by Lee Smolin
So it came down, as things always do, to the details. Does it really work, or is there fine print that diminishes the miracle? If it works, how does such a simple theory actually explain so much? What must we believe about nature if string theory is true? What, if anything, do we lose in the process?
As I learned more about the theory, I began to think of the challenges it posed as very like the ones we confront when buying a new car. You go to the dealer with a list of the options you want. The dealer is overjoyed to sell you a car with those options. Several models are brought out. After a while, you realize that every car you’re being shown has some options that are not on your list. You wanted antilock brakes and a really good sound system with a CD player. The cars with those also have sunroofs, fancy chrome bumpers, titanium hubcaps, eight cupholders, and custom racing stripes.
This is what is known as the package deal. It turns out that you cannot get a car with only the options you want. You have to get a package of options, which includes things you don’t want or need. These extras raise the price quite a bit, but there is no choice. If you want the antilock brakes and the CD player, you have to take the whole package.
String theory, too, seems to be offered only as a package deal. You may want a simple unified theory of all the particles and forces, but what you get includes a few extra features, at least two of which are nonnegotiable.
The first is supersymmetry. There were string theories without supersymmetry, but they were all known to be unstable, because of the presence of those pesky tachyons. Supersymmetry appeared to eliminate the tachyons, but there was a catch. The supersymmetric string theory could be consistent only if the universe has nine dimensions of space. There was no option for a theory that works in a three-dimensional space. If you wanted the other features, you had to take the option with six extra dimensions. Much followed from this. If the theory was not to be ruled out right away, there had to be a way to hide the extra dimensions. There appeared to be no choice but to curl them up so that they were too small to be perceived. Thus we were forced to revive the main ideas of the old unified-field theories.
This gave rise to great opportunities, and great problems. As we saw, earlier attempts to use higher dimensions to unify physics had failed, because there were too many solutions; the introduction of the higher dimensions led to a huge problem of nonuniqueness. It also led to problems of instabilities, because there are processes by which the geometry of the extra dimensions unravels and becomes large and other processes whereby it collapses to a singularity. If string theory was to succeed, it would have to solve these problems.
String theorists soon realized that the problem of nonuniqueness was a fundamental feature of string theory. There were now six extra dimensions to curl up, and there were many ways to do it. Most involved a complicated six-dimensional space, and each gave rise to a different version of string theory. Because string theory is a background-dependent theory, what we understood about it at a technical level was that it gave us a description of strings moving in fixed-background geometries. By choosing different-background geometries, we got technically different theories. They came from the same idea, and the same law was applied in each case. But strictly speaking, each was a different theory.
This is not just splitting hairs. The physical predictions given by all these different theories were different, too. Most of the six-dimensional spaces were described by a list of constants, which were freely specifiable. These denoted different features of the geometry, such as the volumes of the extra dimensions. A typical string theory might have hundreds of these constants. These constants are part of the description of how strings propagate and interact with each other.
Think of an object with a two-dimensional surface, like a sphere. Because it is perfectly spherical, it is described by only one parameter, its circumference. But now imagine a more complicated surface, like a doughnut (see Fig. 7). This surface is described by two numbers. There are two circles that go around the doughnut in two different ways, and they can have different circumferences.
Fig. 7. The hidden dimensions can have different topologies. In this example, there are two hidden dimensions, which have the topology of a doughnut, or torus.
We can imagine more complicated surfaces, with many holes. These take more numbers to describe. But no one (at least, no one I know) can directly visualize a six-dimensional space.
However, we do have tools to describe them, which employ analogs of the holes that can occur in a doughnut and other two-dimensional surfaces. Rather than wrapping a string around a hole, we wrap a higher-dimensional space around it. In each case, the space that is wrapped will have a volume, and that will become a constant describing the geometry. When we work out how the strings move in the extra dimensions, all these extra constants come in. So there is no longer just one constant; there are a large number of constants.
This is how string theory resolves the basic dilemma facing attempts to unify physics. Even if everything comes from a simple principle, you have to explain how the variety of particles and forces arises. In the simplest possibility, where space has nine dimensions, string theory is very simple; all the particles of the same kind are identical. But when the strings are allowed to move in the complicated geometry of the six extra dimensions, there arise lots of different kinds of particles, associated with different ways to move and vibrate in each of the extra dimensions.
So we get a natural explanation for the apparent differences among the particles, something a good unified theory must do. But there is a cost, which is that the theory turns out to be far from unique. What is happening is a trading of constants: The constants that denote the masses of the particles and the strengths of the forces are being traded for constants that denote the geometry of the extra six dimensions. It is then less surprising to find constants that would explain the standard model.
