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It's my understanding that M-Theory necessitates 11 space-time dimensions (10 spatial dimensions plus 1 time dimension) in order work mathematically. This appears to jar with reality, which only appears to have 4 total space-time dimensions; so the most commonly offered solution is that these extra dimensions are compactified near the order of the Plank Length, and so we do not notice these extra dimensions. On one level, the explanation of compactification makes sense (e.g. the common ant-walking-on-a-telephone-wire analogy that Brian Greene uses pretty prolifically), but to me, compactification raises more questions than answers.

One of the big things about our three noticeable spatial dimensions is their large amount of symmetry. If I were in the middle of deep space, with no landmarks, I would not be able to determine my orientation in any meaningful way, because of the symmetry of these three dimensions. In fact, we can go one step further and say that on a conceptual level, a choice of a particular axis as an $x$ axis as opposed to a $y$ axis or $z$ axis is totally arbitrary. However, if I'm comparing one of our spatial dimensions (let's say the $x$ dimension for the sake of convenience) to one of the 7 compactified dimensions (I don't even know if there's a conventional way to arbitrarily designate this dimension) I would be able to tell a difference because of the scale. Is this asymmetry between the 7 compactified dimensions and the 3 observable ones a weakness a problem for M-theory, either mathematically or conceptually?

EDIT IN RESPONSE TO COMMENTS: Some comments suggested that compactified extra dimensions may not, in theory or in practice, be distinguishable from our three observable spatial dimensions (as I asserted above), so I'd like to bring up the ant-on-wire-analogy:

Ant on a Wire

If we were looking from far away, it would appear the ant only has one dimension along which it can travel, labeled in the figure as the "Extended" dimension. However, to the ant, the ant can move forward backward along the "Extended" dimension, or rotate around the "Curled Up" dimension. It seems to be that to the ant, these two dimensons are distinguishable in one major way: the "Extended" dimension is infinite, but the "Curled Up" dimension is closed, such that if the ant were to rotate around the wire, he would eventually return to his starting point. To me, this seems like a major difference that is more than just pedantry. Furthermore, to intentionally stretch this analogy, if the ant were to decide to do a physics experiment on his wire, his choice of co-ordinate system would matter, because the Curled Up dimension is closed, whereas the Extended dimension is not.

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  • $\begingroup$ Two things: 1) You claim you would be able to tell a difference...So, exactly how are you going about telling the difference? 2) Kinda makes you doubt the scientific nature (falsifiability,etc etc) of string theory, eh? $\endgroup$ – hft Mar 3 '15 at 1:46
  • $\begingroup$ Good point. And I had to try really hard in my question to avoid words like "test" and predict" because I'm not sure that there is anything that we can test and predict at the moment. I guess my point was just at least conceptually, there's a difference in terms of scale. $\endgroup$ – Sean Mar 3 '15 at 1:50
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    $\begingroup$ @hft Isnt the fact that we do not see/measure/detect extra dimensions a measurable difference? Of course it is not a validation, one could posit an infinite number of undetectable dimensions or what not. $\endgroup$ – anna v Mar 3 '15 at 5:04
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    $\begingroup$ I don't understand what you are asking or why... $\endgroup$ – hft Mar 3 '15 at 6:35
  • $\begingroup$ @hft, You don't understand what I'm asking or what anna v is asking? At any rate, see my recent edit to the question, to see a little bit more about what i mean when I say distinguishing between dimensions. $\endgroup$ – Sean Mar 3 '15 at 12:46
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I don't think its a weakness in any sense. Because in all string theories, $10+1$ dimensional Lorentz transformations ARE a symmetry of the action itself. However not only in order to agree with phenomenology, but also as an attempt (not completely successful so far) to reproduce the entire structure of the standard model interactions, string theory stipulates that these extra dimensions are compactified in a certain way. This means that the vacuum of the theory $\mathbb{R}^4\times S^7$ doesn't respect the symmetry of the theory itself. This is not strange in physics, e.g. $SU(2)\times U(1)$ of the standard model are a symmetry of the theory, but the vacuum itself is not invariant. In any case, if string theory succeeds in finding the structure of this compactified manifold in such a way as to reproduce the standard model, this should be counted as a strength not a weakness.

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