Eleventh Dimension

by : professor Brian Greene  

      In 1994, a bombshell was dropped. Another breakthrough took place that once again changed the entire landscape. Edward Witten and Paul Townsend of Cambridge University mathematically found that ten-dimensional string theory was actually an approximation to a higher, mysterious, eleven-dimensional theory of unknown origin. Witten, for example, showed that if we take a membranelike theory in eleven dimensions and curl up one dimension, then it becomes ten-dimensional type IIa string theory!

       Soon afterward, it was found that all five string theories could be shown to be the same—just different approximations of the same mysterious eleven-dimensional theory. Since membranes of differ- ent sorts can exist in eleven dimensions, Witten called this new theory M-theory. But not only did it unify the five different string theories, as a bonus it also explained the mystery of supergravity.

         Supergravity, if you’ll recall, was an eleven-dimensional theory that contained just two particles with zero mass, the original Einstein graviton, plus its supersymmetric partner (called the grav- itino). M-theory, however, has an infinite number of particles with different masses (corresponding to the infinite vibrations that can ripple on some sort of eleven-dimensional membrane). But M-theory can explain the existence of supergravity if we assume that a tiny portion of M-theory (just the massless particles) is the old super- gravity theory. In other words, supergravity theory is a tiny subset of M-theory. Similarly, if we take this mysterious eleven-dimensional membranelike theory and curl up one dimension, the membrane turns into a string. In fact, it turns into precisely type II string the- ory! For example, if we look at a sphere in eleven dimensions and then curl up one dimension, the sphere collapses, and its equator becomes a closed string. We see that string theory can be viewed as a slice of a membrane in eleven dimensions if we curl up the eleventh dimension into a small circle.

       Thus, we find a beautiful and simple way of unifying all ten-dimensional and eleven-dimensional physics into a single theory! It was a conceptual tour de force.

      I still remember the shock generated by this explosive discovery. I was giving a talk at Cambridge University at that time. Paul Townsend was gracious enough to introduce me to the audience. But before my talk, he explained with great excitement this new result that in the eleventh dimension, the various string theories can be unified into a single theory. The title of my talk mentioned the tenth dimension. He told me before I spoke that, if this proved to be successful, then the title of my talk would be obsolete.

      I thought silently to myself, “Uh oh.” Either he was raving mad, or the physics community was going to be turned completely upside down.

       I could not believe what I was hearing, so I fired a barrage of questions at him. I pointed out that eleven-dimensional supermembranes, a theory he helped to formulate, were useless because they were mathematically intractable, and worse, they were unstable. He admitted this was a problem, but he was confident that these ques- tions would be solved in the future.

     I also said that eleven-dimensional supergravity was not finite; it blew up, like all the other theories except string theory. That was no longer a problem, he replied calmly, because supergravity was nothing but an approximation of a larger, still mysterious theory, M-theory, which was finite—it was actually string theory reformu- lated in the eleventh dimension in terms of membranes.

      Then I said that supermembranes were unacceptable because no one had ever been able to explain how membranes interact as they collide and re-form (as I had done in my own Ph.D. thesis years agofor string theory). He admitted that was a problem, but he was con- fident it, too, could be solved.

Last, I said that M-theory was not really a theory at all, since its basic equations were not known. Unlike string theory (which could be expressed in terms of the simple string field equations I wrote down years ago that encapsulated the entire theory), membranes had no field theory at all. He conceded this point as well. But he remained confident that the equations for M-theory would eventually be found.

     My mind was sent swimming. If he was right, string theory was once again about to undergo a radical transformation. Membranes, which were once relegated to the dustbin of physics history, sud- denly were being resurrected.

     The origin of this revolution is that string theory is still evolving backward. Even today, no one knows the simple physical principles that underlie the entire theory. I like to visualize this as walking in the desert and accidentally stumbling upon a small, beautiful peb- ble. When we brush away the sand, we find that the pebble is actu- ally the top of a gigantic pyramid buried under tons of sand. After decades of painfully excavating the sand, we find mysterious hieroglyphics, hidden chambers, and tunnels. One day, we will find the ground floor and finally open up the doorway.