by Brian Greene
The rest of this and the next section will be devoted to explaining how Maldacena achieved this breakthrough; the material is the most difficult we will cover. I’ll begin with a short summary, a CliffsNotes version that doubles as a guilt-free pass to jump to the last section should, at any point, the material overwhelm your appetite for detail.
Maldacena’s inspired move was to invoke a new version of the duality arguments discussed in Chapter 5. Recall the branes—the “slice of bread” universes—introduced there. Maldacena considered, from two complementary perspectives, the properties of a tightly stacked collection of three-dimensional branes, as in Figure 9.4. One perspective, an “intrinsic” perspective, focused on strings that move, vibrate, and wiggle along the branes themselves. The other perspective, an “extrinsic” perspective, focused on how the branes influence their immediate environment gravitationally, much as the sun and the earth influence theirs. Maldacena argued that both perspectives describe one and the same physical situation, just from different vantage points. The intrinsic perspective involves strings moving on a stack of branes, while the extrinsic perspective involves strings moving through a region of curved spacetime that’s bounded by the stack of branes. By equating the two, Maldacena found an explicit link between physics taking place in a region and physics taking place on that region’s boundary; he found an explicit realization of holography. That’s the basic idea.
With more color, the story goes like this.
Consider, Maldacena says, a stack of three-branes, so closely spaced that they appear as a single monolithic slab—Figure 9.4—and study the behavior of strings moving in this environment. You’ll recall that there are two types of strings—open snippets and closed loops—and that the endpoints of open strings can move within and through branes but not off them, while closed strings have no ends and so can move freely through the entire spatial expanse. In the jargon of the field, we say that while open strings are confined to the branes, closed strings can move through the bulk of space.
Maldacena’s first step was to confine his mathematical attention to strings that have low energy—that is, ones that vibrate relatively slowly. Here’s why: the force of gravity between any two objects is proportional to the mass of each; the same is true for the force of gravity acting between any two strings. Strings that have low energy have small mass, and so they hardly respond to gravity at all. By focusing on low energy strings, Maldacena was thus suppressing gravity’s influence. That yielded a substantial simplification. In string theory, as we’ve seen (Chapter 5), gravity is transmitted from place to place by closed loops. Suppressing the force of gravity was therefore tantamount to suppressing the influence of closed strings on anything they might encounter—most notably, the open string snippets living on the brane stack. By ensuring that the two kinds of strings, open snippets and closed loops, wouldn’t affect each other, Maldacena was ensuring that they could be analyzed independently.
Figure 9.4 A collection of closely spaced three-branes with open strings confined to the brane surfaces, and closed strings moving through the “bulk.”
Maldacena then changed gears and suggested thinking about the very same situation from a different perspective. Rather than treat the three-branes as a substrate that supports the motion of open strings, he encouraged viewing them as a single object, which has its own intrinsic mass and hence warps space and time in its vicinity. Maldacena was fortunate that previous research, by a number of physicists, had laid the groundwork for this alternative perspective. The earlier works had established that as you stack more and more branes together, their collective gravitational field grows ever stronger. Ultimately, the slab of branes behaves much like a black hole, but one that’s brane-shaped, and so is called a black brane. As with a more ordinary black hole, if you get too close to a black brane, you can’t escape. And, as is also the case with an ordinary black hole, if you stay far away but are watching something approach a black brane, the light you’ll receive will be exhausted from its having fought against the black brane’s gravity. This will make the object appear to have ever less energy and to be moving ever slower.14
From this second perspective, Maldacena again focused on the low-energy features of a universe containing such a black slab. Much as he had when working on the first perspective, he realized that the low-energy physics involved two components that could be analyzed independently. Slowly vibrating closed strings, moving anywhere in the bulk of space, are the most obvious low-energy carriers. The second component relies on the presence of the black brane. Imagine you are far from the black brane and have in your possession a closed string that’s vibrating with an arbitrarily large amount of energy. Then, imagine lowering the string toward the event horizon while you maintain a safe distance. As recalled above, the black brane will make the string’s energy appear ever lower; the light you’ll receive will make the string look as though it’s in a slow-motion movie. The second low-energy carriers are thus any and all vibrating strings that are sufficiently close to the black brane’s event horizon.
Maldacena’s final move was to compare the two perspectives. He noted that because they describe the same brane stack, only from different points of view, they must agree. Each description involves low-energy closed strings moving through the bulk of space, so this part of the agreement is manifest. But the remaining part of each description must also agree.
And that proves astonishing.
The remaining part of the first description consists of low-energy open strings moving on the three-branes. We recall from Chapter 4 that low-energy strings are well described by point particle quantum field theory, and that is the case here. The particular kind of quantum field theory involves a number of sophisticated mathematical ingredients (and it has an ungainly characterization: conformally invariant supersymmetric quantum gauge field theory), but two vital characteristics are readily understood. The absence of closed strings ensures the absence of the gravitational field. And, because the strings can move only on the tightly sandwiched three-dimensional branes, the quantum field theory lives in three spatial dimensions (in addition to the one dimension of time, for a total of four spacetime dimensions).
