It may seem strange to you that some of the vibrational modes on a closed string live in one number of dimensions, while others live in another, much larger set of dimensions. Actually, this little technicality was not lost on the creators of the model, who pointed out an apparently straightforward, if equally bold, solution. Simply curl up sixteen of the dimensions on which the left-moving vibrations operate into very small regions. In this case, then, just as happened in Kaluza-Klein theory to make the fifth dimension invisible, on scales too large to resolve the extra sixteen dimensions, one would appear to be left with only ten remaining leftmoving modes to go along with the ten right-moving modes. In a way this mathematical wizardry is also reminiscent of what happened in the original the Kaluza-Klein model. There, degrees of freedom in the extra curled-up dimension end up looking, in the four-dimensional world, like photons (i.e., particles associated with the gauge symmetry of electromagnetism). In the new model, one could show that the extra sixteen left-moving modes associated with the curled-up sixteen dimensions end up appearing as welcome extra Yang-Mills symmetries and fields on the remaining ten-dimensional closed string.
Incidentally, if this isn’t strange enough for you, it turns out that there is a way to frame the heterotic string in which the extra sixteen left-moving modes are not associated with sixteen extra spatial dimensions at all, but rather with thirty-two extra weird Grassmann anticommuting coordinates on a ten-dimensional string! In string theory, it seems, as we shall see again later, the existence of extra hidden dimensions may actually depend upon the eye of the beholder.
This brings us to the final development of the trilogy in 1984–85 that truly put string theory at the center of the particle physics universe. With the excitement generated by finite, consistent superstrings and the new heterotic possibility of generating large and phenomenologically interesting Yang-Mills symmetries in ten dimensions, there was only one tiny thing left to do: Make some contact with the four-dimensional universe of our experience!
Enter Ed Witten. While some time earlier, Claude Lovelace at Rutgers had begun to examine what might happen if one put strings on spaces that curve up into small balls, a comprehensive analysis of how one might turn these hypothetical hyperdimensional theories of everything into realistic models of our world was carried out first by Witten, and then in a seminal paper by Witten and his collaborators Philip Candelas, Gary Horowitz, and Andy Strominger.
Witten first showed that one could in principle “compactify” six of the ten dimensions associated with superstrings into small, finite volumes in a way that would leave four large dimensions left over while still preserving, in those four dimensions, essential features such as the absence of anomalies. Then, Witten and his colleagues, a second “string quartet,” (the first being Gross and colleagues) explicitly demonstrated how this might be done. The key was to rely on a new type of mathematics, not then well known among physicists, called “Calabi-Yau manifolds,” after the mathematicians who had first described them. A “manifold,” in mathematics, is something like a rubber sheet. Generalizing the properties of such smooth, pliable objects to higher dimensions has allowed mathematicians to invent a host of strange new objects. Calabi-Yau manifolds are one interesting mathematical class of manifolds with exotic curvatures in many dimensions that can be mathematically classified.
Remember that Kaluza and Klein had considered the simple case where their single extra dimension was curled up into a small circle (a very simple one-dimensional manifold). One might likewise imagine that this concept could be applied to the six extra dimensions in string theory, have them curl up into a small six-dimensional sphere. This was the approach first explored by Lovelace, but it turns out not to work. As Witten and collaborators demonstrated, very specific conditions needed to be imposed on this “compactified space” in order for the resulting fourdimensional theory to remain sensible. Such spaces turned out to have been investigated by the mathematicians Eugenio Calabi and Shing-Tung Yau, and Witten (who would later win the most prestigious award in mathematics, the Fields Medal, for his work using string theory to illuminate the detailed mathematics of knot theory) and his collaborators were able to use their results to explore what kind of theories one might expect to produce in four dimensions. The results were encouraging. It appeared to be possible to produce theories with plausible grand unified Yang-Mills symmetries, and with a spectrum of elementary particles, quarks, electrons, muons, and so forth that could bear an eerie resemblance to what we actually observe in our universe.
The reaction to the Candelas and coworkers’ paper by the physics community was astounding. Suddenly the esoteric and mathematically complex field of string theory held the promise of actually making contact with reality—and not just slight contact. It opened up the possibility of providing a fundamental explanation of why the world at its most basic scale looks like it does, and the answer seemed to lie hidden in this extradimensional Calabi-Yau universe. Within two to three years most major physics departments had a group of brilliant young theorists working on string theory, and in turn this group, usually tenured within a few years of getting their PhDs, started training a new generation, many of whom began their training with string theory, and had never heard of such elementary particles as pions, which had started the whole effort off in the first place. It was a common belief at the time that even though the theory was so complex that the approximations that had thus far been performed barely scratched its surface, it was just a matter of time—and not much time, perhaps—before all the details would be worked out and all the big questions answered. For example, in order to approximate the complex Calabi-Yau manifolds, physicists instead explored approximations called “orbifolds,” which on the whole behave like higher-dimensional generalizations of the nice, smooth rubber sheets one can picture in one’s head, but which have, at a discrete number of points, locations where the sheet gets warped into a conelike shape, with a single point of very high (in a strict mathematical sense, infinite) curvature. Thus, all of the complexities of the Calabi-Yau manifolds could be relegated to what might occur at a finite number of weird points in an otherwise smooth and simple space. One hoped that big questions would be insensitive to this dramatic approximation.
