by Lee Smolin
So we had two reasons to look for a background-independent quantum theory of gravity. We already knew that we had to incorporate the dynamical character of geometry given in Einstein’s general theory of relativity. Now we needed it to unify all the different string theories. Doing this would require a new idea but, at least for the time being, it remained out of reach.
One thing that the meta-theory was expected to do was help select which version of string theory was realized physically. Since it was widely believed that string theory was the unique unified theory, many theorists expected that most of the large number of variants would be unstable and that the one truly stable theory would uniquely explain the standard-model constants.
Sometime in the late 1980s, it occurred to me that there was another possibility. Perhaps all string theories were equally valid. This would imply a complete revision of our expectations about physics, in that it would make all the properties of the elementary particles contingent—determined not by fundamental law but by one of an infinite number of solutions to the fundamental theory. There were already indications that this contingency could happen in theories with spontaneous symmetry breaking, but the many versions of string theory opened up the possibility that it was true of essentially all the properties of the elementary particles and forces.
This would mean that the properties of the elementary particles were environmental and could change in time. If so, it would mean that physics would be more like biology, in that the properties of the elementary particles would depend on the history of our universe. String theory would not be one theory, it would be a landscape of theories—analogous to the fitness landscapes that evolutionary biologists study. There might even be a process like natural selection that would select which version applied to our universe. (These thoughts would lead to a 1992 paper titled “Did the Universe Evolve?”10 and a 1997 book called The Life of the Cosmos. Our story later turns on these ideas.)
Whenever I discussed this evolutionary principle with string theorists, they would say, “Don’t worry, there will be a unique version of string theory, selected by a so far unknown principle. When we find it, this principle will correctly explain all the parameters of the standard model and lead to unique predictions for upcoming experiments.”
In any event, progress on string theory slowed, and by the early 1990s string theorists were discouraged. There was no complete formulation of string theory. All we had was a list of hundreds of thousands of distinct theories, each with many free constants. We had no precise idea which of the many versions of the theory corresponded to reality. And while there had been great technical progress, no smoking gun had emerged that would tell us whether string theory was right or wrong. Worst of all, there was not a single prediction made that might be confirmed or falsified by a doable experiment.
There were other reasons for string theorists to be discouraged as well. The late 1980s had been good for the field. Just after the revolution of 1984, the inventors of string theory, like John Schwarz, had had many tempting offers from the best universities. For a few years, young string theorists had moved ahead. But by the early 1990s, this had fallen off, and talented people were again going without job offers.
Some people, young and old, left the field at this point. Luckily, working on string theory had proved to be good intellectual training, and some former string theorists are now flourishing in other areas, such as solid-state physics, biology, neuroscience, computers, and banking.
But others stayed the course. Despite all the reasons for discouragement, many string theorists could not let go of the idea that string theory constituted the future of physics. If there were problems, well, no other approach to unifying the elementary particles was succeeding either. There were a few people working on quantum gravity, but most string theorists remained blissfully unaware of them. For many of them, string theory was simply the only game in town. Even if it was a harder road than they had hoped it would be, no other theory promised to unify all the particles and forces and solve quantum gravity, all within a finite and consistent framework.
The unfortunate result was that the split between believers and skeptics deepened. Each side became more entrenched, and each seemed to have good justification for its position. And it would have stayed like this for a long time, had certain dramatic developments not occurred that radically altered our appreciation of string theory.
9
Revolution Number Two
STRING THEORY INITIALLY proposed to unify all the particles and forces in nature. But as it was studied in the decade following the 1984 revolution, something unexpected happened. The alleged unified theory fractured into many different theories: the five consistent superstring theories in ten-dimensional spacetime, plus millions of variants in the cases where some dimensions were wrapped up. As time went on, it became clear that string theory itself was in need of unification.
The second superstring revolution, which burst on the scene in 1995, gave us just that. The birth of the revolution is often taken to be a talk that Edward Witten gave that March at a string theory conference in Los Angeles, where he proposed a unifying idea. He did not actually present a new unified superstring theory; he simply proposed that it existed and that it would have certain features. Witten’s proposal was based on a series of recent discoveries that had uncovered new facets of string theory and greatly increased our understanding of it. These had further unified string theory with gauge theories and general relativity by exposing additional deep commonalities and relationships among them. These advances, several of which were unprecedented in the history of modern theoretical physics, had eventually won over many skeptics, including me. At first, it had appeared that the five consistent superstring theories described different worlds, but in the mid 1990s we began to understand that they were not as different as they seemed.
