Shadow World

How many dimensions space has could all be a matter of perspective

In a school of thought that teaches the existence of extra dimensions, Juan Maldacena may at first sound a little out of place. String theory is physicists’ still-tentative strategy for reconciling Einstein’s theory of gravitation with quantum physics. Its premise is that the subatomic particles that roam our three-dimensional world are really infinitesimally thin strings vibrating in nine dimensions. According to Maldacena, however, the key to understanding string theory is not to add more dimensions but to cut their number down.

A pattern takes on different appearances depending on the shape of the surface it covers. A 10-year-old mathematical conjecture suggests a similar link between laws of physics as they are perceived in universes with different geometries, even different numbers of dimensions. A. Sandberg
YOU’RE JUST PROJECTING. Universes with different physical laws and even different numbers of dimensions could have an underlying mathematical equivalence, much as a single pattern in this artist’s impression looks different when projected onto surfaces with different geometries. A. Sandberg
SLIPPERY LIQUID. In this computer simulation, collision of two gold nuclei creates a quark-gluon plasma whose behavior mathematically resembles that of seismic waves on the surface of a black hole. Brookhaven National Laboratory

In his vision, the mathematical machinery of strings completely translates into a more ordinary quantum theory of particles, but one whose particles would live in a universe without gravity. Gravity would be replaced by forces similar to the nuclear forces that prevailed in the universe’s first instants. And this would be a universe with fewer dimensions than the realm inhabited by strings.

Just as a hologram creates the illusion of the third dimension by scattering light off a 2-D surface, gravity and the however many dimensions of space could be a higher-dimensional projection of a drama playing out in a flatter world.

In physics parlance, the two theories would be dual to each other—two mathematically equivalent languages for describing the same reality. Physicists could study each phenomenon using whichever language that makes it easier to understand.

Maldacena first presented his conjecture in November 1997, and it quickly became a leading theme in string theory research. Ten years later, physicists still don’t have proof of it, though many have tried and thousands of papers have been written. But hints have been accumulating, and recently experts have found “very strong evidence” that the conjecture is true, says Maldacena, now at the Institute for Advanced Study in Princeton, N.J.

Meanwhile, the work by Maldacena and others has helped clarify a nagging paradox about black holes, gravity’s most extreme phenomena, by translating the problem into ordinary quantum theory. Physicists have also used the dictionary in reverse, turning problems about real-world particles such as quarks into questions about how seismic waves shake black holes. Surprisingly, the black hole calculations have often turned out to be more manageable than the original form of the problem.

But the most important fallout from Maldacena’s intuition has probably been on the field of string theory itself. His work has offered physicists hope that they can make the string idea rigorous by tracing its roots to ordinary quantum physics. Maldacena’s conjecture has energized string theory advocates, occupying the center of a confluence of ideas coming from several branches of physics. “It’s the most incredible discovery in theoretical physics in the last 20 years,” says Harvard University’s Nima Arkani-Hamed.

Stone-cold genius

In 1997, Maldacena was contemplating a stubborn paradox having to do with black holes. Stephen Hawking of the University of Cambridge in England had long ago calculated that black holes would slowly evaporate, eventually disappearing in a burst of gamma rays. Apparently, no record would survive of the shape, size, or history of all the stuff that had fallen into a black hole.

But quantum mechanics does not allow information to be erased from the universe. Physical processes leave traces that could in principle be reversed to reconstruct the past, if accepted principles of quantum theory are correct. But perhaps, Hawking and others suggested, ordinary quantum theory breaks down inside a black hole.

Maldacena attacked the paradox using string theory. But instead of using the extra elbow room afforded by six additional dimensions, he took the opposite approach, suggesting that gravitational phenomena in a stringy universe—including black holes—can have a representation in terms of particles.

So if the quantum physics of particles—where nothing can destroy information—can completely encapsulate the physics of black holes, then a black hole cannot destroy information either. There would have to be some other explanation for Hawking’s paradox, but at least the foundations of quantum theory should be safe.

In 2004 , spurred in part by Maldacena’s work, Hawking admitted that he had changed his mind, and stated that black holes probably don’t destroy information after all (SN: 9/25/04, p. 202).

