The size of extra dimensions Supposing we have a model, like the String one, which predicts (or requires) $N$ spacetimes dimensions, for the precision let's talk only about space dimensions.


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*What is the process, the rule or the method which leads me to understand/calculate the size of these extra dimensions? 
How can I understand or prove that, taking the String example, the additional dimensions are so small? Where does the number of their magnitude come from?

*Second question: is there a way by which it has been proved that our perceived dimensions (the three spatial ones) have magnitude of centimeters, meters or whatever? 
 A: 
What is the process, the rule or the method which leads me to understand/calculate the size of these extra dimensions? How can I understand or prove that, taking the String example, the additional dimensions are so small? Where does the number of their magnitude come from?

At the present moment we have no experimental evidence that there exist the extra ( at minimum) 6 space dimensions of  string theory . This is what forces theorists to a length for the compactified dimensions, similar to the length of the proposed string:

The size of these compactified dimensions is similar to the length of a string, the Planck scale:

$1.6\times10^{-35}m$
This is a very small number, inaccessible to experiment, which sees elementary particles as points, not one dimensional strings. Some phenomonologists have proposed that some of these dimensions are large enough to be accessible to exerimental probes, the so called large extra dimensions. Experiments at LHC are looking for the proposed signatures, with no success at the moment. These proposed large extra dimensions are still smaller than a micrometer.

Second question: is there a way by which it has been proved that our perceived dimensions (the three spatial ones) have magnitude of centimeters, meters or whatever?

Our system of units is axiomatic: we define a mass , a length, and a time unit and keep it consistently within the calculations at hand. So it cannot be proven, it is assumed.
A: Extra dimensions in general have to be compact, since a four-dimensional description fits the world we perceive and measure (so far) very well. It is this compactness that sets a length scale. Usually, we assume that the four "regular" space-time dimensions are not compact, i.e. extended infinitely. In higher dimensions concepts like angular momentum, including Spin, work differently.
The size of this length scale is not set a priori, and a mechanism giving a precise prediction for the size of the compact directions would be a major breakthrough. However, a number of effects in the low-energy description of a theory with extra dimensions depend crucially on this scale - most prominently the spacing of the Kaluza-Klein mass tower, but also e.g. the scale of symmetry breaking in GUT models with extra dimensions. How available these effects are to experiments is highly model-dependent, though. For example, if one introduces brane-localized fields, the corresponding particles do not show a Kaluza-Klein tower of excitations. Therefore, any limits on "the size of extra dimensions" have to be weighted carefully with the model assumptions.
The size and other properties of the extra dimensions become a dynamical quantity if one incorporates General Relativity, which String Theory does inherently. Since these so-called moduli degrees of freedom are usually unstable, a realistic theory must incorporate some mechanism of moduli-stabilization. There are a number of mechanisms on the market, each with benefits and detriments of its own.
So, in conclusion


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*We do not observe the higher Kaluza-Klein states of any particle so far. Even with fully brane-localized matter, the graviton should have KK states and their non-obervation puts an upper limit on the size of extra dimensions that shrinks with higher energies probed. A particularly well-motivated scale is the GUT scale, where a grand unified theory might be broken in the process of compactification.

*The "regular" space-time dimensions are considered infinite. It is not fully excluded that we do live in a compact universe of immense radius, though. These theories are not very popular, since a compact universe immediately implies Lorentz violation (most notably a minimum energy for free photons) and the Lorentz symmetry of our universe is extremely well-tested!


Note that nothing I wrote depends on the actual UV completion, i.e. my statements are independent of String theory.
A: Historically this has always been the problem with the Kaluza-Klein approach. In the original Kaluza-Klein theory there was no mechanism to determine the scale of the compactified dimension, and indeed one of the criticisms of it was that the compact dimension was unstable and would naturally expand to infinity.
In the context of string theory the problem is called moduli stabilisation. This is far outside my comfort zone, but when I last read about this the leading approach was the KKLT paper. Comments I've seen on this site suggest there have been other suggestions since, but I don't think there is currently a definitive solution to the problem.
So the answer, or rather non-answer, to your question is that (a) we don't have a definitive theory and (b) the theories we have so far are so complicated as to be incomprehensible to the layman.
