THEORIES OF THE BRANE

**LISA RANDALL:** Particle physics has contributed to our understanding of many phenomena, ranging from the inner workings of the proton to the evolution of the observed universe. Nonetheless, fundamental questions remain unresolved, motivating speculations beyond what is already known. These mysteries include the perplexing masses of elementary particles; the nature of the dark matter and dark energy that constitute the bulk of the universe; and what predictions string theory, the best candidate for a theory incorporating both quantum mechanics and general relativity, makes about our observed world. Such questions (along with basic curiosity) have prompted my excursions into theories that might underlie currently established knowledge. Some of my most recent work has been on the physics of extra dimensions of space and has proved rewarding beyond expectation.

Particle physics addresses questions about the forces we understand—the electromagnetic force, the weak forces associated with nuclear decay, and the strong force that binds quarks together into protons and neutrons—but we still have to understand how gravity fits into the picture. String theory is the leading contender, but we don't yet know how string theory reproduces all the particles and physical laws we actually see. How do we go from this pristine, beautiful theory existing in ten dimensions to the world surrounding us, which has only four—three spatial dimensions plus time? What has become of string theory's superfluous particles and dimensions?

Sometimes a fruitful approach to the big, seemingly intractable problems is to ask questions whose possible answers will be subject to experimental test. These questions generally address physical laws and processes we've already seen. Any new insights will almost certainly have implications for even more fundamental questions. For example, we still don't know what gives rise to the masses of the fundamental particles—the quarks, leptons (the electron, for example), and electroweak gauge bosons—or why these masses are so much less than the mass associated with quantum gravity. The discrepancy is not small: The two mass scales are separated by sixteen orders of magnitude! Only theories that explain this huge ratio are likely candidates for theories underlying the standard model. We don't yet know what that theory is, but much of current particle physics research, including that involving extra dimensions of space, attempts to discover it. Such speculations will soon be explored at the Large Hadron Collider in Geneva, which will operate at the TeV energies relevant to particle physics. The results of experiments to be performed there should select among the various proposals for the underlying physical description in concrete and immediate ways. If the underlying theory turns out to be either supersymmetry or one of the extra dimension theories I will go on to describe, it will have deep and lasting implications for our conception of the universe.

Right now, I'm investigating the physics of the TeV scale. Particle physicists measure energy in units of electron volts. TeV means a trillion electron volts. This is a very high energy and challenges the limits of current technology, but it is low from the perspective of quantum gravity, whose consequences are likely to show up only at energies sixteen orders of magnitude higher. This energy scale is interesting because we know that the as-yet-undiscovered part of the theory associated with giving elementary particles their masses should be found there.

Most of us, however, suspect that a prerequisite for progress will be a worked-out theory that relates gravity to the micro-world. Back at the very beginning the entire universe could have been squeezed to the size of an elementary particle—quantum fluctuations could shake the entire universe, and there would be an essential link between cosmology and the micro-world. Of course, string theory and M-theory are the most ambitious and currently-fashionable attempts to do that. When we have that theory we at least ought to be able to formulate some physics for the very beginning of the universe. One question, of course, is whether we will find that space and time are so complicated and screwed that we can't really talk about a beginning in time. We've got to accept that we will have to jettison more and more of our commonsense concepts as we go to these extreme conditions.

The main stumbling block at the moment is that the mathematics involved in these theories is so difficult that it's not possible to relate the complexity of this 10- or 11-dimensional space to anything we can actually observe. In addition, although these theories may appear aesthetically attractive, and although they give us a natural interpretation of gravity, they don't yet tell us why our three dimensional world contains the types of particles that physicists study. We hope that one day this theory, which already deepens our insight into gravity, will gain credibility by explaining some of the features of the microworld that the current 'standard model' of particle physics does not..

Although Roger Penrose can probably manage four dimensions, I don't think any of these theorists can in any intuitive way imagine the extra dimensions. They can, however, envision them as mathematical constructs, and certainly the mathematics can be written down and studied. The one thing that is rather unusual about string theory from the viewpoint of the sociology and history of science—is that it's one of the few instances where physics has been held up by a lack of the relevant mathematics. In the past, physicists have generally taken fairly old-fashioned mathematics off the shelf. Einstein used 19th century non-Euclidean geometry, and the pioneers in quantum theory used group theory and differential equations that had essentially been worked out long beforehand. But string theory poses mathematical problems that aren't yet solved, and has actually brought math and physics closer together.

