Why does string theory require 10 or 11 spacetime dimensions? The answer at a technical level is well-known, but it's hard to bring it down to earth. By reading economics blogs by people who check out political theory blogs, I stumbled across an attempt at making it clear -- by frequent CV commenter Moshe Rozali, writing in Scientific American. After explaining a bit about supersymmetry, Moshe concludes:
A guide in this pursuit is a theorem devised/put forth by physicists Steven Weinberg and Edward Witten, which proves that theories containing particles with spin higher than 2 are trivial. Remember each supersymmetry changes the spin by one half. If we want the spin to be between -2 and 2, we cannot have more than eight supersymmetries. The resulting theory contains a spin -2 boson, which is just what is needed to convey the force of gravitation and thereby unite all physical interactions in a single theory. This theory--called N=8 supergravity--is the maximally symmetric theory possible in four dimensions and it has been a subject of intense research since the 1980s. Another type of symmetry occurs when an object remains the same despite being rotated in space. Because there is no preferred direction in empty space, rotations in three dimensions are symmetric. Suppose the universe had a few extra dimensions. That would lead to extra symmetries because there would be more ways to rotate an object in this extended space than in our three-dimensional space. Two objects that look different from our vantage point in the three visible dimensions might actually be the same object, rotated to different degrees in the higher-dimensional space. Therefore all properties of these seemingly different objects will be related to each other; once again, simplicity would underlie the complexity of our world. These two types of symmetry look very different but modern theories treat them as two sides of the same coin. Rotations in a higher-dimensional space can turn one supersymmetry into another. So the limit on the number of supersymmetries puts a limit on the number of extra dimensions. The limit turns out to be 6 or 7 dimensions in addition to the four dimensions of length, width, height and time, both possibilities giving rise to exactly eight supersymmetries (M-theory is a proposal to further unify both cases). Any more dimensions would result in too much supersymmetry and a theoretical structure too simple to explain the complexity of the natural world.
This is reminiscent of Joe Polchinski's argument (somewhat tongue-in-cheek, somewhat serious) that all attempts to quantize gravity should eventually lead to string theory. According to Joe, whenever you sit around trying to quantize gravity, you will eventually realize that your task is made easier by supersymmetry, which helps cancel divergences. Once you add supersymmetry to your theory, you'll try to add as much as possible, which leads you to N=8 in four dimensions. Then you'll figure out that this theory has a natural interpretation as a compactification of maximal supersymmetry in eleven dimensions. Gradually it will dawn on you that 11-dimensional supergravity contains not only fields, but two-dimensional membranes. And then you will ask what happens if you compactify one of those dimensions on a circle, and you'll see that the membranes become superstrings. Voila!
