One might imagine that these structures are called D-branes because they need not be two-dimensional, but can be any number of dimensions, less than or equal to the total number of dimensions of space-time itself. That would be too simple, however. It turns out that they are called D-branes because of the special mathematical conditions (called “Dirichlet boundary conditions”) that one imposes, which Polchinski realized could exist when a string ends on a surface. The different dimensional D-branes are actually called “p-branes” (since the letter D was taken already), where p refers to the dimensionality and D to Dirichlet. A one-brane looks like a string, a two-brane looks like a familiar membrane (like a rubber sheet), a three-brane like our own three-dimensional space, and so on. What is more notable about these new objects than their names is the fact that they have their own type of dynamics. Recall that years earlier, when dual strings were first being explored, physicists had wondered whether one might generalize the underlying concepts to yet higherdimensional objects. In a sense, Polchinski’s D-branes are just these generalizations, but more interestingly, he showed that they are required to arise when one attempts to consider the full dynamics of string theories. They had been previously missed for two reasons. First, almost all of the previous analyses of strings had dealt with the simplest approximation to the theory, the so-called weak coupling limit—namely, when strings are almost noninteracting and their wiggles are minimal. Second, fixing the ends of strings to lie on some surface spoils some of the space-time symmetries of the theory in ten dimensions. Physicists had tacitly assumed that keeping such symmetries was essential. But they seemed to forget that the world we experience is only four-dimensional, and what is important is that the resulting theory have the observed space-time symmetries in four dimensions that Einstein ultimately incorporated into general relativity. D-branes, through the mathematical conditions that occur when strings are connected to them, preserve these latter symmetries, if not the full tendimensional symmetries. Once D-branes are included in the theory, it becomes much richer and more complex than it was before, with a host of possible new phenomena. One might imagine that it was somewhat of an embarrassment that string theorists had previously proclaimed that they were on the verge of victory in creating a “theory of everything,” when they had in fact virtually missed “almost everything” in the theory. But in the everoptimistic string worldview, there are no embarrassments. On a slightly less facetious note, it is important to realize that devoting literally decades of one’s career to a theoretical struggle, with unknown odds for success, requires those who engage in it to have a deep underlying faith in the validity of what they are attempting. For these “true believers,” every new development provides an opportunity to confirm one’s expectations that these ideas ultimately reflect reality. What separates this from religion, or what should separate this from religion, however, is the willingness to give up these expectations if it turns out that the theory makes predictions that disagree with observations, or if it turns out that the theory is impotent and makes no predictions.
In any case, what made D-branes a cause for celebration rather than sullenness, was that they allowed a full demonstration that the various consistent string theories in ten dimensions were in fact different aspects of the same theory. In order to establish this, the previously discovered “duality” of open strings on donut-shaped toroidal spaces—in which large and small radii of the different compactified dimensions are exchanged—was essential. Once D-branes are included in the picture, going to the small-radii limit in one type of string theory could be seen as producing the same physics as the large-radii limit of another theory.
D-branes are also of great interest because charges can exist on them, like electric charges, that are the source of fields like the electromagnetic field. Since D-branes are the surfaces on which the two ends of open strings are fixed, and it turns out that Yang-Mills charges can exist on the ends of open strings, these charges are then fixed to the D-dimensional surface of the brane. However, remember that closed strings, which have no end points and thus are not fixed to branes, also incorporate all the physics associated with gravitons, the particles that convey gravitational forces. Thus gravity can operate throughout the “bulk” ten-dimensional space both on and between the branes, while the charges that are the source of Yang-Mills fields live on the branes themselves. As we shall see, this can have dramatic implications.
In any case, the presence of D-branes in string theory also allowed theorists to explore the all-important domain where strings might interact strongly with one another, an area that could not be addressed using conventional techniques developed to try to understand the theory. This was especially critical because it was known that considering only the possibilities where strings might interact more feebly with one another would yield a picture of the theory that was not fully accurate, quantitatively or qualitatively. In particular, it was discovered that there is a new kind of “duality” in string theories with D-branes. Recall once more that for strings living on toroidal (i.e., donut-shaped spaces) the large-radius physics is equivalent, and thus “dual” to the small-radius physics. Now, when D-branes are introduced into the picture, a new and different sort of duality results that connects what otherwise may seem to be disparate physical extremes, obtained by interchanging strings and branes in the theory. This interchange maps a part of the theories where strings may be interacting strongly with each other, and where one cannot perform calculations, with a part of the theories where the strings are more weakly interacting, and their behavior can be more simply followed. In this way, not only might one hope to explore new features of the various different string theories, but it becomes possible to demonstrate how different theories might be related.