String theory is the dominant approach right now, and it has some successes already, but the question is whether it will develop to the stage where we can actually solve problems that can be tested observationally. If one can't bridge the gap between this ten-dimensional theory and anything that we can observe it will grind to a halt. In most versions of string theory the extra dimensions above the normal three are all wrapped up very tightly, so that each point in our ordinary space is like a tightly wrapped origami in six dimensions. We see just three dimensions: the rest are invisible to us because they are wrapped up very tightly. If you look at a needle it looks like a one-dimensional line from a long distance, but really it's three-dimensional. Likewise, the extra dimensions above our three could be seen if you looked at things very closely. Space on a very tiny scale is grainy and complicated—its smoothness is an illusion of the large scale. That's the conventional view in these string theories.

An extra idea which has become popular in the last two or three years is that not all the extra dimensions are wrapped up, but there might be at least one extra dimension which exists on a large scale. Lisa Randall and Raman Sundrum have developed this idea in their work on branes. According to their theory there could be other universes, perhaps separated from ours by just a microscopic distance. However, that distance is measured in some fourth spatial dimension of which we are not aware. Because we are imprisoned in our three dimensions we can't directly detect these other universes. It's rather like a whole lot of bugs crawling around on a big, two-dimensional sheet of paper, who would be unaware of another set of bugs that might be crawling around on another sheet of paper that could be only a short distance away in the third dimension. In a different way, this concept features in a rather neat model that Paul Steinhardt and Neil Turok have discussed, which allows a perpetual and cyclic universe, These ideas, again, may lead to new insights. They make some not-yet-testable predictions about the fluctuation of gravitational waves, but the key question is whether they have the ring of truth about them. We may know that when they've been developed in more detail.

Two of the potential explanations for the huge disparity in energy scales are supersymmetry and the physics of extra dimensions. Supersymmetry, until very recently, was thought to be the only way to explain physics at the TeV scale. It is a symmetry that relates the properties of bosons to those of their partner fermions (bosons and fermions being two types of particles distinguished by quantum mechanics). Bosons have integral spin and fermions have half-integral spin, where spin is an internal quantum number. Without supersymmetry, one would expect these two particle types to be unrelated. But given supersymmetry, properties like mass and the interaction strength between a particle and its supersymmetric partner are closely aligned. It would imply for an electron, for example, the existence of a corresponding superparticle—called a selectron, in this case—with the same mass and charge. There was and still is a big hope that we will find signatures of supersymmetry in the next generation of colliders. The discovery of supersymmetry would be a stunning achievement. It would be the first extension of symmetries associated with space and time since Einstein constructed his theory of general relativity in the early twentieth century. And if supersymmetry is right, it is likely to solve other mysteries, such as the existence of dark matter. String theories that have the potential to encompass the standard model seem to require supersymmetry, so the search for supersymmetry is also important to string theorists. Both for these theoretical reasons and for its potential experimental testability, supersymmetry is a very exciting theory.

However, like many theories, supersymmetry looks fine in the abstract but leaves many questions unresolved when you get down to the concrete details of how it connects to the world we actually see. At some energy, supersymmetry must break down, because we haven't yet seen any "superpartners." This means that the two particle partners—for example, the electron and the selectron—cannot have exactly the same mass; if they did, we would see both. The unseen partner must have a bigger mass if it has so far eluded detection. We want to know how this could happen in a way consistent with all known properties of elementary particles. The problem for most theories incorporating supersymmetry-breaking is that all sorts of other interactions and decays are predicted which experiment has already ruled out. The most obvious candidates for breaking supersymmetry permit the various kinds of quarks to mix together, and particles would have a poorly defined identity. The absence of this mixing and the retention of the various quark identities is a stringent constraint on the content of the physical theory associated with supersymmetry-breaking, and is one important reason that people were not completely satisfied with supersymmetry as an explanation of the TeV scale. To find a consistent theory of supersymmetry requires introducing physics that gives masses to the supersymmetric partners of all the particles we know to exist, without introducing interactions we don't want. So it's reasonable to look around for other theories that might explain why particle masses are associated with the TeV energy scale and not one that is sixteen orders of magnitude higher.

There was a lot of excitement when it was first suggested that extra dimensions provide alternative ways to address the origin of the TeV energy scale. Additional spatial dimensions may seem like a wild and crazy idea at first, but there are powerful reasons to believe that there really are extra dimensions of space. One reason resides in string theory, in which it is postulated that the particles are not themselves fundamental but are oscillation modes of a fundamental string. The consistent incorporation of quantum gravity is the major victory of string theory. But string theory also requires nine spatial dimensions, which, in our observable universe, is obviously six too many. The question of what happened to the six unseen dimensions is an important issue in string theory. But if you're coming at it from the point of view of the relatively low-energy questions, you can also ask whether extra dimensions could have interesting implications in our observable particle physics or in the particle physics that should be observable in the near future. Can extra dimensions help answer some of the unsolved problems of three-dimensional particle physics?

