call themselves and be called by others as "string theorists") use the Polyakov action (for mostly vacuous calculations) is the fact that it has the quadratic form (so what) and therefore they have a huge hope that it is much more easier than the Nambu-Goto action to be "quantized" (whatever that word means to you because they "could see" the above mentioned string coordinates Z^{\mu}(\sigma) as "scalar fields" (the so-called "BOSONIC string coordinates") over 2d surface with coordinates \sigma^{a}).

If quantum (particle) mechanics is 2-tier quantum theory:

http://www.springer.com/philosophy/book/978-0-7923-3565-8

i.e. 1d QFT then quantum (super)string kinematics has to be 3-tier quantum theory:

http://www.math.harvard.edu/~lurie/papers/cobordism.pdf

which could not only be 2d (super)conformal QFT (it has to be much more richer than that). By the way to be said, it "could be shown" that the above mentioned induced metric and world-sheet metric after "the standard quantization procedure" are related in "the same way" as in the classical (non-quantum) case only if the target flat space is 26-dimensional.

That is why:

Albert Einstein: "We can't solve problems by using the same kind of thinking we used when we created them.",

dear "lionelbrits", a string theorist did not elaborate about the above mentioned.

Thus, the above mentioned actions (Polyakov's and Nambu-Goto's) could not be a starting point for quantum strings of any sort but it is up to you gays to do something about that bad thing in the current mathematization attempts for fundamental laws of nature.