The remaining part of the second description consists of closed strings, executing any vibrational pattern, as long as they are close enough to the black branes’ event horizon to appear lethargic—that is, to appear to have low energy. Such strings, although limited in how far they stray from the black stack, still vibrate and move through nine dimensions of space (in addition to one dimension of time, for a total of ten spacetime dimensions). And because this sector is built from closed strings, it contains the force of gravity.
However different the two perspectives might seem, they’re describing one and the same physical situation, so they must agree. This leads to a thoroughly bizarre conclusion. A particular nongravitational, point particle quantum field theory in four spacetime dimensions (the first perspective) describes the same physics as strings, including gravity, moving through a particular swath of ten spacetime dimensions (the second perspective). This would seem as far-fetched as claiming … Well, honestly, I’ve tried, and I can’t come up with any two things in the real world more dissimilar than these two theories. But Maldacena followed the math, in the manner we’ve outlined, and ran smack into this conclusion.
The sheer strangeness of the result—and the audacity of the claim—isn’t lessened by the fact that it takes but a moment to place it within the line of thought developed earlier in this chapter. As schematically illustrated in Figure 9.5, the gravity of the black brane slab imparts a curved shape to the ten-dimensional spacetime swath in its vicinity (the details are secondary, but the curved spacetime is called anti–De Sitter five-space times the five sphere); the black brane slab is itself the boundary of this space. And so, Maldacena’s result is that string theory within the bulk of this spacetime shape is identical to a quantum field theory living on its boundary.15
This is holography
come to life.
Maldacena had built a self-contained mathematical laboratory in which, among other things, physicists could explore in concrete detail a holographic realization of physical law. Within a few months, two papers, one by Edward Witten and one by Steven Gubser, Igor Klebanov, and Alexander Polyakov, supplied the next level of understanding. They established a precise mathematical dictionary for translating between the two perspectives: given a physical process on the brane boundary, the dictionary showed how it would appear in the bulk interior, and vice versa. In a hypothetical universe, then, the dictionary rendered the holographic principle explicit. On the boundary of this universe, information is embodied by quantum fields. When the information is translated by the mathematical dictionary, it reads as a story of stringy phenomena happening in the universe’s interior.
Figure 9.5 A schematic illustration of the duality between string theory operating in the interior of a particular spacetime and quantum field theory operating on the boundary of that spacetime.
Figure 9.6 The holographic equivalence applied to a black hole in the bulk of spacetime yields a hot bath of particles and radiation on the region’s boundary.
The dictionary itself renders the holographic metaphor all the more appropriate. An everyday hologram bears no resemblance to the three-dimensional image it produces. On its surface appear only various lines, arcs, and swirls etched into the plastic. Yet a complex transformation, carried out operationally by shining a laser through the plastic, turns those markings into a recognizable three-dimensional image. Which means that the plastic hologram and the three-dimensional image embody the same data, even though the information in one is unrecognizable from the perspective of the other. Similarly, examination of the quantum field theory on the boundary of Maldacena’s universe shows that it bears no obvious resemblance to the string theory inhabiting the interior. If a physicist were presented with both theories, not being told of the connections we’ve now laid out, he or she would more than likely conclude that they were unrelated. Nevertheless, the mathematical dictionary linking the two—functioning as a laser does for ordinary holograms—makes explicit that anything taking place in one has an incarnation in the other. At the same time, examination of the dictionary reveals that just as with a real hologram, the information in each appears scrambled on translation into the other’s language.
As a particularly impressive example, Witten investigated what an ordinary black hole in the interior of Maldacena’s universe would look like from the perspective of the boundary theory. Remember, the boundary theory does not include gravity, and so a black hole necessarily translates into something very unlike a black hole. Witten’s result showed that much as the Wizard of Oz’s frightening visage was produced by an ordinary man, a rapacious black hole is the holographic projection of something equally ordinary: a bath of hot particles in the boundary theory (Figure 9.6). Like a real hologram and the image it generates, the two theories—a black hole in the interior and a hot quantum field theory on the boundary—bear no apparent resemblance to each other, and yet they embody identical information.*
In Plato’s parable of the cave, our senses are privy only to a flattened, diminished version of the true, more richly textured, reality. Maldacena’s flattened world is very different. Far from being diminished, it tells the full story. It’s a profoundly different story from the one we’re used to. But his flattened world may well be the primary narrator.
Parallel Universes or Parallel Mathematics?
Maldacena’s result, and the many others it has spawned in the years since, is deemed conjectural. Because the mathematics is tremendously difficult, fashioning an airtight argument remains elusive. But the holographic ideas have been subject to a great many stringent mathematical tests; having come through unscathed, they’ve been propelled into mainstream thought among physicists searching for the deep roots of natural laws.