People entertained the idea of extra dimensions before string theory came along, although such speculations were soon forgotten or ignored. It's natural to ask what would happen if there were different dimensions of space; after all, the fact that we see only three spatial dimensions doesn't necessarily mean that only three exist, and Einstein's general relativity doesn't treat a three-dimensional universe preferentially. There could be many unseen ingredients to the universe. However, it was first believed that if additional dimensions existed they would have to be very small in order to have escaped our notice. The standard supposition in string theory was that the extra dimensions were curled up into incredibly tiny scales—10 33 centimeters, the so-called Planck length and the scale associated with quantum effects becoming relevant. In that sense, this scale is the obvious candidate: If there are extra dimensions, which are obviously important to gravitational structure, they'd be characterized by this particular distance scale. But if so, there would be very few implications for our world. Such dimensions would have no impact whatsoever on anything we see or experience.

From an experimental point of view, though, you can ask whether extra dimensions really must be this ridiculously small. How large could they be and still have escaped our notice? Without any new assumptions, it turns out that extra dimensions could be about seventeen orders of magnitude larger than 10-33 cm. To understand this limit requires more fully understanding the implications of extra dimensions for particle physics.

If there are extra dimensions, the messengers that potentially herald their existence are particles known as Kaluza-Klein modes. These KK particles have the same charges as the particles we know, but they have momentum in the extra dimensions. They would thus appear to us as heavy particles with a characteristic mass spectrum determined by the extra dimensions' size and shape. Each particle we know of would have these KK partners, and we would expect to find them if the extra dimensions were large. The fact that we have not yet seen KK particles in the energy regimes we have explored experimentally puts a bound on the extra dimensions' size. As I mentioned, the TeV energy scale of 10-16 cm has been explored experimentally. Since we haven't yet seen KK modes and 10-16 cm would yield KK particles of about a TeV in mass, that means all sizes up to 10-16 are permissible for the possible extra dimensions. That's significantly larger than 10 33 cm, but it's still too small to be significant.

This is how things stood in the world of extra dimensions until very recently. It was thought that extra dimensions might be present but that they would be extremely small. But our expectations changed dramatically after 1995, when Joe Polchinski, of the University of California at Santa Barbara, and other theorists recognized the importance of additional objects in string theory called branes. Branes are essentially membranes—lower-dimensional objects in a higher-dimensional space. (To picture this, think of a shower curtain, virtually a two-dimensional object in a three-dimensional space.) Branes are special, particularly in the context of string theory, because there's a natural mechanism to confine particles to the brane; thus not everything need travel in the extra dimensions even if those dimensions exist. Particles confined to the brane would have momentum and motion only along the brane, like water spots on the surface of your shower curtain.

Branes allow for an entirely new set of possibilities in the physics of extra dimensions, because particles confined to the brane would look more or less as they would in a three-plus-one-dimension world; they never venture beyond it. Protons, electrons, quarks, all sorts of fundamental particles could be stuck on the brane. In that case, you may wonder why we should care about extra dimensions at all, since despite their existence the particles that make up our world do not traverse them. However, although all known standard-model particles stick to the brane, this is not true of gravity. The mechanisms for confining particles and forces mediated by the photon or electrogauge proton to the brane do not apply to gravity. Gravity, according to the theory of general relativity, must necessarily exist in the full geometry of space. Furthermore, a consistent gravitational theory requires that thegraviton, the particle that mediates gravity, has to couple to any source of energy, whether that source is confined to the brane or not. Therefore, the graviton would also have to be out there in the region encompassing the full geometry of higher dimensions—a region known as the bulk—because there might be sources of energy there. Finally, there is a string-theory explanation of why the graviton is not stuck to any brane: The graviton is associated with the closed string, and only open strings can be anchored to a brane.

A scenario in which particles are confined to a brane and only gravity is sensitive to the additional dimensions permits extra dimensions that are considerably larger than previously thought. The reason is that gravity is not nearly as well tested as other forces, and if it is only gravity that experiences extra dimensions, the constraints are much more permissive. We haven't studied gravity as well as we've studied most other particles, because it's an extremely weak force and therefore more difficult to precisely test. Physicists have showed that even dimensions almost as big as a millimeter would be permitted, if it were only gravity out in the higher-dimensional bulk. This size is huge compared with the scales we've been talking about. It is a macroscopic, visible size! But because photons (which we see with) are stuck to the brane, too, the dimensions would not be visible to us, at least in the conventional ways.

