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As I understand it, string theory has only one fundamental dimensionful quantity - the string tension $T = 1/2 \pi \alpha'$, where $\alpha'$ is the Regge slope (in units where $c$ and $\hbar$ equal $1$ but Newton's constant $G$ remains dimensionful). From $T$, we can define a "string length" $L_\text{string} := \sqrt{\alpha'}$ (let's put aside the question of whether this is really the "length" of the string in the colloquial sense, and just agree that it's the natural length scale for describing elementary strings).

Of course, the natural length scale for describing quantum gravitational physics is the Planck length $l_\text{Planck} = \sqrt{G}$, which is essentially equivalent to Newton's constant. Since the only fundamental length scale in string theory is $l_\text{string}$, we have $l_\text{Planck} = k\ l_\text{string}$, where $k$ is a dimensionless constant that could in principle be computed directly from the theory (using the fact that the emergent value of Newton's constant $G$ is in one-to-one correspondence with the expectation value of the dilaton field).

If I understand the history, people originally thought that there would be a unique theory of string theory with no dimensionless free parameters (discrete or continuous). If this were the case, than $k$ would simply be a mathematical constant, which would presumably be $o(1)$ since there are no parameters on which it could depend. In this case the string and Planck lengths would indeed be essentially guaranteed to be of the same order of magnitude.

On the other hand, in practice perturbative string theory is usually formulated with one dimensionless free parameter - the string coupling constant $g$ - which relates the Planck and string lengths. For example, for type IIb string theory on $\text{AdS}_5 \times \text{S}_5$, we have $L_\text{Planck} = 8^{1/4} \pi^{3/8} g^{1/4} L_\text{string}$ (under the appropriate normalization conventions), as can be seen by dividing equations (1.9) and (1.10) in these notes. In this case, the ratio of the Planck and string lengths is a completely free parameter, and there is no reason that it must be $o(1)$. Nevertheless, people often claim that the string length is presumably on the order of magnitude of the Planck length. Why is this? Is it just the usual philosophical preference for "natural" theories (in the technical sense of the word), or is there an actual mathematical motivation for the claim that $l_\text{string} \sim l_\text{Planck}$?

One possible resolution that occurs to me - although I am far out of my comfort zone here - is the possibility that the value of $g$ is not actually a free parameter, but is determined by the choice of Calabi-Yau compactification of the extra dimensions in the string theory landscape. If this is the case (which I'm not sure if it is), then by assuming every compactification is a priori equally likely, we could in principle calculate a "probability distribution" for the value of $g$ given by the "density of states" of $g$-values over the $o \left (10^{500} \right)$ different compactifications. (Strictly speaking, this a probability mass function, but a discrete random variable that ranges over $10^{500}$ possible values is effectively continuous in practice.) I assume that in practice, we can't even begin to actually calculate this probability distribution, but do we have any reason to believe that it is peaked around $g \sim 1$, which would lead to $l_\text{string} \sim l_\text{Planck}$?

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    $\begingroup$ Related 315133. $\endgroup$ Aug 1, 2017 at 22:53
  • $\begingroup$ Yeah, I saw that question, but I thought my question was sufficiently more specific that it merited a new question $\endgroup$
    – tparker
    Aug 2, 2017 at 0:23
  • $\begingroup$ I suspect you might well be overthinking it. The statement is that there is no other basic scale around; not that you always get a length within x orders of magnitude of the Planck length. To the extent the mooring is on gravitational physics, the heuristics cannot but yield that rough scale. But since this is all academic with absolutely no phenomenological input, I don't see any firm precise statements on identity made anywhere... $\endgroup$ Aug 2, 2017 at 0:34
  • $\begingroup$ @CosmasZachos Fair enough. I just thought it was interesting (and counterintuitive) that we can very simply and explicitly construct a string theory in which the string scale is arbitrarily far away from the Planck scale (in either direction). $\endgroup$
    – tparker
    Aug 2, 2017 at 0:38
  • $\begingroup$ Which people? (As in your title). I have read a lot of accounts that are basically guesses (Forgive me, but although I know the problems string theory has, the actual nitty grittity details seem to depend on who you read). I am certainly one of the least qualified here to judge, but I do know a bad ST popular + mathy account from a good one at this stage). FWIW, I think your question is unanswerable, as it is unverifiable, AFAIK. $\endgroup$
    – user163104
    Aug 2, 2017 at 1:38

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The string lenght can not be of the order of the Planck lenght.

Just recall the starting point of perturbative string theory, the embedding of strings on a classical background. After quantization you can learn that all the target space physics is encoded in the worldsheet, the target is now the moduli space of worldsheet scalars (and fermion superparthners in the superspace formalism) and more than that, all possible quantum spacetime processes can be computed (and inferred) from "the collision" of vertex operators over worldsheets with appropriate topology. That´s the magic of perturbative string theory.

Stringy effects start to become relevant at the point ath which the "inner structure of the string is measurable" (states associated to the string spectrum can be proved from the target perspective), notice that this is the regime where the string lenght is finite but the Planck lenght is still zero. That's why there should be a scale separation in between the Planck and string lenghts, otherwise perturbative string theory can not be formulated in the way that Polchinski does in his textbook.

In simple words, there is no known way to formulate perturbative string theory (the relevant physics at the string scale) around a topology changing quantum/foamy/highly curved(planckian) background (the expected behaviour of spacetime at the Planck scale) and that can be seen as circumstantial evidence for the aforementioned scale separation.

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