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I'm reading this introduction to string theory by Mariana Grana and Hagen Triendl and I have understanding problems around the method of compactification in string theoretical sense. The compactification scheme works as follows (see pages 14/15):

The ten-dimensional (or say abstractly $n$-diml) spacetime is divided into an external non-compact spacetime $M_{10-d}$ and an internal compact space $M_d$, so that it reads

$$M_{10} =M_{10-d} \times M_d $$

The physically interesting case is of course $d = 6$. The compactification scale is $M_c = 1/R$ ( here $R$ the typical length scale associated with the internal space) and is considered much smaller than the string scale $M_s = 1/l_s$.

Question 1: What is precisely a 'scale' in this setting? How is it precisely defined? I not understand what it does precisely mean to say that 'compactification scale is $M_c = 1/R$' or that 'string scale $M_s = 1/l_s$'. Equivalently which meaning have lenghts $R$ and $L_s$?

Question 2: Assume we know the answer of question 1, that is we know what scales are. What is now precisely the 'compactification'? Just $M_{10} =M_{10-d} \times M_d $ with additional assumption that $M_c \ll M_s$?

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The term "scale" is thrown around a lot in physics as a way of describing the size of things. For example, the distance scale of football fields is 100 yds, human scale is 2 meters, bacteria scale is micormeters.

So, when a string theorist says the string scale is $x$, they mean that the energies necessary to probe the string nature of the universe are similar to $x$. In most of particle physics, we use units where the speed of light is set to $1$. Because of the famous relationship $E = m c^2$, this makes units of mass and energy equivalent. Slightly less obvious, we now have a relationship between energy and distances and times. In particular, distance and time have units of $1/$Energy. This means that big energies correspond to small distances etc. (If you want to play around with this relationship, try putting every day scales in gutcalc.com with the units set to "natural".)

So, when they say the string scale is $M_s = 1/l_s$, they're using this fact that energies and distances are related. The compactification scale $M_c$ tells us how much energy we need in order to see features in our universe smaller than the compactification radius (the size of the extra dimensions) $R \sim 1/M_c$. (Also, just to add, compactification refers to extra dimensions that are compact, i.e. have finite spatial extent).

To summarize, $R$ is the size of the extra dimensions, and $l_s$ is the length of the strings. Similarly, $M_c$ is the energy you need to probe the compactified extra dimensions and $M_s$ is the energy you need to probe the strings.

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  • $\begingroup$ Hi, thank you for the answer. One question: you wrote: "Similarly, $M_c$ is the energy you need to probe the compactified extra dimensions". What do you mean precisely by "energy you need to probe the compactified extra dimensions"? So assume you can generate enough energy $E_c \sim M_c$. What does it mean that you can probe the compactified extra dimensions with it? $\endgroup$ Aug 15 '20 at 17:40
  • $\begingroup$ Does it mean in the sense of a thought experiment that one assumes the existence of an abstract "detector" such that it can only "detect" objects / effects of scale length $l_s$ consuming energy of the order of magnitude $\sim M_c$? $\endgroup$ Aug 15 '20 at 17:46
  • $\begingroup$ What I mean by "you need $M_c$ energy to probe the extra dimension" is the following. Suppose $M_c$ is much larger than the energy of every day objects, and that $R = 1/M_c$ is much smaller than the size of every day objects. Additionally, suppose we have a mathematical theory for how every day objects should behave, and it does a very good job making predictions for distances larger than $R$ and energies smaller than $M_c$. If I now build, say, a particle collider that smashes stuff together with energy greater than $M_c$, I will start to observe deviations from my theoretical predictions. $\endgroup$
    – David
    Aug 15 '20 at 18:21
  • $\begingroup$ Similarly, if I built a small enough instrument that it was roughly the same size or smaller than the extra dimensions, it would start to observe deviations from our low-energy large-distance theory. $\endgroup$
    – David
    Aug 15 '20 at 18:22
  • $\begingroup$ I understand. And about question 2. Can the compactification principle be though exactly as a product decomposition of a manifold $\mathcal{M}= \mathcal{M}_s \times \mathcal{M}_c$ with compact part $\mathcal{M}_c$ and nocompact $\mathcal{M}_s$ with additional requirement $M_c \ll M_s$ as definition? $\endgroup$ Aug 15 '20 at 18:36

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