When two different ways of looking at the same phenomenon arise, we refer to this as a duality. Ask the members of a couple to tell you, separately, the story of their relationship. They will not be the same stories, but each important event in one will correspond to an important event in the other. If you talk to them long enough, you will be able to predict how the two stories relate and differ. For example, a husband’s perception of his wife’s assertiveness might map to the wife’s perception of an instance of her husband’s passivity. One can say that the two descriptions are dual to each other.
String theorists, in their efforts to relate the five theories to one another, began to speak of several kinds of dualities. Some dualities are exact: that is, the two theories are not really different but are simply two ways of describing the same phenomenon. Other dualities are approximate. In these cases, the two theories really are different, but there are phenomena in one that are similar to phenomena in the other, leading to approximations in which certain features of one theory can be understood by studying the other.
The simplest duality that holds among the five superstring theories is called T-duality. “T” stands for “topological,” because this duality has to do with the topology of space. It occurs when one of the compactified dimensions is a circle. In this case, a string can wind around the circle; in fact, it can wind a number of times (see Fig. 9). The number of times the string wraps around the circle is called the winding number.
Fig. 9. Strings can wind around a hidden dimension. In this case, space is one-dimensional and the hidden dimension is a small circle. Pictured are strings that wind around the circle zero, one, and two times.
Another number measures how a string is vibrating. A string has overtones, just like a piano string or a guitar string, and natural numbers denote the various levels of vibration. T-duality is a relationship between two string theories both of which wrap around a circle. The radii of the two circles differ but are related to each other; one is equal to the inverse of the other (in units of string length). In such cases, the winding states of the first string theory behave exactly the same as the levels of vibration of the
second string theory. This kind of duality turns out to exist between certain pairs of the five string theories. They appear to be different theories to start with, but when you wrap their strings around circles, they become the same theory.
There is a second kind of duality that is also conjectured to be exact, although this has not yet been proved. Recall from chapter 7 that there is a constant in each string theory that determines how probable it is that strings will break and join. This is the string coupling constant, conventionally denoted by the letter g. When g is small, the probability for strings to break and join is small, so we say the interactions are weak. When g is large, they break and join all the time, so we say the interactions are strong.
Now, it can happen that two theories are related in the following way: Each theory has a coupling g. But when the g of the first theory is equal to 1/g of the second theory, the two theories appear to behave identically. This is called S- (for strong-weak) duality. If g is small, meaning the strings interact weakly, 1/g is big, so the strings in the second theory interact strongly.
How can these two theories behave identically if their coupling constants are different? Can’t we tell if the probability for strings to join and break is large or small? We can, if we know what the strings are. But what is believed to happen in cases of S-duality is that these two theories have more strings than they are supposed to.
This proliferation of strings is an example of the familiar but rarely understood phenomenon of emergence, a term that describes the arising of new properties in large and complex systems. We may know the laws that the elementary particles satisfy, but when many particles are bound together, all kinds of new phenomena become apparent. A bunch of protons, neutrons, and electrons may combine to produce a metal; others, of equal number, may combine to produce a living cell. Both the metal and the living cell are just collections of protons, neutrons, and electrons. How, then, do we describe what makes a metal a metal and a bacterium a bacterium? The properties that distinguish them are called emergent properties.
Here’s an example: Perhaps the simplest thing a metal can do is vibrate; if you hit one end of a metal bar, a sound wave will travel through it. The frequency at which the metal vibrates is an emergent property, as is the speed that sound travels in the metal. Recall the wave/particle duality of quantum mechanics, which asserts that there is a wave associated with every particle. The reverse is also true: There is a particle associated with every wave, including a particle associated with the sound wave traveling through the metal. It is called a phonon.
A phonon is not an elementary particle. It is certainly not one of the particles that make up the metal, for it exists only by virtue of the collective motion of huge numbers of the particles that do make up the metal. But a phonon is a particle just the same. It has all the properties of a particle. It has mass, it has momentum, it carries energy. It behaves precisely the way quantum mechanics says a particle should behave. We say that a phonon is an emergent particle.
Things like this are believed to happen to strings as well. When the interactions are strong, there are many, many strings breaking and joining, and it becomes difficult to follow what happens to each individual string. We then look for some simple emergent properties of large collections of strings—properties that we can use to understand what is going on. Now comes something really fun. Just as the vibrations of a whole bunch of particles can behave like a simple particle—a phonon—a new string can emerge out of the collective motion of large numbers of strings. We can call this an emergent string.
The behavior of these emergent strings is the exact opposite of that of ordinary strings—let’s call the latter the fundamental strings. The more the fundamental strings interact, the less the emergent strings do. To put this a bit more precisely: If the probability for two fundamental strings to interact is proportional to the string coupling constant g, then in some cases the probability for the emergent strings to interact is proportional to 1/g.
