• Choose a category
• All
• Classic-Fiction

By Root 547 0

only weak interactions. He named the new eleven-dimensional theory M-theory—the theory I mentioned at the beginning of this chapter. You can get any known version of superstring theory from M-theory. But M-theory also extends beyond the known versions to domains we have yet to understand. M-theory has the potential to give a more unified, coherent picture of the superstring and to fully realize string theory’s potential as a theory of quantum gravity. However, more pieces or patterns are needed before string theorists understand M-theory sufficiently well to pursue these goals. If the known versions of superstring theory are shards taken from an archeological dig, M-theory is the sought-after enigmatic artifact that would piece them all together. No one yet knows the best way to formulate M-theory. But string theorists now think of it as their primary goal.

More On Duality

This section gives a little more detail about the particular duality I mentioned above, the one between ten-dimensional superstring theory and eleven-dimensional supergravity. I won’t use these explanations later, so feel free to skip ahead to the next chapter if you like. But since this is a book about dimensions, a digression about the duality between two theories with different dimensions doesn’t seem entirely out of place.

One feature that makes duality a little more reasonable is that one of the two theories always contains strongly interacting objects. If the interactions are strong, only rarely can you directly deduce the physical implications of the theory. Although it’s strange to think of a theory that looks ten-dimensional being best described by another, totally different, eleven-dimensional theory, it seems less strange when you remind yourself that your ten-dimensional theory contained such strongly interacting objects that you couldn’t predict what was happening there in the first place. All bets were off anyway.

There are nonetheless many baffling features about a duality between theories with different numbers of dimensions. And in the particular case of the duality between ten-dimensional superstring theory and eleven-dimensional supergravity, at first glance there appears to be an extremely basic problem. Ten-dimensional superstring theory contains strings, whereas eleven-dimensional supergravity does not.

Physicists used branes to solve this puzzle. Even though eleven-dimensional supergravity does not contain strings, it does contain 2-branes. But unlike strings, which have only one spatial dimension, 2-branes have two (as you might have guessed). Now, suppose that one of the eleven dimensions is rolled up into an extremely tiny circle. In that case, a 2-brane that encircles the rolled-up circular dimension looks just like a string. The rolled-up brane appears to have only one spatial dimension, as illustrated in Figure 69. This means that eleven-dimensional supergravity theory with a rolled-up dimension appears to contain strings, even though the original eleven-dimensional theory does not.

Figure 69. A brane with two spatial dimensions that is rolled up on a very small circle looks like a string.

This might sound like a cheat, since we have already argued that a theory with a curled-up dimension always appears to have fewer dimensions at long distances and low energies, so you wouldn’t be surprised to find that an eleven-dimensional theory with a rolled-up dimension acts like a ten-dimensional theory. If you want to show that these ten-and eleven-dimensional theories are equivalent, why should it be sufficient to study the eleven-dimensional theory when one of the dimensions is rolled up?

The key to the answer is that in Chapter 2 we only showed that a rolled-up dimension is invisible at long distances or low energies. Edward Witten went further at Strings ’95. He demonstrated the equivalence of the ten-and eleven-dimensional theories by showing that the eleven-dimensional supergravity theory with one of the dimensions curled up is completely equivalent to ten-dimensional superstring theory, even at short distances. When