Even so, this scheme might have been compelling if it had led to unique predictions for the constants of the standard model. If by translating the standard model’s constants into constants denoting the geometry of the extra dimensions, we had found out something new about the standard model’s constants, and if these findings had agreed with nature, that would have constituted strong evidence that string theory must be true.
But this is not what happened. The constants that could be freely varied in the standard model were translated into geometries that could be freely varied in string theory. Nothing was constrained or reduced. And because there were a huge number of choices for the geometry of the extra dimensions, the number of free constants went up, not down.
Furthermore, the standard model was not completely reproduced. It is true that we can derive its general features, such as the existence of fermions and gauge fields. But the exact combinations seen in nature did not come out of the equations.
From here it got worse. All the string theories predicted extra particles—particles not seen in nature. Along with them came extra forces. Some of these extra forces came from variations in the geometry of the extra dimensions. Think of a sphere attached to every point in space, as in Fig. 8. The radius of the sphere can change as we move around in space.
So the radius of each sphere can be seen as a property of the point to which it is attached. That is, it is something like a field. Just like the electromagnetic field, such fields propagate in space and time, and this gives rise to extra forces. This is clever, but there was a danger that these extra forces would disagree with observation.
Fig. 8. The geometry of the hidden dimensions can vary in space and time. In this example, the radii of the spheres vary.
We have been speaking of generalities, but there is one world. If string theory was to succeed, it had to not just model possible worlds but also explain our world. A key question, then, was: Is there a way to curl up the extra six dimensions so that the standard model of particle physics is completely reproduced?
One way was to have a world with supersymmetry. While string theory had
supersymmetry, how exactly that symmetry was manifested in our three-dimensional world turned out to be dependent on the geometry of the extra dimensions. One could arrange them so that the supersymmetry seemed to be broken in our world. Or it could be the case that there was much more supersymmetry than could be accommodated in a realistic theory.
So there arose an interesting problem: Could the geometry of the extra six dimensions be chosen so as to achieve exactly the right amount of supersymmetry? Could we arrange it so that our three-dimensional world has a version of particle physics described by the supersymmetric versions of the standard model?
This question was solved in 1985 in a very important paper, written by a quartet of string theorists: Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten.6 They were lucky, because two mathematicians, Eugenio Calabi and Shing-tung Yau, had already solved a mathematical problem that gave the answer. They had discovered and studied a particularly beautiful form of six-dimensional geometry that we now call Calabi-Yau spaces. The four string theorists were able to show that the conditions needed for string theory to reproduce a version of the supersymmetric standard model were the same as the conditions that defined a Calabi-Yau space. They then proposed that nature is described by a string theory in which the extra six dimensions are chosen to be a Calabi-Yau space. This cut down on the possibilities and gave the theory more structure. For example, they showed explicitly how you could trade constants in the standard model, such as those that determine the masses of the different particles, for constants describing the geometry of a Calabi-Yau space.
This was great progress. But there was an equally great problem. Had there been only one Calabi-Yau space, with fixed constants, we would have had the unique unified theory we yearned for. Unfortunately, there turned out to be many Calabi-Yau spaces. No one knew how many, but Yau himself was quoted as saying there were at least a hundred thousand. Each of these spaces gave rise to a different version of particle physics. And each space came with a list of free constants governing its size and shape. So there was no uniqueness, no new predictions, and nothing was explained.
In addition, the theories involving Calabi-Yau spaces had lots of extra forces. It turns out that as long as string theory is supersymmetric, many of these forces will have infinite range. This was unfortunate, because there are strict experimental limits on the existence of any infinite-range force besides gravity and electromagnetism.
There remained other problems. The constants that give the geometry of the extra dimensions can vary continuously. This could give rise to instabilities, as in the old Kaluza-Klein theories. Unless there is some mysterious mechanism that freezes the geometry of the extra dimensions, these instabilities lead to catastrophe, such as singularities coming from the collapse of the extra dimensions.
On top of this, even if our world was described by one of the Calabi-Yau geometries, there was no explanation for how it got that way. String theory came in many other versions besides the Calabi-Yau spaces. There were versions of the theory in which the number of curled-up dimensions varied from none all the way up to nine. Those geometries that had dimensions that weren’t curled up were called flat; they defined worlds that large beings like us would experience. (In investigating the implications for particle physics, we could ignore gravity and cosmology, in which case the non-curled-up dimensions had a geometry described by the special theory of relativity.)