Once branes are included in the picture, you can start talking about crazily large extra dimensions. If the extra dimensions are very large, that might explain why gravity is so weak. (Gravity might not seem weak to you, but it's the entire earth that's pulling you down; the result of coupling an individual graviton to an individual particle is quite small. From the point of view of particle physics, which looks at the interactions of individual particles, gravity is an extremely weak force.) This weakness of gravity is a reformulation of the so-called hierarchy problem—that is, why the huge Planck mass suppressing gravitational interactions is sixteen orders of magnitude bigger than the mass associated with particles we see. But if gravity is spread out over large extra dimensions, its force would indeed be diluted. The gravitational field would spread out in the extra dimensions and consequently be very weak on the brane—an idea recently proposed by theorists Nima Arkani Hamed, Savas Dimopoulos, and Gia Dvali. The problem with this scenario is the difficulty of explaining why the dimensions should be so large. The problem of the large ratio of masses is transmuted into the problem of the large size of curled-up dimensions.

Raman Sundrum, currently at Johns Hopkins University, and I recognized that a more natural explanation for the weakness of gravity could be the direct result of the gravitational attraction associated with the brane itself. In addition to trapping particles, branes carry energy. We showed that from the perspective of general relativity this means that the brane curves the space around it, changing gravity in its vicinity. When the energy in space is correlated with the energy on the brane so that a large flat three-dimensional brane sits in the higher-dimensional space, the graviton (the particle communicating the gravitational force) is highly attracted to the brane. Rather than spreading uniformly in an extra dimension, gravity stays localized, very close to the brane.

The high concentration of the graviton near the brane—let's call the brane where gravity is localized the Planck brane—leads to a natural solution to the hierarchy problem in a universe with two branes. For the particular geometry that solves Einstein's equations, when you go out some distance in an extra dimension, you see an exponentially suppressed gravitational force. This is remarkable because it means that a huge separation of mass scales—sixteen orders of magnitude—can result from a relatively modest separation of branes. If we are living on the second brane (not the Planck brane), we would find that gravity was very weak. Such a moderate distance between branes is not difficult to achieve and is many orders of magnitude smaller than that necessary for the large-extra-dimensions scenario just discussed. A localized graviton plus a second brane separated from the brane on which the standard model of particle physics is housed provides a natural solution to the hierarchy problem—the problem of why gravity is so incredibly weak. The strength of gravity depends on location, and away from the Planck brane it is exponentially suppressed.

This theory has exciting experimental implications, since it applies to a particle physics scale—namely, the TeV scale. In this theory's highly curved geometry, Kaluza-Klein particles—those particles with momentum in the extra dimensions—would have mass of about a TeV; thus there is a real possibility of producing them at colliders in the near future. They would be created like any other particle and they would decay in much the same way. Experiments could then look at their decay products and reconstruct the mass and spin that is their distinguishing property. The graviton is the only particle we know about that has spin 2. The many Kaluza-Klein particles associated with the graviton would also have spin 2 and could therefore be readily identified. Observation of these particles would be strong evidence of the existence of additional dimensions and would suggest that this theory is correct.

As exciting as this explanation of the existence of very different mass scales is, Raman and I discovered something perhaps even more surprising. Conventionally, it was thought that extra dimensions must be curled up or bounded between two branes, or else we would observe higher-dimensional gravity. The aforementioned second brane appeared to serve two purposes: It explained the hierarchy problem because of the small probability for the graviton to be there, and it was also responsible for bounding the extra dimension so that at long distances (bigger than the dimension's size) only three dimensions are seen.

The concentration of the graviton near the Planck brane can, however, have an entirely different implication. If we forget the hierarchy problem for the moment, the second brane is unnecessary! That is, even if there is an infinite extra dimension and we live on the Planck brane in this infinite dimension, we wouldn't know about it. In this "warped geometry," as the space with exponentially decreasing graviton amplitude is known, we would see things as if this dimension did not exist and the world were only three-dimensional.

