by Lee Smolin
This situation is reminiscent of the strong-weak duality conjecture, in that it is possible to demonstrate the strongest results only on a very special subspace of states where there is a lot of extra symmetry. As in the strong-weak case, pessimists worried that the extra symmetry forced the theories to agree in a way they would not otherwise, whereas optimists were confident that the extra symmetry allowed us to achieve results that revealed a relationship that was true more generally.
Ultimately, it matters a lot which version of the Maldacena conjecture is true. One place it matters is in the description of black holes. Black holes can arise in universes with a negative dark energy, so one can try to use the Maldacena conjecture to study how the black-hole information paradox posed by Stephen Hawking is resolved. Depending on whether the correspondence between the two theories is exact or approximate, the resolution of the paradox could be different.
Suppose there is only a partial correspondence between the gravity theory in the interior of a black hole and the gauge theory. In that case, a black hole can trap information forever—or even pass the information on to a new universe born from the singularity at the center of a black hole, as some theorists, such as John Archibald Wheeler and Bryce DeWitt, long ago speculated. Thus the information is not lost after all, for it lives on in the new universe, but the information is lost forever to an observer at the black hole’s boundary. This loss is possible if the gauge theory at the boundary contains only partial information about the interior. But suppose the correspondence between the two theories is exact. The gauge theory has neither horizons nor singularities and there is no place in which information can be lost. If it corresponds exactly to a spacetime with a black hole, no information can be lost there, either. In the first case, the observer loses information; in the second, he retains it. As of this writing, this issue has yet to be resolved.
As we have seen more than once, supersymmetry plays a fundamental role in string theory. String theories built without supersymmetry have instabilities; left alone, they will take off, emitting more and more tachyons in a process that has no end, until the theory breaks down. This is very unlike our world. Supersymmetry eliminates this behavior and stabilizes the theories. But in some respects, it does that too well. This is because supersymmetry implies that there is a symmetry in time, the upshot being that a supersymmetric theory cannot be built on a spacetime that is evolving in time. Thus, the aspect of the theory required to stabilize it also makes it difficult to study questions we would most like a quantum theory of gravity to answer, like what happened in the universe just after the Big Bang, or what happens deep inside the horizon of a black hole. Both are circumstances where the geometry is evolving rapidly in time.
This is typical of what we learned about string theory during the second superstring revolution. Our understanding expanded greatly, following a set of fascinating, unprecedented results. They gave us tantalizing hints of what might be true, if one could only peer behind an ever present veil and see the real thing. But try as we might, many of the calculations we wanted to do remained out of reach. To get any results, we had to choose special examples and conditions. In many instances, we were left not knowing whether the calculations we could do gave results that were a true guide to the general situation or not.
I personally found this situation very frustrating. We were either making fast progress toward the theory of everything, or we were off on a wild-goose chase, unwisely overinterpreting results, always taking the most optimistic reading from the calculations we were able to do. When I complained about this to some of the leaders of string theory in the mid 1990s, I was told not to worry, it was just that the theory was smarter than we were. We cannot, I was told, ask the theory questions directly and expect answers. Any direct attempt to solve the big problems was bound to fail. Instead, we should trust the theory and follow it, content to explore the parts it was willing to reveal using our imperfect methods of calculation.
There is only one catch. A genuine quantum version of M-theory would have to be background-independent, for the same reason that any quantum theory of gravity must be. But in addition to the reasons I spelled out earlier, M-theory must be background-independent because the five superstring theories, with all their different manifolds and geometries, are supposed to be part of M-theory. This includes all the different ways those geometries could be wrapped up, in all spatial dimensions from one to ten. They all provide backgrounds for strings and branes to move. But if they are part of one unified theory, that theory cannot be built on any one background, because it must encompass all backgrounds.
The key problem in M-theory, then, is to make a formulation of it that is consistent with quantum theory and background independence. This is an important issue, perhaps the most important open question in string theory. Unfortunately, not much progress has been made on it. There have been some fascinating hints, but we still do not know what M-theory is, or whether there is any theory deserving of the name.
There was some progress on an approach to a quantum-mechanical M-theory but, again, in a particular background. This was an attempt to make a quantum theory of the eleven-dimensional membrane theory, back in the 1980s. Three European physicists, Bernard de Wit, Jens Hoppe, and Hermann Nicolai, found that this could be done by a trick in which the membrane is represented by a two-dimensional table, or array, of numbers—called by mathematicians a matrix. Their formulation required that there be nine such tables, and from it they got a theory that approximates the behavior of the membrane.6
De Wit and his colleagues had found that you could make their matrix theory consistent with quantum theory. There was only one hitch, which was that to describe the membranes, the matrix had to extend infinitely, whereas the quantum theory could be shown to make sense only if the matrix was finite. So we were left with a conjecture that if the quantum theory could be consistently extended to infinite arrays of numbers, it would give a quantum theory of the membranes.