In the meantime, the world of elementary particle physics underwent a sea change after 1984. In particular, an interesting sociological phenomenon began to take place that still has repercussions for the field today. The largely mathematical questions underlying the new theories became for a number of young physicists new to the field much more interesting than trying to figure out such “trivial” low-energy details as how grand unification might account for the actual physics that resulted in a universe full of matter instead of antimatter, or why the proton is two thousand times heavier than the electron. In short, the as-of-yet hypothetical world of hidden extra dimensions had, for many who called themselves physicists, ultimately become more compelling than the world of our experience.
C H A P T E R 1 5
M IS FOR MOTHER
I never think of the future. It comes soon enough.
—Albert Einstein
The theoretical discoveries of 1984–85 energized theoretical particle physicists as nothing had done in a long while. At the same time they produced a remarkable optimism in those who had already begun to work on string theory that the long-sought goal of a consistent unified theory of all the fundamental interactions in nature was at hand, if only the theory could be fully understood. What began as an investigation of an idea that might incorporate gravity and quantum mechanics had, precisely because of its enforced necessity of extra dimensions, begun to appear as if it might explain why everything else existed as well. The concluding sentence of the original heterotic superstring paper stated, “Although much work remains to be done, there seem to be no insuperable obstacles to deriving all known physics from the . . . heterotic superstring.” (Italics mine.) With the realization that the heterotic superstring literally required, for its intern
al consistency, precisely those Yang-Mills symmetries that appeared most promising to describe the real world, it seemed as if nature was saying, “Build a string, and they will come.” If the requirements for a consistent string theory in turn required a specific Yang-Mills symmetry that might explain all of the observed distribution of particles and forces in our four-dimensional universe, then maybe we could finally resolve Einstein’s long-ago query, “Did God have any choice in the creation of the Universe?” The answer would be “No, not if she chose to create it via strings!”
Along with the optimism came a sense of astonishment: Within the course of less than a year it seemed as if an almost insurmountable problem had largely been resolved. So it was that one often heard the remark that, by means of a fortunate accident (the development of dual string models to attempt to explain the strong interactions) we had discovered what rightfully should have been considered twenty-first-or twentysecond-century physics in the twentieth century. We were truly living in the future!
And if life were an impressionist painting, we would have been. Seen with broad brush strokes, everything appeared to be in order. However, there were still a number of nagging details, not to mention the growing recognition that the theory was nowhere near to being fully explored, let alone understood. Indeed, it was not quite clear precisely what string theory actually was. In a prescient paper written in 1983, shortly before the great string revolution, in which he guessed that string theories might be candidates for a consistent theory of quantum gravity, Ed Witten admitted, “What is really unsatisfactory about string theory at the moment is that it isn’t yet a theory.”
Unfortunately, the closer one looked, the greater the problems became. The very richness of the string models and compactification schemes, for example, appeared to undermine claims for uniqueness and with it the hope that string theory would prescribe a universe that simply had to look precisely like the one we live in. Shin-Tung Yau had, for example, elucidated over a hundred thousand different Calabi-Yau manifolds, and compactifying six dimensions on each of them would produce a different four-dimensional theory. Moreover, detailed analysis of the approximations used to compactify the theory from ten dimensions to four suggested that these operations might not be well controlled, invalidating the attractive phenomenological pictures that had first been presented. In an effort to check whether the four-dimensional theories that appeared to result from compactification really were consistent, theorists began to analyze string theories in four dimensions from a new perspective. It turns out that because a string is a one-dimensional object moving in time, its “world sheet”—that is, the region of space-time it maps out as it moves—is a two-dimensional surface. This is the case whether the string is moving in four dimensions, ten dimensions, or twenty-six dimensions. Adding new fields onto the world sheet, which is what happens when fermions and Yang-Mills fields are added to strings, therefore involves studying how fields behave on two-dimensional surfaces.
Interestingly, this is an area of intense interest in condensed matter physics, which studies the bulk properties of real material, whether boiling water, superconductors, or magnets. When such materials undergo a change of phase—for example, water begins to boil, magnets become magnetized—then near the point of this change the properties of the material become particularly interesting and simple. The physics turns out in some cases to depend almost entirely on phenomena associated with twodimensional surfaces, such as bubble walls form the boundary between different phases of boiling water. As a result, condensed matter physicists have become experts on studying such surfaces. Moreover, it turns out that as materials approach the conditions where such phase transitions can occur, their nature begins to look self-similar (i.e., the same phenomena like bubbles seem to appear on all scales). This “scale invariance” is similar to the conformal symmetry of the string theories, which implied that the physics looked the same regardless of over what scales one might stretch the strings.
