by Lee Smolin
Until recently, there was a saving grace: At least the observed value of the cosmological constant was zero—that is, there was no evidence of an accelerating universal expansion rate. This was comforting, because we could hope that a new principle would be found that would eliminate the embarrassment from the equations altogether and make the cosmological constant exactly zero. It would have been far worse if the observed value had been some tiny non-zero number, because it’s much harder to imagine a new principle cutting a number down to a much smaller but still non-zero number. Thus, for decades we thanked our various gods that at least we did not have that problem.
The cosmological constant posed a problem for all of physics, but the situation appeared a bit better for string theory. String theory could not explain why the cosmological constant was zero, but at least it explained why it was not a positive number. One of the few things we could conclude from the string theories then known was that the cosmological constant could only be zero or negative. I don’t know of any particular string theorist who predicted that the cosmological constant could not be a positive number, but it was widely understood to be a consequence of string theory. The reasons are too technical to do justice to them here.
In fact, string theories with negative cosmological constants had been studied. The famous Maldacena conjecture, for example, involved a spacetime with a negative cosmological constant. There were a number of difficulties, and to this day no one has explicitly written down the details of a string theory in a world with a negative cosmological constant. But this lack of explicitness is believed to be a technical issue—there is no known reason why it should not be possible in principle.
You can imagine the surprise, then, in 1998, when the observations of supernovas began to show that the expansion of the universe was accelerating, meaning that the cosmological constant had to be a positive number. This was a genuine crisis, because there appeared to be a clear disagreement between observation and a prediction of string theory. Indeed, there were theorems indicating that universes with a positive cosmological constant—at least, as long as quantum effects were neglected—could not be solutions of string theory.
Edward Witten is not someone given to pessimism, yet he flatly declared in 2001 that “I don’t know any clear-cut way to get de Sitter space [a universe with a positive cosmological constant] from string theory or M-theory.”2
Philosophers and historians of science, among them Imre Lakatos, Paul Feyerabend, and Thomas Kuhn, have argued that one experimental anomaly is rarely enough to kill a theory. If a theory is believed deeply enough, by a large enough group of experts, they will go to ever more extreme measures to save it. This is not always bad for science, and occasionally it can be very good. Sometimes the theory’s defenders succeed, and when they do, great and unexpected discoveries can be made. But sometimes they fail, and then lots of time and energy is wasted as theorists dig themselves deeper and deeper into a hole. The story of string theory in the last few years is one that Lakatos or Feyerabend would have understood well, for it is the story of a large group of experts doing what they can to save a cherished theory in the face of data that seem to contradict it.
What saved string theory—if indeed it has been saved—was the solution to an entirely different problem: how to make the higher dimensions stable. Recall that in higher-dimensional theories the curling up of the extra dimensions produces many solutions. The ones that could possibly reproduce the world we observe are very special, in that certain aspects of the geometry of the higher-dimensional spaces have to be kept frozen. Otherwise, once the geometry starts to evolve, it may just keep going, resulting in either a singularity or a fast expansion that makes the curled-up extra dimensions as big as the dimensions we observe.
String theorists called this the problem of moduli stabilization, “moduli” being a general name for the constants that denote the properties of the extra dimensions. This was a problem that string theory had to solve, but for a long time it was not clear how. As in other cases, the pessimists fretted, while the optimists were confident that sooner or later we’d discover the solution.
In this case, the optimists were right. Progress began in the 1990s, when several theorists in California understood that the key was to use the branes to stabilize the higher dimensions. To understand how, we have to appreciate one feature of the problem, which is that the geometry of the higher dimensions can vary continuously while remaining a good background for a string theory. In other words, you can vary the volume or the shape of the higher dimensions and, by doing so, have them flow through a space of different string theories. This means that there is nothing to stop the geometry of the extra dimensions from evolving in time. To avoid this evolution, we had to find a class of string theories that were impossible to move seamlessly among. One way to do this was to find string theories for which every change is a discrete step—that is, instead of flowing smoothly among theories, you have to make big, abrupt changes.
Joseph Polchinski told us that there were indeed discrete objects in string theory: branes. Recall that there are string backgrounds in which branes are wrapped around surfaces in the extra dimensions. Branes come in discrete units. You can have 1, 2, 17, or 2,040,197 branes but not 1.003 branes. Since branes carry electric and magnetic charges, this gives rise to discrete units of electric and magnetic flux.
So in the late 1990s, Polchinski, working with an imaginative postdoc named Raphael Bousso, began to study string theories in which large numbers of units of electric flux are wrapped around the extra dimensions. They were able to get theories in which some parameters could no longer vary continuously.
But could you freeze all the constants this way? This required a much more complicated construction, but the answer had an added benefit. It made a string theory with a positive cosmological constant.
