Compactified extra dimensions and symmetry

Question

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:

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.

Answer 1

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.

Answer 2

Time scalar is cardinal -but not a vector. Any direction is a vector. Future lies in all directions. Time can not be relegated to a direction. It is abstract. It has no physical size = to a singularity: Calabi-Yao. As an abstract -imaginary numbers are allowed.The perimeter of Hibert space CAN be turned backwards if time is an abstract. Multiply by -1: ok.Then an infinite time value associates with the beginning point of Hilbert space. Zero with the outer perimeter. Hence singularities are a limit to reality having required infinite vectors. I think time is a quantum energy. The clock of reality was fast but is slowing. The present slides into future states= arrow of time. Distant stars at the beginning of Hilbert space were dense & hot. Time or clock energy was high: time flow was higher. (Being in that clock, at the moment, the beginning might not seem a bang at all.) I think all volume dimensions exist together by definition. The hidden dimensions within Hilbert space may be just too small to see. They might be curved too small to be seen. But in either case you still have a 3D model... Sans time dimension. Smaller atomic particles have had shorter half-lives, could shorter temporal dimensions and smaller spaces be opposite sides of the same coin? Approaching inifinity what of the half-lives? Perhaps the clock of time is what is all that is shorter-quicker. Then the temporal dimensions would perhaps be the missing component of Hilbert space. There is a analog smear of time with much of the energy of reality just beyond our moment of now. Hidden. Dark. The same Hilbert trigonometry in a curved time. (A curve in time is the clock rate. If you wanted a 2D depiction with a timeline on x the units would increase in value left to right. And negative values would be impossible due to the approach to infinity at zero. The history of any Y, z & ? parameter would be curved.) But time as an axis is mixing apples and oranges. The Calabi-Yao-snapshot in-time-depiction is perhaps the right direction. If you spin that sucker: in any moment the pointy projections will trace out internal shells of time dimensions = a digital view of my analog curve of time. Calbi & Yao need to ditch mixing apples and oranges in their thinking. I can imagine that the time separations could account for 9 more 3d spin directions in the atom,(same 3 separated by time Ala Hilbert space time variation). In these quantum separations restoration of energy, (reverse action potential) to the nucleus by 9 intervals accounts for quark flipping,(re-emergance of electrons), partial vs. whole spins hence chromodynamics (AND atomic decay for larger atoms),tunneling and probably spooky action at a distance. This -1 business: It must be associated causally with storage of energy taken from quantum space into Hilbert space- Enthalpy. A negative Feigenbaum constant? Accident or designed? What if reality is multifocal in the beginning. Consider two foci and enthalpy approaching the infinty, (or something connected to the fine particle constant). True vacuum and false vacuum compete from 2 or more sources. Do these Hilbert spaces add or subtract? Interference? Can the wave interference create the false vacuum,(space) and subsequently in like manner foci for particles of mass? Does this storage of quantum energy cause the slowing of time? Create the arrow of time? Is that the true vacuum? I hope you alow me to be read and criticized.