# Why do people say that the string length is on the order of the Planck length?

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 $l_\text{string} \sim l_\text{Planck}$?

• Related 315133. – Cosmas Zachos Aug 1 '17 at 22:53
• Yeah, I saw that question, but I thought my question was sufficiently more specific that it merited a new question – tparker Aug 2 '17 at 0:23
• 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... – Cosmas Zachos Aug 2 '17 at 0:34
• @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). – tparker Aug 2 '17 at 0:38
• 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. – user163104 Aug 2 '17 at 1:38