The crucial breakthrough was made in early 2003, by a group of scientists from Stanford, including Renata Kallosh, a pioneer of supergravity and string theory; Andrei Linde, who is one of the discoverers of inflation; and two of the best young string theorists, Shamit Kachru and Sandip Trivedi.3 Their work is complicated even by the standards of string theory; it has been characterized by their Stanford colleague Leonard Susskind as a “Rube Goldberg contraption.” But it had a huge impact, because it solved both the problem of stabilizing the extra dimensions and the problem of making string theory consistent with the observations of dark energy.
Here’s a simplified version of what the Stanford group did. They started with a much-studied kind of string theory—a flat four-dimensional spacetime with a small six-dimensional geometry over each point. They chose the geometry of the six wrapped-up dimensions to be one of the Calabi-Yau spaces (see chapter 8). As noted, there are at least a hundred thousand of these, and all you have to do is pick a typical one whose geometry depends on many constants.
Then they wrapped large numbers of electric and magnetic fluxes around the six-dimensional spaces over each point. Because you can wrap only discrete units of flux, this tends to freeze out the instabilities. To further stabilize the geometry, you have to call on certain quantum effects not known to arise directly from string theory, but they are understood to some extent in supersymmetric gauge theories, so it is possible that they play a role here. Combining these quantum effects with the effects from the fluxes, you get a geometry in which all the moduli are stable.
This can also be done so that there appears to be a negative cosmological constant in the four-dimensional spacetime. It turns out that the smaller we want the cosmological constant to be, the more fluxes we must wrap, so we wrap huge numbers of fluxes to get a cosmological constant that is tiny but still negative. (As noted, we don’t know explicitly how to write the details of a string theory on such a background, but there’s no reason to believe it doesn’t exist.) But the point is to get a positive cosmological constant, to match the new observations of the universe’s expansion rate. So the next step is to wrap other branes around the ge
ometry, in a different way, which has the effect of raising the cosmological constant. Just as there are antiparticles, there are antibranes, and the Stanford group used them here. By wrapping antibranes, energy can be added so as to make the cosmological constant small and positive. At the same time, the tendency of string theories to flow into one another is suppressed, because any change requires a discrete step. Thus, two problems are solved at once: The instabilities are eliminated and the cosmological constant is small and positive.
The Stanford group may have saved string theory, at least for the time being, from the crisis generated by the cosmological constant. But the way they did it had such weird and unintended consequences that it has split the string community into factions. Before this, the community had been remarkably in accord. Going to a string theory conference in the 1990s was like going to China in the early 1980s, in that almost everyone you talked with seemed to fervently hold the same point of view. For better or worse, the Stanford group destroyed the party unity.
Recall that the particular string theory we are discussing comes from wrapping fluxes around the compact geometries. To get a small cosmological constant, you have to wrap many fluxes. But there’s more than one way to wrap a flux; in fact, there are a lot of ways. How many?
Before answering this question, I have to emphasize that we don’t know if any of the theories made by wrapping fluxes around the hidden dimensions give good consistent quantum string theories. The question is too hard to answer using the methods we have. So what we do is apply tests, which give us necessary but insufficient conditions for good string theories to exist. The tests require that the string theories, if they exist, have strings that interact weakly. This means that if we could do calculations in the string theories, the results would be very close to predictions of the approximate calculations we are able to do.
A question we can answer is how many string theories pass these tests, which involve wrapping fluxes around the six hidden dimensions. The answer depends on what value of the cosmological constant we want to come out. If we want to get a negative or zero cosmological constant, there are an infinite number of distinct theories. If we want the theory to give a positive value for the cosmological constant, so as to agree with observation, there are a finite number; at present there is evidence for 10500 or so such theories.
This is of course an enormous amount of string theories. Moreover, each one is distinct. Each will give different predictions for the physics of the elementary particles and different predictions for the values of the parameters of the standard model.
The idea that string theory gave us not one theory but a landscape consisting of many possible theories had been proposed in the late 1980s and early 1990s, but it had been rejected by most theorists. As noted, Andrew Strominger had found in 1986 that there was a huge number of apparently consistent string theories, and a few string theorists had continued to worry about the resulting loss of predictivity, while most of them had remained confident that a condition would emerge that would settle on a unique and correct theory. But the work of Bousso and Polchinski and the Stanford group finally tipped the balance. It gave us an enormous number of new string theories, as Strominger had, but what was new was that these numbers were needed to solve two big problems: that is, to make string theory consistent with the observations of a positive vacuum energy and to stabilize the theories. Probably for these reasons, the vast landscape of theories finally came to be seen not as a freak result to be ignored but as a means of saving string theory from being falsified.
Another reason the landscape idea took hold was, quite simply, that theorists were discouraged. They had spent a long time searching for a principle that would select a unique string theory, but no such principle had been discovered. Following the second revolution, string theory was now much better understood. The dualities, in particular, made it more difficult to argue that most string theories would be unstable. Thus, string theorists began to accept the vast landscape of possibilities. The question driving the field was no longer how to find a unique theory but how to do physics with such a huge collection of theories.