In 1996, four American string theorists revived this idea, but with a twist. Thomas Banks, Willy Fischler, Stephen Shenker, and Leonard Susskind proposed that on a background of eleven-dimensional flat spacetime, the same matrix theory gave not just the eleven-dimensional membrane theory but all of M-theory.7 This matrix model doesn’t give a full answer to what M-theory is, because it is in a particular background. It works in a few other backgrounds, but it cannot give sensible answers when more than four dimensions of space are wrapped up. If M-theory is right, our world has seven wrapped dimensions, so this isn’t good enough. Moreover, we still don’t know whether it leads to a completely consistent quantum theory if the matrix becomes infinite.
Unfortunately, M-theory remains a tantalizing conjecture. It’s tempting to believe it. At the same time, in the absence of a real formulation, it is not really a theory—it is a conjecture about a theory we would love to believe in.
When I think of our relationship to string theory over the years, I am reminded of an art dealer who represented a friend of mine. When we met, he mentioned that he was also a good friend of a young writer whose book I had admired; we can call her “M.” A few weeks later, he called me and said, “I was speaking to M. the other day, and, you know, she is very interested in science. Could I get you two together sometime?” Of course I was terribly flattered and excited and accepted the first of several dinner invitations. Halfway through a very good meal, the art dealer’s cell phone rang. “It’s M.,” he announced. “She’s nearby. She would love to drop by and meet you. Is that OK?” But she never came. Over dessert, the dealer and I had a great talk about the relationship between art and science. After a while, my curiosity about whether M. would actually show up lost out to my embarrassment over my eagerness to meet her, so I thanked him and went home.
A few weeks later he called, apologized profusely, and invited me to dinner again to meet her. Of course I went. For one thing, he ate only in the best restaurants; it seems that the managers of some art galleries have expens
e accounts that exceed the salaries of academic scientists. But the same scene was repeated that time and at several subsequent dinners. She would call, then an hour would go by, sometimes two, before his phone rang again: “Oh, I see, you’re not feeling well” or “The taxi driver didn’t know where the Odeon is? He took you to Brooklyn? What is this city coming to? Yes, I’m sure, very soon . . . ” After two years of this, I became convinced that the picture of the young woman on her book jacket was a fake. One night I told him that I finally understood: He was M. He just smiled and said, “Well, yes . . . but she would have so enjoyed meeting you.”
The story of string theory is like my forever postponed meeting with M. You work on it even though you know it’s not the real thing, because it’s as close as you know how to get. Meanwhile the company is charming and the food is good. From time to time, you hear that the real theory is about to be revealed, but somehow that never happens. After a while, you go looking for it yourself. This feels good, but it, too, never comes to anything. In the end, you have little more than you started with: a beautiful picture on the jacket of a book you can never open.
10
A Theory of Anything
IN THE TWO STRING revolutions, observation played almost no role. As the numbers of string theories grew, most string theorists continued to believe in the original vision of a unique theory that gave unique predictions for experiments, but there were no results that pointed in this direction, and a few theorists had worried all along that the unique theory would never emerge. Meanwhile, the optimists insisted that we must have faith and follow where the theory led. String theory appeared to do so much that was required of a unified theory that the rest of the story would surely reveal itself in time.
In the last several years, however, there has been a complete turnaround in how many string theorists think. The long-held hopes for a unique theory have receded, and many of them now believe that string theory should be understood as a vast landscape of possible theories, each of which governs a different region of a multiple universe.
What led to this reversal in expectations? Paradoxically, it was a confrontation with the data. But these were not the data we’d hoped for—they were data that most of us had never expected.
A good theory should surprise us; it means that whoever invented it was doing their job. But when an observation surprises us, theorists worry. No observation in the last thirty years has been more upsetting than the discovery of the dark energy in 1998. What we mean when we say that energy is dark is that it seems to differ from all forms of energy and matter previously known, in that it is not associated with any particles or waves. It is just there.
We do not know what the dark energy is; we know about it only because we can measure its effects on the expansion of the universe. It manifests itself as a source of gravitational attraction spread uniformly through space. Since it is distributed evenly, nothing falls toward it, for there is the same amount everywhere. The only effect it can have is on the average speed at which the galaxies move away from one another. What happened in 1998 was that observations of supernovas in distant galaxies indicated that the expansion of the universe was accelerating in a way that could best be explained by the existence of dark energy.1
One thing that the dark energy might be is something called the cosmological constant. This term refers to a form of energy with a remarkable feature: The properties of the energy, such as its density, appear exactly the same to all observers, no matter where they are in space and time and no matter how they are moving. This is highly unusual. Normally, energy is associated with matter, and there is a preferred observer, who moves with the matter. The cosmological constant is different. It is called a constant because you get the same universal value for it no matter where and when it is measured and how the observer is moving. Because it seems to have no origin or explanation in terms of particles or waves moving in space, it is called cosmological—that is, it is a feature of the whole universe and not any particular thing in it. (I should note that we are not yet sure that the dark energy is in fact in the form of a cosmological constant; all the evidence we have at present points that way, but we will know far better in the next few years whether the energy density is really unchanging in space and time.)