One factor contributing to the difficulty of rigorously proving that the boundary and bulk worlds are disguised versions of one another highlights why the result, if true, is so powerful. I described in Chapter 5 how physicists more often than not rely on approximation techniques, the perturbative methods that I outlined (recall the lottery example with Ralph and Alice). I also emphasized that such methods are accurate only if the relevant coupling constant is a small number. In analyzing the relationship between quantum field theory on the boundary and string theory in the bulk, Maldacena realized that when the coupling of one theory was small, that of the other was large, and vice versa. The natural test, and a possible means of proving that the two theories are secretly identical, is to perform independent calculations in each theory and then check for equality. But this is difficult to do, since when perturbative methods work for one, they fail for the other.16
However, if you accept Maldacena’s more abstract argument, as outlined in the previous section, the perturbative vice becomes a calculational virtue. Much as we found with the string dualities in Chapter 5, the bulk-boundary dictionary translates daunting calculations, beset by a large coupling, in one framework into straightforward calculations, with a small coupling in the other. In recent years, this has been parlayed into results that may be experimentally testable.
At the Relativistic Heavy Ion Collider (RHIC) in Brookhaven, New York, gold nuclei are slammed into each other at just shy of light speed. Because the nuclei contain many protons and neutrons, the collisions create a commotion of particles that can be more than 200,000 times as hot as the sun’s core. That’s hot enough to melt the protons and neutrons into a fluid of quarks and the gluons that act between them. Physicists have exerted great effort to understand this fluidlike phase, called the quark gluon plasma, because it’s likely that matter briefly assumed this form soon after the big bang.
The challenge is that the quantum field theory (quantum chromodynamics) describing the hot soup of quarks and gluons has a large value for its coupling constant, and that compromises the accuracy of perturbative methods. Ingenious techniques have been developed to skirt this hurdle, but experimental measurements continue to controvert some of the theoretical results. For example, as any fluid flows—be it water, molasses, or the quark gluon plasma—each layer of the fluid exerts a drag force on the layers flowing above and below. The drag force is known as shear viscosity. Experiments at RHIC measured the shear viscosity of the quark gluon plasma, and the results are far smaller than those predicted by the perturbative quantum field theory calculations.
Here’s a possible way forward. In introducing the holographic principle, the perspective I’ve taken is to imagine that everything we experience lies in the interior of spacetime, with the unexpected twist being processes, mirroring those experiences, which take place on a distant boundary. Let’s reverse that perspective. Imagine that our universe—or, more precisely, the quarks and gluons in our universe—lives on the boundary, and so that’s where the RHIC experiments take place. Now invoke Maldacena. His result shows that the RHIC experiments (described by quantum field theory) have an alternative mathematical description in terms of strings moving in the bulk. The details are involved but the power of the rephrasing is immediate: difficult calculations in the boundary description (where the coupling is large) are translated into easier calculations in the bulk description (where the coupling is small).17
Pavel Kovtun, Andrei Starinets, and Dam Son did the math, and the results they found come impressively close to the experimental data. This pioneering work has motivated an army of theoreticians to undertake many other string theory calculations in an effort to make contact with RHIC observations, driving forward a vigorous interplay between theory and experiment—a welcome novelty for string theorists.
Bear in mind that the boundary theory doesn’t model our universe fully since, for example, it doesn’t contain the gravitational force. This doesn’t compromise contact with RHIC data because in those experiments the particles have such small mass (even when traveling near light speed) that the gravitationa
l force plays virtually no role. But it does make clear that in this application string theory is not being used as a “theory of everything”; instead, string theory provides a new calculational tool for breaking through obstacles that have impeded more traditional methods. Conservatively, analyzing quarks and gluons by using a higher dimensional theory of strings can be viewed as a potent string-based mathematical trick. Less conservatively, one can imagine that the higher dimensional string description is, in some yet to be understood way, physically real.
Regardless of perspective, conservative or not, the resulting confluence of mathematical results with experimental observations is extremely impressive. I am not a fan of hyperbole, but I view these developments as among the most exciting advances in decades. Mathematical manipulations that utilize strings moving through a particular ten-dimensional spacetime tell us something about quarks and gluons living in a four-dimensional spacetime—and the “something” the calculations tell us seems to be borne out by experiments.
Coda: The Future of String Theory
The developments we’ve covered in this chapter transcend evaluations of string theory. From Wheeler’s emphasis on analyzing the universe in terms of information, to the recognition that entropy is a measure of hidden information, to the reconciliation between the Second Law of Thermodynamics and black holes, to the realization that black holes store entropy on their surface, to the understanding that black holes set a maximum for the amount of information that can occupy a given region of space, we’ve followed a winding road across many decades and traversed an intricate web of results. The journey has been full of remarkable insights, and has led us to a new unifying idea—the holographic principle. The principle, as we’ve seen, suggests that the phenomena we witness are mirrored on a thin, distant bounding surface. Looking to the future, I suspect that the holographic principle will be a beacon for physicists well into the twenty-first century.