How do you tell the fundamental strings from the emergent strings? It turns out that you can’t—at least, in some cases. In fact, you can turn the picture around and see the emergent strings as fundamental. This is the fantastic trick of strong-weak duality. It is as if we could look at a metal and see the phonons—the quantum sound waves—as fundamental and all the protons, neutrons, and electrons making up the metal as emergent particles made up of phonons.
Like T-duality, this kind of strong-weak duality turned out to relate certain pairs of the five superstring theories. The only question was whether this relationship applied just to some states of the theories or was deeper. This was an issue, because to show the relationship at all, you had to study special states of the paired theories—states constrained by a certain symmetry. Otherwise you didn’t have enough control of the calculations to get good results.
There were, then, two possible paths for theorists. The optimists—and in those days most string theorists were optimists—went beyond what could be shown, to a conjecture that the relationship between the special symmetric states they could examine in the paired theories extended to all five of the theories. That is, they posited that even without the special symmetries there were always emergent strings and that they always behaved just like the fundamental strings of another theory. This implied that S-duality does not just relate some aspects of the theories but demonstrates their complete equivalence.
On the other hand, the few pessimists worried that perhaps the five theories really were different from one another. They thought it pretty wonderful that there were even a few cases in which emergent strings of one theory behaved like fundamental strings of another, but they realized that such a thing might be true even if the theories were all different.
A lot rested (and continues to rest) on whether the optimists or the pessimists were right. If the optimists turn out to be right, then all five of the original superstring theories really are just different ways of describing a single theory. If the pessimists are right, then they really all are different theories, and therefore there is no uniqueness, no fundamental theory. As long as we do not know whether strong-weak duality is approximate or exact, we do not know whether string theory is unique or not.
One piece of evidence in favor of the optimistic view was that similar dualities were known to exist in theories that were simpler and better understood than string theories. One example is a version of Yang-Mills theory called N = 4 super-Yang-Mills theory, which has as much supersymmetry as possible. For short, we’ll call it the maximally super theory. There is good evidence that this theory has a version of S-duality. It works roughly like this. The theory has in it a number of electrically charged particles. It also has some emergent particles that carry magnetic charges. Now, normally there are no magnetic charges, there are only magnetic poles. Every magnet has two, and we refer to them as north and south. But in special situations there may be emergent magnetic poles that move independently of each other—they are known as monopoles. What happens in the maximally super theory is that there is a symmetry within which electric charges and magnetic monopoles trade places. When that happens, if you change the value of the electric charge to 1 divided by the original value, you don’t change anything in the physics described by the theory. The maximally super theory is a remarkable theory, and it was to play a central role in the second superstring revolution, as we will see shortly. But now that we understand a little about different kinds of dualities, I can explain the conjecture that Witten discussed in his celebrated talk in Los Angeles.
As I mentioned, the key idea in Witten’s talk was that the five consistent superstring theories were all actually the same theory. But what was this single theory? Witten didn’t tell us, but he did describe a dramatic conjecture about it, which was that the theory unifying the five superstring theories would require one more dimension, so that space now had ten dimensions and spacetime eleven.1
This particular conjecture had been first made by two British physicists,
Christopher Hull and Paul Townsend, a year earlier.2 Witten had found a great deal of evidence for the conjecture, based on dualities that had been found not just between the five theories but between string theories and theories in eleven dimensions.
Why should a unification of string theories have one more dimension? A property of an extra dimension—the radius of the extra circle in Kaluza-Klein theory—can be interpreted as a field varying over the other dimensions. Witten used this analogy to suggest that a certain field in string theory was actually the radius of a circle extending in the eleventh dimension.
How did this introduction of yet one more spatial dimension help? After all, there wasn’t a consistent supersymmetric string theory in eleven spacetime dimensions. But there was a supersymmetric gravity theory in eleven spacetime dimensions. This, you may recall from chapter 7, is the highest-dimensional of all the supergravity theories, a veritable Mount Everest of supergravity. So Witten conjectured that the eleven-dimensional world whose existence the extra field pointed to could be described—in the absence of quantum theory—by eleven-dimensional supergravity.
Moreover, although there isn’t a string theory in eleven dimensions, there is a theory of two-dimensional surfaces moving in an eleven-dimensional spacetime. This theory is quite beautiful, at least at the classical level. It was invented in the early 1980s and is called, imaginatively, the eleven-dimensional supermembrane theory.
The supermembrane theory had been ignored by most string theorists until Witten, and for good reason. It is not known whether the theory can be made consistent with quantum mechanics. Some people had tried to combine it with quantum theory and failed. When the first superstring revolution took off in 1984, based on magical properties of theories in ten dimensions, these eleven-dimensional theories were given up by most theorists.