Because the graviton has such a small probability of being located away from the Planck brane, anything going on far away from the Planck brane should be irrelevant to physics on or near it. The physics far away is in fact so entirely irrelevant that the extra dimension can be infinite, with absolutely no problem from a three-dimensional vantage point. Because the graviton makes only infrequent excursions into the bulk, a second brane or a curled-up dimension isn't necessary to get a theory that describes our three-dimensional world, as had previously been thought. We might live on the Planck brane and address the hierarchy problem in some other manner—or we might live on a second brane out in the bulk, but this brane would not be the boundary of the now infinite space. It doesn't matter that the graviton occasionally leaks away from the Planck brane; it's so highly localized there that the Planck brane essentially mimics a world of three dimensions, as though an extra dimension didn't exist at all. A four-spatial-dimensions world, say, would look almost identical to one with three spatial dimensions. Thus all the evidence we have for three spatial dimensions could equally well be evidence for a theory in which there are four spatial dimensions of infinite extent.

It's an exciting but frustrating game. We used to think the easiest thing to rule out would be large extra dimensions, because large extra dimensions would be associated with low energies, which are more readily accessible. Now, however, because of the curvature of space, there is a theory permitting an infinite fourth dimension of space in a configuration that so closely mimics three dimensions that the two worlds are virtually indistinguishable.

If there are differences, they will be subtle. It might turn out that black holes in the two worlds would behave differently. Energy can leak off the brane, so when a black hole decays it might spit out particles into the extra dimension and thus decay much more quickly. Physicists are now doing some interesting work on what black holes would look like if this extra-dimensional theory with the highly concentrated graviton on the brane is true; however, initial inquiries suggest that black holes, like everything else, would look too similar to distinguish the four- and three dimensional theories. With extra dimensions, there are an enormous number of possibilities for the overall structure of space. There can be different numbers of dimensions and there might be arbitrary numbers of branes contained within. Branes don't even all have to be three-plus-one-dimensional; maybe there are other dimensions of branes in addition to those that look like ours and are parallel to ours. This presents an interesting question about the global structure of space, since how space evolves with time would be different in the context of the presence of many branes. It's possible that there are all sorts of forces and particles we don't know about that are concentrated on branes and can affect cosmology.

In the above example, physics everywhere—on the brane and in the bulk—looks three-dimensional. Even away from the Planck brane, physics appears to be three dimensional, albeit with weaker gravitational coupling. Working with Andreas Karch (now at the University of Washington), I discovered an even more amazing possibility: Not only can there be an infinite extra dimension but physics in different locations can reflect different dimensionality. Gravity is localized near us in such a way that it's only the region near us that looks three-dimensional; regions far away reflect a higher-dimensional space. It may be that we see three spatial dimensions not because there really are only three spatial dimensions but because we're stuck to this brane and gravity is concentrated near it, while the surrounding space is oblivious to our lower-dimensional island. There are also some possibilities that matter can move in and out of this isolated four-dimensional region, seeming to appear and disappear as it enters and leaves our domain. These are very hard phenomena to detect in practice, but theoretically there are all sorts of interesting questions about how such a construct all fits together.

Whether or not these theories are right will not necessarily be answered experimentally but could be argued for theoretically, if one or more of them ties into a more fundamental theory. We've used the basic elements found in string theory—namely, the existence of branes and extra dimensions—but we would really like to know if there is a true brane construction. Could you take the very specific branes given by string theory and produce a universe with a brane that localizes gravity? Whether you can actually derive this from string theory or some more fundamental theory is important. The fact that we haven't done it yet isn't evidence that it's not true, and Andreas and I have made good headway into realizing our scenario in string theory. But it can be very, very hard to solve these complicated geometrical set-ups. In general, the problems that get solved, although they seem very complicated, are in many ways simple problems. There is much more work to be done; exciting discoveries await, and they will have implications for other fields.

In cosmology, for instance. Alan Guth's mechanism whereby exponential expansion smooths out the universe works very well, but another possibility has been suggested: a cyclic universe, Paul Steinhardt's idea, wherein a smaller amount of exponential expansion happens many times. Such a theory prompts you to ask questions. First of all, is it really consistent with what we see? The jury's out on that. Does it really have a new mechanism in it? In some sense, the cyclic idea still uses inflation to smooth out the universe. Sometimes it's almost too easy to come up with theories. What grounds your theories? What ties them down? What restricts you from just doing anything? Is there really a new idea there? Do we really have a new mechanism at work? Does it connect to some other, more fundamental theoretical idea? Does it help make that work? Recently I have been exploring the implications of extra dimensions for cosmology. It seems that inflation with extra dimensions works even better than without! What's so nice about this theory is that one can reliably calculate the effect of the extra dimension; no ad hocassumptions are required. Furthermore, the theory has definite implications for cosmology experiments. All along, I've been emphasizing what we actually see. It's my hope that time and experiments will distinguish among the possibilities.