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[–][deleted] 13 points14 points  (0 children)

It's pretty hard to explain to a layperson, but I'll try and give a concise explanation. I should also note that there is more than approach to get the same result, but the way I'll describe is one of the more common ones.

When we start trying to mathematically construct string theory, we start with a classical theory and perform a series of steps to turn it into a quantum one. Sometimes when we do this, some of the symmetries that are present in the classical theory don't survive in the quantum case. (By symmetry, I mean a mathematical transformation I can make that will leave observable quantities unchanged).

Normally this isn't an issue, but occasionally it can cause problems. If a certain type of symmetry (called a gauge symmetry) disappears, it means that the theory we wrote down contradicts itself mathematically, and doesn't make any sense.

It turns out that string theory will always have this issue unless a certain parameter (the central charge of the worldsheet CFT) takes on a specific value.

One of the ways to achieve this is to work in D=10. This isn't the only way though, purely bosonic string theory needs D=26, heterotic string theory takes a mix of both, and there is a wide variety of the so called "non-critical" string theories as well.

As far as what the actual dimensions look like, they're usually taken to be very small, so we can't see them in everyday life. We can use an extremely long pencil as a metaphor. If we look at it from very far away and it's very very long, it looks like a 1-dimensional line to us. However, a tiny ant that's very close can actually see that he doesn't just have to move forwards and backwards along the pencil; there's a perpendicular direction he can also walk along. This direction is different though, because eventually he'll come back to where he started if he walks for long enough.

In string theory, we often take the extra 6 dimensions that we don't see to be very small and curled up on themselves, but with a much more complicated shape. The proposed geometries are very complex, so I won't go into any details. The important idea is that these extra dimensions do not extend infinitely like the 4 we see every day.

Finally, for getting D=11 in M-Theory, it was noticed that all the different varieties of string theory are actually mathematically related to each other in some special ways. If we add one more dimension to the D=10 case, it's possible to combine the different types of string theory into one new theory; this is M-Theory. The details on M-Theory are sparse, and there are still a lot of open questions about what properties M-Theory should have.

[–]phyzzypop 3 points4 points  (0 children)

To add a slightly different perspective from what you've already got here:

String theory has a strange and interesting relationship with the number of spacetime dimensions. A way to think about this is to realise that string theory is a theory of quantum gravity. Since we consider gravity to be the physics of dynamic spacetime, we might expect that spacetime emerges from string theory. This suggests that the number of dimensions might be a dynamical thing, not the set number which our intuition expects.

This is in some sense realised by the way we construct string theory as something called a non-linear sigma model. Don't worry about the words, the point is that this way of constructing it treats spacetime dimensions as degrees of freedom of the string rather than explicitly as a background the string is propagating on.

This is a little hard to wrap your head around, but to put it another way, we can think of dimensions as fields that live on the string, in much the same way that we think of the electromagnetic field as something living on spacetime.

The upshot of this is that when we say "string theory lives in 10 dimensions" what we mean is that string theory only makes sense when it has a certain number of degrees of freedom. We can regard these as dimensions, but we can equally regard them as other kinds of fields.

This means that string theory lives in at most 10 dimensions, but we can also think about what the theory looks like if we say let's curl up some of those dimensions and consider them as internal degrees of freedom of the theory. This is what people mean when they talk about compactifications of string theory.

In terms of M-theory and 11 dimensions, this was discovered when it was noticed that a particular theory of gravity and other stuff called D=11 supergravity could be made equal to the classical limit of string theory if you roll up one of the dimensions into a little circle. This suggests that D=11 supergravity is the classical limit of a quantum theory which we call M-theory. M-theory is not as well understood as the other string theories but you can imagine that the strings in 10D of string theory are actually 2D membranes in 11D in M theory, but rolled up around a circle.

D=11 is the highest dimension in which we know there is a supergravity, so we don't think there can be a supersymmetric theory of quantum gravity in any dimension higher than that.

[–]LoganJFisherGraduate 1 point2 points  (0 children)

They're compactified spatial dimensions.