A hundred thousand Calabi-Yau manifolds were only the tip of the iceberg. In 1986, Andrew Strominger discovered a way to construct a vast number of additional supersymmetric string theories. It will be useful to keep in mind what he wrote in the conclusion of his paper describing that construction:
[T]he class of supersymmetric superstring compactifications has been enormously enlarged. . . . [I]t does not seem likely that [these] solutions . . . can be classified in the foreseeable future. As the constraints on [these] solutions are relatively weak, it does seem likely that a number of phenomenologically acceptable . . . ones can be found. . . . While this is quite reassuring, in some sense life has been made too easy. All predictive power seems to have been lost.
All of this points to the overwhelming need to find a dynamical principle for determining [which theory describes nature], which now appears more imperative than ever.7 (Italics mine.)
Thus, by taking on the strategy of the older higher-dimensional theories, string theory took on their problems as well. There were lots of solutions, and a few of them gave rise to a description of something roughly like the real world, but most didn’t. There were lots of instabilities, which manifested themselves in lots of extra forces and particles.
This was bound to create controversy, and it did. Few could disagree that the list of good features was long and impressive. It really did seem that the idea of particles as vibrations of strings was the missing link that could work powerfully to resolve many open problems. But the price was high. The extra features we were forced to buy took away some of the beauty of the original proposal—at least, for a few of us. Others found the geometry of the extra dimensions the most beautiful thing about the theory. No wonder theorists came down strongly on either side.
Those who believed tended to believe in the whole package. I knew many physicists who were sure that supersymmetry and the extra dimensions were there, waiting to be discovered. I knew as many who jumped ship at that point, because it meant accepting too much that had no foundation in experiment.
Among the detractors was Richard Feynman, who explained his reluctance to go along with the excitement as follows:
I don’t like that they’re not calculating anything. I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation—a fix-up to say “Well, it still might be true.” For example, the theory requires ten dimensions. Well, maybe there’s a way of wrapping up six of the dimensions. Yes, that’s possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, there’s no reason whatsoever in superstring theory that it isn’t eight of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesn’t produce anything; it has to be excused most of the time. It doesn’t look right.8
These sentiments were shared by many of the older generation of particle physicists, who knew that the success of particle theory had always required a continual interaction with experimental physics. Another dissenter was Sheldon Glashow, Nobel Prize winner for his work on the standard model:
But superstring physicists have not yet shown that their theory really works. They cannot demonstrate that the standard theory is a logical outcome of string theory. They cannot even be sure that their formalism includes a description of such things as protons and electrons. And they have not yet made even one teeny-tiny experimental prediction. Worst of all, superstring theory does not follow as a logical consequence of some appealing set of hypotheses about nature. Why, you may ask, do the string theorists insist that space is nine-dimensional? Simply because string theory doesn’t make sense in any other kind of space. . . . 9
Beyond the controversy, however, there was a clear need to understand the theory better. A theory that came in so many different versions did not seem like a single theory. If anything, the different theories seemed like different solutions to some other, as yet unknown theory.
We are used to the idea that one theory has many different solutions. Newton’s laws describe how a particle moves in response to forces. Suppose we fix the forces—for example, we want to describe a ball being thrown in Earth’s gravitational field. Newton’s equations have an infinite number of solutions, corresponding to the infinite number of trajectories the ball can take: It can be thrown higher or lower, faster or slower. Each
way of throwing the ball gives rise to a different trajectory, each of which is a solution of Newton’s equations.
General relativity also has an infinite number of different solutions, each of which is a spacetime—that is, a possible history of the universe. Since the geometry of spacetime is a dynamical entity, it can exist in an infinity of different configurations and evolve into an infinity of different universes.
Each of the backgrounds on which a string theory is defined is a solution of Einstein’s equation or some generalization of it. Thus, it began to occur to people that the growing catalog of string theories meant that we weren’t actually studying a fundamental theory. What we were doing, perhaps, was studying the solutions to some deeper, still unknown theory. We might call this a meta-theory, because each of its solutions is a theory. This meta-theory is the real fundamental law. Each solution of it will give rise to a string theory.
Thus, it would be more compelling if we could think not of an infinity of string theories but of an infinity of solutions arising from one fundamental theory.
Recall that each of the many string theories is a background-dependent theory that describes strings moving in a particular background spacetime. Since the various approximate string theories live on different spacetime backgrounds, the theory that unifies them must not live on any spacetime background. What is needed to unify them is a single, background-independent theory. The way to do this was thus clear: Invent a meta-theory that would itself be background-independent, then derive all the background-dependent string theories from this single meta-theory.