Maldacena first posted his proposal online in November 1997, barely a year after earning his Ph.D. degree at Princeton University. Within a few weeks, some of the leading string theory experts, including Edward Witten of the Institute for Advanced Study and Igor Klebanov of Princeton University, helped write a more explicit dictionary for Maldacena’s duality. By the following June, when physicists met for a string theory conference in Santa Barbara, Calif., many were already unraveling the implications of Maldacena’s idea.

At the meeting’s banquet, physicists sang and danced to a song entitled “The Maldacena,” a spoof of the then-popular “Macarena.” “In some ways, it really took over the field,” Klebanov says of the conjecture, which another leading researcher calls the work of a “stone-cold genius.”

The sky’s the limit

Since 1997, physicists have proposed countless variations on Maldacena’s theme, all of which interpret a string as a swarm of particles living in a small number of dimensions. Perhaps the easiest case to visualize is when that number is two. In such a scenario, anything that takes place in your many-dimensional, stringy universe has a sort of shadow representation in terms of particles moving on that universe’s “sphere at infinity.” This esoteric-sounding concept is actually similar to the familiar celestial sphere of the night sky as seen from Earth: It’s the two-dimensional surface spanning all possible directions one can point to infinitely far in space.

But on the face of it, neither of the universes involved in the duality has anything even remotely to do with the actual physical world. At one end of the duality are particles living in, say, two dimensions. The physics they obey, called conformal field theory, is vaguely similar to the physics of quarks, but not quite the same. The strong nuclear force between real quarks actually gets relatively weak when the quarks get extremely close to each other. But in conformal field theory, forces are the same at any distance.

At the other end is a stringy universe that has an eternal tendency to contract (even though it doesn’t get any smaller because it’s infinitely large to begin with). That’s quite the opposite from the universe in which we live, which seems to contain a sort of antigravity called dark energy that makes the universe expand at an accelerating pace (SN: 1/3/98, p. 4).

Unfortunately, the equations of conformal field theory seem a good match only for the mathematics of strings living in a contracting universe. Still, many physicists remain hopeful that they will find an appropriate version of the duality that will do the trick for a universe like ours. If proved true, such a correspondence would offer a road map for building a complete string theory for the laws of nature.

Aside from the need to find a way of testing their ideas with experiments, string theorists’ ultimate goal is to reconcile Einstein’s theory of gravity with quantum physics. Gravity is the only fundamental force of nature that hasn’t been “quantized,” or subjected to the weird rules of quantum theory. As Arkani-Hamed puts it, if we lived in an eternally contracting universe, “the problem of quantizing gravity would have been solved.”

Soon after Maldacena’s first proposal, physicists realized that his duality could already shed light on the real world. For example, physicists believe that Maldacena’s arguments on black holes, while formulated for the black holes of a contracting universe, are probably also relevant to black holes living in a universe like ours. In that case, a problem that seemed intractable on the strings side became much easier on the particles side. But the converse can also happen.

Black hole near New York!

When physicists smash heavy atomic nuclei together with sufficient energy, the atoms’ protons and neutrons break up. For less than a sextillion of a second they melt into a blob called a quark-gluon plasma. It’s similar to the state of all matter in the first microseconds after the big bang.

Beginning in 2000, Dam Son, now at the University of Washington in Seattle, and his collaborators wanted to calculate a quark-gluon plasma’s viscosity—roughly speaking, a measure of how quickly the plasma will dampen turbulence within it. In principle, one should be able to do such calculations using the known equations of particle physics. When quarks are not bound together, though, those equations become extremely hard to solve.

But in a quark-gluon plasma, quarks will experience extremely intense forces, whose strength does not vary appreciably as the particles move. That makes the plasma’s behavior a good approximation of the conformal field theory that rules Maldacena’s sphere at infinity. Starting from that assumption, Son showed that Maldacena’s duality translates the physics of plasma turbulence into that of black hole earthquakes.

A gravitational disturbance, Son says, will alter a black hole’s shape, which is otherwise that of a perfect sphere. In response, the black hole will “oscillate, radiate energy, and settle down to be spherical again.” Son and his collaborators calculated how quickly the seismic waves on the black hole’s surface will dampen down. Translated back, the calculation suggested that the viscosity of a quark-gluon plasma could be much smaller than physicists thought possible.

