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According to Brian Greene it is possible to calculate the physical constants from the shape of the extra dimensions. Is it possible to do the inverse, so predict the shape of these dimensions from the physical constants we experimentally measured (assuming they really exist)?

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When Brian Greene talks about the shape of the extra dimensions he is using a simple word for some exceedingly complicated mathematics. For example suppose you are trying to compactify just two dimensions - call them $x$ and $y$ for convenience. Compactifying the dimensions means forming them into a loop, but starting from a flat sheet you could loop the two dimensions to make a sphere, a torus or some bizarre object like a Klein bottle:

Compactification

In string theory you are compactifying six dimensions, so you can have a six dimensional sphere, a six dimensional torus or a large (probably infinite) number of other six dimensional shapes. All these different shapes give rise to different 4D theories of physics, and the sizes of the different shapes, e.g. the two axes of the 2D torus, also give different physics.

Historically the need to preserve a property called supersymmetry has restricted the type of shapes to a type called Calabi-Yau manifolds, but it isn't known if the number of these is finite or infinite, and in any case low energy supersymmetry seems to be going out of fashion which means we need to consider all manifolds not just Calabi-Yau manifolds.

And even when you've selected a manifold calculating the corresponding physical theory is a long and complicated task.

The point of all this is that while Brian Greene is correct to say we can calculate physical constants from the shape of the extra dimensions, the numbers we get are the end point of a long and complicated process. The complexity means there is no (known!) way to start with the constants and work backwards to the corresponding compactification.

In fact (at the time of writing) there is no compactification known that exactly reproduces the Standard Model, though I believe several people have got close.

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According to Brian Greene it is possible to calculate the physical constants from the shape of the extra dimensions.

This isn't scientific fact I'm afraid. It's perhaps presented as such, but there's absolutely no evidence for string theory, and hasn't been for fifty years.

Is it possible to do the inverse, so predict the shape of these dimensions from the physical constants we experimentally measured (assuming they really exist)?

No, because the "physical constants" aren't constant. For example take a look at the fine structure constant on NIST:

"Thus α depends upon the energy at which it is measured, increasing with increasing energy, and is considered an effective or running coupling constant. Indeed, due to e+ e- and other vacuum polarization processes, at an energy corresponding to the mass of the W boson (approximately 81 GeV, equivalent to a distance of approximately 2 x 10-18 m), α(mW) is approximately 1/128 compared with its zero-energy value of approximately 1/137. Thus the famous number 1/137 is not unique or especially fundamental."

It's a running constant. Which means it isn't constant. And note that it's defined in terms of other fundamental physical constants. So some or all of them can't be constant either. For example, people usually say the speed of light is constant, but take a look at what Einstein said in 1920:

enter image description here

Some people even think Planck's constant varies, see this physicsworld article: Can GPS find variations in Planck's constant?.

So all in all, I'm afraid what Brian Greene is doing is promoting a hypothesis, and of course his popscience books, without telling you about the real-world science and the bona-fide physics. I would urge you to gather further input on this, and read articles like this.

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