String theory did not predict the dark energy; even worse, the value detected was very hard for string theory to accommodate. Consequently, its discovery precipitated a crisis for the field. To understand why, we have to go back and tell the cosmological constant’s strange, sordid story.
The story begins around 1916, with Einstein’s refusal to believe the most dramatic prediction of his then new general theory of relativity. He had embraced the big lesson of general relativity, which was that the geometry of space and time evolves dynamically. So when people began applying his new theory to models of the universe, he should not have been surprised by what they found, which was that the universe, too, evolves dynamically in time. The model universes they studied expanded and contracted; they even seemed to have beginnings and ends.
But Einstein was surprised by these results—and dismayed. From Aristotle up until that point, the universe had always been thought to be static. It might have been created by God, but if so, it hadn’t changed since. Einstein was the most creative and successful theoretical physicist of the preceding two centuries, but even he could not imagine the universe as anything but eternal and immutable. We are tempted to say that if Einstein had been a real genius, he might have believed his theory more than his prejudice and predicted the expansion of the universe. But a more productive lesson is just how hard it is for even the most adventurous thinkers to give up beliefs that have been held for millennia.
We, who are so used to the idea now, can only speculate about how hard it was to accept the notion that the universe might have had a beginning. In any case, there was at the time no evidence that the universe changed or evolved in time, so Einstein took the predictions of an expanding universe as a sign that his theory was flawed and sought to reconcile it with his conception of an eternal universe.
He noticed that his equations for gravitation allowed a new possibility, which was that the energy density of empty space might have a value—in other words, it might not be zero. Furthermore, this universal energy density would be the same for all observers, no matter where or when they made observations, no matter how they moved. So he named it the cosmological constant. He found that the effect of the constant depended on its sign. When it was a positive number, it would cause the universe to expand—not only expand but do so at an accelerating rate. This is different from the effect of ordinary matter, which would cause the universe to contract because of the mutual gravitational attraction of all the matter it contains. So Einstein realized that he could use the expansive tendency of the new term to balance the contraction due to gravitational force, thus achieving a universe that was static and eternal.
Einstein later called the cosmological constant his biggest blunder. Actually, it was a blunder twice over. First, it didn’t work very well; it didn’t really keep the universe from contracting. You could balance the contraction caused by matter against the expansion caused by the cosmological constant, but only momentarily. The balance was inherently unstable. Tickle the universe and it would start to grow or shrink. But the real blunder was that the idea of a static universe was wrong to begin with. A decade later, an astronomer named Edwin Hubble began to find evidence that the universe was expanding. Since the 1920s, the cosmological constant has been an embarrassment, something to get rid of. But as time went on, this got harder and harder to do, at least theoretically. One could not just set it at zero and ignore it. Like the elephant in the corner, it was there even if you pretended it wasn’t.
People soon began to understand that quantum theory had something to say about the cosmological constant. Unfortunately, it was the opposite of what we wanted to hear. Quantum theory—in particular, the uncertainty principle—appeared to require a huge cosmolog
ical constant. If something is exactly still, it has a definite position and momentum, and this contradicts the uncertainty principle, which says that you cannot know both these things about a particle. A consequence is that even when the temperature is zero, things keep moving. There is a small residual energy associated with any particle and any degree of freedom, even at zero temperature. This is called the vacuum, or ground-state, energy. When quantum mechanics is applied to a field, such as the electromagnetic field, there is a vacuum energy for every mode of vibration of the field. But a field has a huge number of modes of vibration; hence, quantum theory predicts a huge vacuum energy. In the context of Einstein’s general theory of relativity, this implies a huge cosmological constant. We know this is wrong, because it implies that the universe would have expanded so fast that no structure at all could have formed. The fact that there are galaxies puts very strong limits on how big the cosmological constant can be. Those limits are some 120 orders of magnitude smaller than the predictions given by quantum theory; it might just qualify as the worst prediction ever made by a scientific theory.
Something is badly wrong here. A reasonable person could take the view that a radically new idea is needed and that no progress can be made in the unification of gravity and quantum theory until this discrepancy is explained. Several of the most sensible people feel this way. One of them is the German theoretical physicist Olaf Dreyer, who argues that the incompatibility between quantum theory and general relativity can be resolved only if we give up the idea that space is fundamental. He proposes that space itself emerges from a more fundamental description that is quite different. This point of view is also argued by several theorists who did great work in the field of condensed-matter physics, such as the Nobel laureate Robert Laughlin and the Russian physicist Grigori Volovik. But most of us who work on fundamental physics simply ignore this question and go on studying our different approaches, even if at the end of the day they do nothing to resolve it.