Initially, some nuclear physicists were nonplussed, to say the least, about the idea of doing nuclear physics using black holes. “The first time I heard about it, I literally thought it was crazy,” says William Zajc of Columbia University in New York City.

In 2005, however, physicists at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory in Upton, N.Y., announced the results of an experiment that collided nuclei of gold atoms, melting them into a quark-gluon plasma (SN: 4/23/05, p. 259). The stuff’s viscosity seemed close to Son’s prediction, says Larry McLerran, a RHIC (pronounced “rick”) theorist.

Many physicists working at RHIC—Zajc being one of them—changed their minds about Son’s calculation. “It’s far more useful than we ever imagined,” he says. “The fact that it was done in some higher-dimensional space and it involved black holes—well, that just added to the intrigue.”

Since then, some of the RHIC physicists have revisited certain theoretical assumptions used to interpret the experiment’s data. As a result, some say it’s no longer so clear that the viscosity is as low as Son claimed it could be. Not everyone buys the black hole model of a quark-gluon plasma. “It’s certainly interesting, but you have to be very skeptical about it,” he says.

More recently, Subir Sachdev of Harvard University and his team have extended Son’s ideas to study transitions between certain exotic—but real—states of matter. As Sachdev and coauthors describe in the October Physical Review B, the team applied its new methods to the motion of electrons inside a superconductor when the temperature goes up just enough that the material becomes an electrical insulator. Instead of estimating viscosity, as Son did, the researchers calculated how long it will take for vortices of electrons to stop whirling. In their case, Sachdev says, the relevant dual phenomenon was the damping of electromagnetic disturbances that ensue when a photon falls into a black hole.

Ariadne’s thread

The power of the string—particle duality, Maldacena says, lies in the fact that one can frame a problem in whichever mathematical language makes it easier to solve.

Calculations about particles are more manageable when the particles interact weakly. But the duality translates strongly interacting particles into weakly interacting strings. “When one of the descriptions becomes hard, the other one becomes easy, and vice versa,” Maldacena says.

At least that is the prevailing belief, even though it has not been rigorously proved. In all cases in which physicists have been able to calculate two dual quantities independently, they got the same result, which is encouraging. But until recently, in all those examples the interaction strengths were at the extremes—infinitesimally small or infinitely large.

In the past 2 years, Niklas Beisert, now at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) in Potsdam, Germany, and his collaborators have found the first examples that work at all possible interaction strengths. “If this was the theory of the real world, we would in some sense describe the mass of the proton and of all other composite particles,” he says. What they found is that the two theories make the same predictions for those values. The calculations have created a kind of Ariadne’s thread that can be followed from one theory to the other.

“The work they did is really wonderful,” Maldacena says. “It’s an incredible test” for Maldacena’s conjecture, says Klebanov, who recently helped corroborate the results with numerical calculations. Still, the conjecture “certainly hasn’t been proven in mathematical terms,” Beisert warns. However, most experts now say they are virtually sure that it eventually will be.

But even if Maldacena’s conjecture is true, does it mean that string theory is correct? Most string theorists would bet on it. It would be too much of a coincidence, they say, if such a seemingly miraculous mathematical duality were to apply to a particular kind of abstract universe but not to our own. “I believe that nature uses the same small set of ideas over and over,” says Joseph Polchinski of the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara.

Others are not so sure, and point out that there have been times in history when physicists have promoted hypotheses on the basis of their aesthetic appeal, only to be contradicted by the experimental evidence. A classic example, says Abhay Ashtekar of Pennsylvania State University in University Park, is Lord Kelvin’s idea of vortices. In the 1860s, Kelvin pointed out that many of the known properties of chemical elements could arise naturally if atoms were knotted vortices in the fabric of the ether. The uncanny coincidence went away once physicists demonstrated that the ether probably didn’t exist.

For now, Maldacena’s duality ideas have become an engine for motivating and inspiring string theory research. “It’s been a very good run,” Klebanov says. “But we’re still just kind of scratching the surface.”

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