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Just got done with my third reading of "The Elegant Universe" and I have had a somewhat interesting thought on the difficulty in coming up with the actual Calabi-Yau shape that the Kaluza-Klein curled up spaces take.
I don't think there is just one.
I look at it this way. Suppose that what causes our quantum foam is a continuous and never-ending orbifolding of Calabi-Yau shapes. Any time a specific Calabi-Yau shape comes about that is not consistent with the universe as we observe it, the virtual pair particles it produces don't work; they simply would not be visible to our universe. However, any time the Calabi-Yau shape from this continuous orbifolding is produced that is consistent with our universe, then they hang around and we can therefore observe them. So, that might explain the quantum foam. The question then arises, how then does the universe manage to exist with all these incompatible Calabi-Yau shapes instead of just one? Well, why should there be just one? Heizenberg's uncertainty principle says we can't know anything about the universe at less than the Planck scale, so we really don't know if there is more than one Calabi-Yau shape or not. Perhaps the orbifolding happens in such a way that they all tend toward a certain shape that determines the universe we see. Or perhaps any time a Calabi-Yau shape is produced by orbifolding that is not compatible with our universe, it just spits out particle pairs into another universe...and maybe this is a way to get to the concept of the multiverse that Linde talks about.
Perhaps there exists String Theory mathematics (that we haven't discovered yet) that will explain orbifolding in this way. Maybe there are vast quantities of Calabi-Yau shapes instead of just one. Maybe Calabi-Yau shapes are like potato chips...you can't have just one.
I don't know - I got the impression that every "Planck pixel" of our universe had to have a valid compactification at that point. Is there some indication that the different Calabi-Yau configurations would have detectable properties at all? There would probably also needs to be something else to explain why, for example, virtual particles would come out of this in pairs.
That said... not a big fan of string theory as it stands. The ten - or eleven - dimensions are for its own symmetry needs and the Calabi-Yau shapes are a consequence of having to deal with the extra six dimensions in a non-violating manner.
LQG is so far not experimentally verifiable, it seems, but it manages to do its work in the normal 3 space dimensions and 1 of time.
The real deal is probably not going to come out of either of them, I'd bet. There would be a lot more hypotheses to play with had we not decided to take the everyone-on-board tack that had actually worked so well back in Standard Model experimental days.
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