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Is the only way that string theory can respect the principles of quantum mechanics, and Einstein's special theory of relativity, is to formulate it in a hypothetical nine dimensional space?

You could use string compactification for $d$ number of non compact directions, all that is required to have $d$ non-compact directions is to choose an appropriate compact space of dimension $9-d$.

The question that arises from that is: Why does our Universe have 3 large spatial dimensions?

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M - Theory suggests that the universe we live in is a D-brane. D branes comes in different dimensionalities from the D25 to D0. There is no known compelling reason why a particular configuration of D branes, compact spaces and the laws of physics are prefered over the others. The current view is that all possibilities can happen in the multiverse, so maybe is kind of lucky that this particular configuration, that we call the universe, is stable and have all the constants tuned for us to observe it.

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The question "why" is always a bit tricky. Elsewhere on PSE there is a good answer from Jerry Schirmer (Why are extra dimensions necessary?), it explains that other dimensions are compact because of the high energy requirement to make them grow bigger (my words). This seems to be a variation on the principle in GR that space only exists because of the mass/energy in it.

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  • $\begingroup$ No on the contrary. Energy is needed to keep the dimensions rolled up, they naturally want to decompactify and energy would be released in this process. $\endgroup$ – Dilaton Jul 2 '14 at 16:42

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