In any case, studies of such condensed matter systems had classified essentially all two-dimensional field theories, and demonstrated that many of them had the properties that one guessed they might have if they instead described string world sheets obtained by compactifying from higher dimensional theories. That was the good news. But at the same time it suggested that perhaps one could consider string theories in four dimensions without ever worrying about their ten-dimensional roots. Indeed, are the ten dimensions necessary at all, or are the extra dimensions just mathematical artifacts? This is the central question that continues to haunt us.
It was clear that to go beyond the impressionistic connection to the real world, one was going to have to understand string theory a lot better than it was understood thus far. And this was going to be hard work, involving the development of new mathematics that could handle systems far more complex than anything that had been heretofore studied. An army of bright new physicists immediately launched a campaign to scour every cave where interesting possibilities might lurk. Over the next four years the line that had previously tended to separate articles that appeared in physics journals from those that were published in mathematics journals began to blur. Ed Witten, in particular, worked furiously on a host of remarkable ideas.
But, in spite of this plethora of talent and output, progress in actually answering questions about our four-dimensional world was distinctly lacking. New insights about the possible nature of string theory, field theory, and Yang-Mills theories might have been accumulating, but solid physical predictions were not.
Most embarrassing (from my point of view, at least) was the apparent inability of string theory to address the key physical paradoxes that seemed to be associated with a quantum theory of gravity. Sure, the theory appeared to get rid of infinities that might otherwise render predictions nonsensical, but when it came to predicting such things as what the energy of empty space might be (i.e., why the cosmological constant must be zero or extremely small), the theory appeared to make no useful predictions. Another area where strings had thus far shed no light was the very question that Stephen Hawking raised that appeared to result in a direct challenge to quantum theory itself in a world of gravity. What happens to the information about what falls into black holes if the black holes can ultimately evaporate away and disappear? While not much had happened on these fronts, theoretical progress in trying to understand the different varieties of consistent string theories had begun to suggest that the five different types of consistent string theories explored in ten dimensions, might be related.
Might these apparently different theories merely be different manifestations of some single “über” theory? As early as 1985, in fact, several researchers had suggested this possibility. After all, this is precisely the trend that had worked so well to simplify the physics of the known world: Electricity and magnetism had been shown to be different reflections of the same force, the weak and electromagnetic interactions had been shown to be different reflections of the same underlying physics, and so on. Interestingly, however, when physicists began to explore such a possible new connection between the different string theories, hints began to appear that these different theories might well be unified—but not in ten dimensions. Rather, they seemed as if they might be different tendimensional reflections of an underlying eleven-dimensional theory!
Alert readers may remember that eleven dimensions had previously appeared in the grab bag of theoretical physics, associated with a special theory of supergravity. In eleven dimensions, all interactions and particles are specified by gravity and supersymmetry alone, while in ten dimensions there is much more freedom to choose extra Yang-Mills symmetries, fields, and so on. Perhaps an eleven-dimensional theory might be unique, even if a ten-dimensional theory wasn’t.
The first step on this road came from work by Witten and collaborators in 1995, which suggested that all five known consistent string theories were merely different versions of a single underlying, more expansive theory. The next major development in
understanding this possible unification came from a remarkable and unexpected observation in 1995 by Joe Polchinski, at the Kavli Institute for Theoretical Physics at Santa Barbara. Polchinski changed the whole nature of our understanding of what was possible in string theory because he demonstrated that what people had been exploring up to that point—indeed, the theory that had been claimed to be a theory of everything—had in fact overlooked an infinite number of things, including new objects in higher dimensions. For reasons that will become clear, he called them D-branes.
His observation derived from considerations of how open strings might behave in toroidal (i.e., donut-shaped) spaces. As you will recall, insuch spaces it appeared that shrinking one radius of the donut produced a theory that, for closed strings that might wrap in different directions around the donut, looked identical to one in which the same closed strings were wrapping around a radius that became very large.
Open strings—that is, strings that do not close back upon themselves, forming loops, but have two end points like a regular piece of string—however, end up in this case leading to another interesting phenomenon. Their ends are free to move about, and it turns out that the surfaces comprising the set of points along which their ends can move can themselves form a whole new type of mathematical object, behaving like a sort of (mem)brane. In three spatial dimensions, for example, a two-dimensional brane could be a plane or a membrane surface like a rubber sheet. Open strings would be attached at either end to this plane (as the diagram shows). They could wiggle and move in the extra dimension, but their ends would by definition, move about on the plane (brane).
Hiding in the Mirror: The Quest for Alternate Realities, From Plato to String Theory (By Way of Alicein Wonderland, Einstein, and the Twilight Zone) Page 21