The good news is that a new relation between formerly disparate theories was uncovered. The bad news is that while previously there had been five distinct consistent string theories—suggesting that string theory in ten dimensions, with six dimensions ultimately being compactified to leave four large dimensions, was not unique—there now appeared to be a continuum of theories. Specifically, these different theories were related to one another, but each theory represented a distinctly different physical limit. These different theories could be continuously transformed into each other, implying a continuously infinite number of intermediate physical possibilities. There was a ray of hope, however. When examining one of the string theories with branes when the string interaction strength became large, the number of states grew in such a way that it appeared as if some new, hidden dimension was appearing. Recall that in the original Kaluza-Klein theory, as long as one was considering distances much larger than the radius of the circular fifth dimension, all the extra five-dimensional degrees of freedom remained hidden. However, as the radius of the fifth dimension becomes larger in this model, the energy required to resolve these new states decreases. Ultimately, as the radius goes to infinity, the infinite tower of new states makes its presence known. Such behavior was precisely what was being observed for the number of D-branes in this string theory as one tuned up the string interaction strength. Suddenly an eleventh dimension began to suggest itself. This apparent extra dimension was not observed in the first decade following the superstring revolution in 1984 precisely because the analysis of weakly interacting strings could only reveal a small part of the theory. It was now understood that this “weak-coupling” approximation was really very similar to what our four-dimensional world is in the original Kaluza-Klein model—namely, an approximation to reality obtained when the size of the extra dimension is very small compared to anything one might measure. It would have been missed, just as a fifth dimension would be forever missed in the original Kaluza-Klein model, if one always did experiments on scales much larger than the extra compact dimension. This is as close as anything can come to “physics irony.” Here we had an apparently remarkable new paradigm for physical theory that in some sense had ultimately been motivated by the suggestion of Kaluza and
Klein that the physics of our world might derive from the hidden physics of extra dimensions. Yet hidden within the theory itself apparently lies hidden physics of yet another hidden dimension!
The key questions then become: What is this new hidden physics, and does the propagation of dimensions continue? The answer to the first question was, and to some extent still is, “Anyone’s guess.” Clearly the theory will in some limit in eleven dimensions resemble supergravity, which forms the basis of much of string theory. But at higher energies it is unlikely to resemble either supergravity or string theory, but perhaps something even more miraculous.
One thing is clear, however. If this picture is correct, what string theorists had previously claimed were fundamental tiny strings wiggling in tiny extra dimensions deep inside what we otherwise thought were fundamental elementary particles, would in fact perhaps be tiny membranes wrapped around yet other tiny extra dimensions, with yet even more fundamental objects. They would be masquerading as strings because, in the approximations that had been used to define the string theories in question, the extra dimension was curled up on a scale smaller than the string scale, so that a two-dimensional surface would look like a one-dimensional string. Strings, in this respect, need not therefore be the truly elementary objects in the theory.
Even when a new theory might not be understood fully, at least it can be labeled. This new eleven-dimensional theory has become known as M-theory. What does the M stand for? Well, first we must recognize that the term M-theory has evolved to encompass not just the theory that the ten-dimensional theories each approach as some parameters are varied, but the theory that encompasses all the theories in all their limits! Thus, it is only partially facetious to claim that the name stands for “mother of all theories.” I am told that Ed Witten introduced the term and said it stands for magical, or mysterious, but that may be apocryphal. Other proposals exist: Membrane theory? Marvelous theory?
A somewhat more informed guess, however, suggests that perhaps the M stands for matrix. The argument for this is based on the fact that if one takes one of the string theories that appears to suggest this hidden extra dimension, then as the string interaction strength is varied, the quantities that would normally be the coordinates describing the motion of the strings and branes are not simple numbers but are instead described by mathematical objects called matrices.
A matrix is like a table of numbers, arrayed in rows and columns. Here are two examples:
Matrices can be treated like ordinary numbers in that one can define for them operations such as multiplication and addition. However, unlike normal numbers, matrix multiplication is not commutative. That is, while 3 × 4 equals 12 whether or not one multiplies 3 times 4 or 4 times 3, the product of two matrices A and B is not in general equal to the product of B times A. This is because the rules for multiplying matrices are complicated. One multiplies each term in the first row of one matrix times the term in the corresponding column and then adds the sum to get the corresponding term (upper-left-hand corner) in the new matrix. Thus, for example, for the two matrices given above, the first term in the corresponding matrix if I multiply the first matrix times the second is [(1 × 2) + (5 × 1) + (7 × 4)] = 35. However, if I multiply the second matrix times the first, the first term in the new matrix is [(2 × 1)+ (4 × 3) + (5 × 2)] = 24.
What is interesting and at the same time odd about this is that if matrices are the fundamental objects describing the eleven-dimensional universe of M-theory, then each point in the space is described by a matrix and not a mere number. This means the eleven-dimensional universe of M-theory bears little or no resemblance to the universe we experience. The coordinates that describe where you are in this space don’t commute with each other! As if eleven-dimensional ordinary space was not complicated enough to think about. Equally important is the fact that in this new eleven-dimensional space, neither strings nor D-branes may be the truly fundamental objects. If this picture is correct, strings in ten dimensions are just as much an approximate illusion of reality as elementary particles in four dimensions were supposed to be in the original string picture.
