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There are two questions here:

One of the confusing points about String Theory is the existence of extra dimensions. These are explained by saying that the these extra dimensions are compactified.

  1. Does this they are something similar to a principal bundle? For example electromagnetism is described as a circle bundle; is string theory similar, except that instead of a circle, the standard fibre is something else, for example a Calabi-Yau space?

Now, there is another place where extra dimensions are found and thats Quantum Mechanics. Here, not only do we have extra dimensions, we have an infinite number of them - the Hilbert space of states.

  1. Where are they - or is this the wrong question to ask because they are a mathematical artifact - an artifact of this particular description?
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In many instances the number of eigenvalues and eigenfunctions of the Schrodinger equation is infinite. Because the Schrodinger equation is linear, the (normalizable) eigenfunctions can be taken as a basis set, and the dimension of this space is thus infinite. This cardinality has nothing to do with the actual number of physical dimension of the ambient space (i.e. the number of spatial coordinates).

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A particle that lives in $3$-dimensional physical space has a position and momentum that together comprise a $6$-dimensional time-dependent vector living in a mathematical structure called a phase space. If you take Einstein's work on relativity into account, the particle lives in an $8$-dimensional phase space; the extra dimensions are for time and space.

Now consider the electric field, a function of time and space. The possible functions line in an infinite-dimensional space. It's not that physical space has extra dimensions; it's that phase space expands due to extra degrees of freedom. You can specify the field by its values everywhere on a boundary, but each of the infinitely many points contribute to the phase space's dimension.

The wavefunctions of quantum mechanics are in this respect analogous to the classical electric field. So that takes care of your second question. As for your first, string theory expands the finite number of dimensions of the first kind of phase space I discussed. I recommend you read about Kaluza-Klein theory, which only adds one small dimension to space. This is a primer to string theory, which adds several with a similar motivation.

The Calabi-Yau shape of the extra dimensions it's not what you might have guessed at; why not a toroidal shape? A CY has a complicated motivation in terms of supersymmetry that lies beyond the scope of this answer; there has also been some string theory that tries other geometries. For now, it's more important to focus on why several extra dimensions are added: because anomaly cancellation requires more spatial dimensions than the small number of large spatial dimensions we know.

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  • $\begingroup$ Answers use full HTML URLs <A HREF="http://url.com">link text</A> $\endgroup$ – Sean E. Lake Dec 10 '17 at 21:27
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    $\begingroup$ @SeanE.Lake Thanks for that alternative. I don't know why the phase space hyperlink was broken, but I couldn't repair it without the href tagging format. $\endgroup$ – J.G. Dec 10 '17 at 21:28
  • $\begingroup$ A 1d Calabi-Yau manifold is an elliptic curve, and this is a torus: I guessed calabi-yau manifolds since I've heard of them in relation to string compactifications. $\endgroup$ – Mozibur Ullah Dec 10 '17 at 21:52
  • $\begingroup$ @MoziburUllah Just as centuries ago we realised closed planetary orbits can't be expected to be circles but there are theoretical reasons to restrict them to the important generalisation that are ellipses, so now we can say that, instead of having some reason the shape needs to be a hypertoroid, there's a theoretical reason (albeit a controversial one) to restrict to an important generalisation of that. $\endgroup$ – J.G. Dec 10 '17 at 21:56
  • $\begingroup$ Those elliptical orbits are very close to circles though... $\endgroup$ – Mozibur Ullah Dec 10 '17 at 22:02
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When you're dealing with the "infinite dimensions" of quantum mechanics, those dimensions are orthogonal to space-time. In ordinary quantum mechanics, those dimensions are the degrees of freedom of the wave function. Thus, this would be like, in classical physics, looking at the electromagnetic vector potential and counting each component of the 4-potential at each point in space as a separate dimension. The wave function is orthogonal to space-time in the sense that it changes independently of it. In reality, it's tangent to the space because it transforms as part of something called the "Dirac spinor" under Lorentz transformations. Same for the electromagnetic 4-potential, it just transforms like a vector instead of a scalar or spinor.

Bottom line, "where" are they? There's a set of dimensions at every point in space - some of which are tangent to the space (spinors, vectors, etc), others are completely orthogonal to it (scalars). Relevant terminology to look into: tangent space, tangent bundle, and fiber bundle.

The extra dimensions of string theory are very different animals. Those are literal extra spatial dimensions through which the strings can move. In order to make their existence consistent with a macroscopic observation of 3 spatial dimensions we need them to be "compactified" so that movement along their directions is frozen out at energies we have observed so far. In other words, the first excited state for movement along them has an energy gap with the ground state that is so great we don't see it popping up.

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  • $\begingroup$ If they were “orthogonal” to space time, wouldn’t this imply $\langle x\vert \psi\rangle=0$ always by orthogonality? $\endgroup$ – ZeroTheHero Dec 10 '17 at 21:40
  • $\begingroup$ @ZeroTheHero No. $\psi(x)\equiv\langle x|\psi\rangle$ is the dimension that is orthogonal to space. They're orthogonal in the sense that the length of a block and the color it's painted are orthogonal (or number of Skittles and number of M&Ms in a jar). $\endgroup$ – Sean E. Lake Dec 10 '17 at 21:42
  • $\begingroup$ Maybe “orthogonal” is not the ideal word... $\psi(x)$ is a scalar... maybe I’m just being picky... $\endgroup$ – ZeroTheHero Dec 10 '17 at 21:44
  • $\begingroup$ @ZeroTheHero I can't think of a better term. It's stronger than "linearly independent", and better than its synonyms on "Big Huge Thesaurus". $\endgroup$ – Sean E. Lake Dec 10 '17 at 21:47
  • $\begingroup$ Your clarification of “orthogonal” is close enough... granted a better word is not obvious. $\endgroup$ – ZeroTheHero Dec 10 '17 at 21:50

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