One response is to say that it’s impossible. Even if we limit ourselves to theories that agree with observation, there appear to be so many of those that some of them will almost certainly give you the outcome you want. Why not just take this situation as a reductio ad absurdum? That sounds better in Latin, but it’s more honest in English, so let’s say it: If an attempt to construct a unique theory of nature leads instead to 10500 theories, that approach has been reduced to absurdity.
This is painful for many who have invested years and even decades of their working lives in string theory. If it is painful for me, having devoted a certain amount of time to the effort, I can only imagine how some of my friends who have staked their whole careers on string theory must feel. Still, even if it hurts like hell, acknowledging the reductio ad absurdum seems a rational and honest response to the situation. It is a response that a few people I know have chosen. But it is not one that most string theorists choose.
There is another rational response: Deny the claim that a vast number of string theories exist. The arguments for the new theories with positive cosmological constant are based on drastic approximations; perhaps they lead theorists to believe in theories that do not exist mathematically, let alone physically.
In fact, the evidence for a vast number of string theories with a positive cosmological constant is based on very indirect arguments. We do not know how to actually describe strings moving in these backgrounds. Moreover, we can define some necessary conditions for a string theory to exist, but we don’t know whether these conditions are also sufficient for the theory’s existence. There is, then, no proof that a theory of strings really exists in any of these backgrounds. So a rational person might say that perhaps they don’t. Indeed, there are recent results—from Gary Horowitz, who is one of the discoverers of the Calabi-Yau spaces, and two younger colleagues, Thomas Hertog and Kengo Maeda—that raise questions about whether any of these theories describe stable worlds.4 One can either take such evidence seriously or ignore it, which is what many string theorists are doing. The possible instability found by Horowitz and his collaborators afflicts not just the landscape of new theories found by the Stanford group but all solutions that involve the six-dimensional Calabi-Yau spaces. If these solutions are indeed all unstable, it means that most of the work aimed at connecting string theory with the real world will have to be thrown out. There is also currently a debate over the validity of some of the Stanford group’s assumptions.
At the beginning of the first superstring revolution, it was miraculous that any string theory existed at all. That there were eventually five was even more surprising. The sheer improbability cemented our belief in the project. If at first it was unlikely to work and then it did work—well, this was nothing less than wonderful. Today string theorists are ready to accept the existence of a landscape containing a vast number of theories, based on much less evidence than we needed twenty years ago to convince ourselves that a single theory existed.
So one place to draw the line is simply to say, “I need to be persuaded that these theories exist, using the same standards that were required decades ago to evaluate the original five.” If you insist on those standards, then you will not believe in the vast number of new theories, because the evidence for any theory in the current landscape is pretty minimal according to the old standards. This is the point of view I find myself leaning toward, most of the time. It just seems to me the most rational reading of the evidence.
11
The Anthropic Solution
MANY PHYSICISTS I know have lowered their expectations that string theory is the fundamental theory of nature—but not everyone. In the last few years, it has become fashionable to argue that the problem lies not with string theory but with our expectations of what any physical theory should look like. That argument was introduced a couple of years ago by Leonard Susskind in
a paper called “The Anthropic Landscape of String Theory”:
Based on the recent work of a number of authors, it seems plausible that the landscape is unimaginably large and diverse. Whether we like it or not, this is the kind of behavior that gives credence to the Anthropic Principle. . . . The [theories in the Stanford group’s landscape] are not at all simple. They are jury-rigged, Rube Goldberg contraptions that could hardly have fundamental significance. But in an anthropic theory simplicity and elegance are not considerations. The only criteria for choosing a vacuum is utility, i.e. does it have the necessary elements such as galaxy formation and complex chemistry that are needed for life. That together with a cosmology that guarantees a high probability that at least one large patch of space will form with that vacuum structure is all we need.1
The anthropic principle that Susskind refers to is an old idea proposed and explored by cosmologists since the 1970s, dealing with the fact that life can arise only in an extremely narrow range of all possible physical parameters and yet, oddly enough, here we are, as though the universe had been designed to accommodate us (hence the term “anthropic”). The specific version that Susskind invokes is a cosmological scenario that has been advocated by Andrei Linde for some time, called eternal inflation. According to this scenario, the rapidly inflating phase of the early universe gave rise not to one but to an infinite population of universes. You can think of the primordial state of the universe as a phase that is exponentially expanding and never stops. Bubbles appear in it, and in these places the expansion slows dramatically. Our world is one of those bubbles, but there are an infinite number of others. To this scenario, Susskind adds the idea that when a bubble forms, one of the vast number of string theories is chosen by some natural process to govern that universe. The result is a vast population of universes, each of which is governed by a string theory randomly chosen from the landscape of theories. Somewhere in the so-called multiverse is every possible theory in the landscape.