One might, of course, wonder if all of this rampant breeding of new dimensions is any different from the earlier rampant breeding of new elementary particles at ever higher energies, which seemed so confusing and complex in the 1960s, and which led, in a sense, to the original proposal for dual string theories.
Nevertheless, there are reasons to suspect that eleven dimensions are as far as one need go. After all, one cannot have sensible supergravity symmetries in higher dimensions, and supergravity is one of the hallmarks that is supposed to characterize feasible and consistent string theories as candidates for quantum gravitational theories. Readers with a fantastic memory and remarkable attention for detail may remember that another feature of eleven-dimensional supergravity theories was that gravity determined all of the matter fields in the theory, and that there was no room left over for Yang-Mills fields and all the other paraphernalia that makes our world so interesting. So, what is the difference in M-theory? It is that M-theory contains many more objects than merely elementary particles and fields. It contains things that look, in some limits, like strings and D-branes, and in other limits, like matrices. And who knows what else? Finally, after this seemingly miraculous convergence on an unknown M-theory (I remind you that for some people everything in string theory is miraculous), you might think that this fiddling with extra dimensions would be over with. However, the next, and up to the present time, last string miracle was yet to occur.
In 1997, a young Princeton graduate student turned Harvard professor Juan Maldacena made a daring conjecture, which once again completely changed the face of string theory. Remember that strings in ten dimensions can host Yang-Mills gauge fields, while in eleven dimensions at low enough energies, gravitational degrees of freedom associated with supergravity are all that can be detected. Maldacena suggested another kind of dramatic correspondence appropriate for our understanding of Yang-Mills theories in four dimensions (i.e., the world of our experience). Using ideas based in ten-dimensional string theory, Maldacena proposed that perhaps our fourdimensional world, full of Yang-Mills gauge symmetries, might have a hidden five-dimensional meaning. Specifically, he conjectured that a fourdimensional flat space with quantum Yang-Mills fields and supersymmetry, which our world might contain, could be completely equivalent to a somewhat strange five-dimensional universe with just classical (super)gravity and nothing else. If this sounds suspiciously like déjà-vu all over again—namely, like a modern reframing of the original Kaluza proposal of 1919, in which electromagnetism in four dimensions arose from an underlying theory involving just gravity in five dimensions—you are not that far off. But there is a fundamental and critical difference. In Kaluza-Klein theory, and all subsequent theories with extra dimensions, our four-dimensional universe is merely the tip of the iceberg. We only see four dimensions because our microscopes cannot resolve those tiny extra dimensions. However, in Maldecena’s conjecture, four-dimensional space is not just some large-distance approximation of the underlying five-dimensional space. Rather, the two are precisely the same! All the physical laws of one universe are equivalent to those of the other universe!
Before wondering what this idea might imply regarding the actual meaning of extra dimensions, you might wonder how it could be possible that four dimensions could contain all the physical information of a fivedimensional universe? After all, if one has extra dimensions, there are extra physical degrees of freedom available. In our own world, for example, it is hard to ignore the extra freedom offered by being able to access the third dimension to jump over obstacles on the ground, or the second dimension to go around obstacles in front of you. If four dimensions are somehow to encompass five, then somehow the extra five-dimensional physical degrees of freedom have to be encoded—obviously in a different form—in the lower-dimensional space. Perhaps it is simplest to think of the four-dimensional universe as the surface of a fivedimen
sional volume. Then the question becomes: How could one encode all the information associated with some volume on a surface bounding that volume?
Framed in these terms, there is a well-known example of precisely this phenomenon in three dimensions: holograms. A hologram, stored on a piece of film or plate, is a two-dimensional record of a three-dimensional scene. But when you look at or through the holographic sheet, depending upon its type and the source of light, you see the entire original threedimensional image. If you move your head, you can look around foreground objects to see objects in the background. Unlike a photograph, which simply stores a two-dimensional projection of the three-dimensional image a hologram stores all the information in an image. The reason a hologram allows this degree of image reconstruction is reminiscent of the information loss problem when material falls into a black hole. If the black hole evaporates, then all the energy that fell into it may be radiated away by Hawking radiation, but the question of whether the information can be retrieved comes down to delicate issues having to do with measurement, and what can be reconstructed from subsequent detection of this radiation. When an ordinary camera records an image, it simply records the intensity of light of each color impinging on the photographic film, or the electronic digital recording media, in the case of digital cameras. Because light is a wave, however, not only does it have an intensity, but its electromagnetic fields at any point oscillate in time as the wave passes by. Different light rays, associated with different electromagnetic waves, will cause electromagnetic field oscillations which will in general be out of phase with one another when they pass different points. This phase information is not recorded when the light intensity alone is measured at any given point. However, holograms manage to use sophisticated techniques to capture this additional information. When this phase data is stored on a twodimensional piece of film, it turns out that a full three-dimensional image can be reconstructed.
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 22
