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In string theory, it is assumed that a string is about the size of a Planck length, $$\ell_{string} \sim \ell_{Pl} \simeq 10^{-35}\,\text m.$$ Why that length? Why not for example a hundred times larger?

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  • $\begingroup$ If they were bigger we would see their stringy structure... which we don't... $\endgroup$
    – hft
    Mar 22, 2022 at 18:36

2 Answers 2

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In string theory, the string length is not taken to be the Planck length,

$$\ell_P = \sqrt{\frac{\hbar G}{c^3}}.$$

Rather, we expect strings, should they exist, to be approximately around this scale but from my knowledge of the literature and many introductory texts, there is no claim it is precisely $\ell_P$.

The energy scale of the quantum theory itself is believed to also be around the Planck mass, $M_P$. The Einstein-Hilbert action can be expanded in powers of $M_P^{-1}$ and this is believed to be the relevant coupling in the quantum theory.

The reason quantum gravity does not appear to affect everyday life is that the scale the LHC probes, that is, the electroweak scale $M_{EW}$ has $M_{EW}/M_P \sim 10^{-15}$.

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  • $\begingroup$ @hamza The insertion of a comma is really debatable since different journal style guides for example would not all agree a comma should be placed there. The "the" in "the quantum theory" at the end is important, because I'm talking about the specific quantum theory you pass to after quantising the Einstein-Hilbert action. Your only valid edit was "every day" to "everyday." $\endgroup$
    – JamalS
    Mar 22, 2022 at 18:40
  • $\begingroup$ I actually only initially tried to edit to "everyday", but apparently edits have to be at least 6 characters long for some reason. In any case, you're right about "the", but I'm pretty sure that the comma was actually necessary when combining two independent clauses with a coordinating conjunction. You might be confusing this with the use of a subordinating conjunction instead, which "and" is not. $\endgroup$
    – hamza
    Mar 23, 2022 at 19:16
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@JamalS 's point that the association is a purely dimensional argument cannot get you more precisely than a factor of 100 to $\ell_{Pl}$. The "freshman physics" standard Nambu 1970 string picture, repurposed to gravity by Schwarz-Scherk, is that the string's Regge slope is $\alpha'\sim \ell_{Pl}^2 \sim \frac{1}{2\pi T}$.

The $10^{40}$ton string tension T is related to the Regge slope of the gravity-applied Regge trajectory $$ \frac {dJ}{dE^2}=\alpha' $$ if you consider a freely spinning string of length L, since $E\sim TL$, and $J\sim TL^2\sim \alpha' E^2$. While L can grow indefinitely with E, for particle spins, such as J ~ 2, ..., we need $L\sim \ell_{Pl}$.

(You might flesh the argument out in more details in a paper of mine & collaborators, 1986; or String Theory and M-Theory, A Modern Introduction, by Becker, Becker & Schwarz, Cambridge UP, 2007, Ch. 2)

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