Why Beauty is Truth

by Ian Stewart

  The common ancestor of elephants and wombats is more recent than that of, say, elephants and swans. So this divergence constitutes the most recent branching of the tree of these four species. Before that, the common ancestor of elephants and wombats split off from some ancestor of the swan. Earlier still, the common ancestor of these three species split from that of the gnat.

  Speciation can be viewed as a kind of symmetry-breaking. A single species is (approximately) symmetric under any permutation of its organisms; every wombat resembles every other wombat. When there are two distinct species—wombats and elephants—you can permute the wombats among themselves, and permute the elephants, but you can’t change an elephant into a wombat without someone noticing.

  The physicists’ explanation of the underlying unity of the four forces is similar. The role of DNA, however, is played by the temperature of the universe—that is, its energy level. Although the underlying laws of nature are the same at all times, they lead to different behavior at different energies—just as the same laws cause water to be solid at low temperatures, liquid at medium ones, and a gas at high ones. At very high temperatures, the water molecules break up to form a plasma, composed of separate particles. At higher temperatures still, the particles themselves break to form a quark–gluon plasma.

  How four species diverge as time passes.

  How the four fundamental forces diverge as time passes.

  When the universe first came into being at the Big Bang, 13 billion years ago, it was enormously hot. At first, all four forces acted in exactly the same way. But as the universe cooled, its symmetry broke, and the forces split into individuals with distinguishable characteristics. Our present universe, with its four forces, is an imperfect shadow of that elegant original—the result of three broken symmetries.

  14

  THE POLITICAL JOURNALIST

  In June 1972, during the run-up to the U.S. presidential election, a security guard at the Watergate complex noticed that a door had been taped open. He removed the tape, thinking it must have been left accidentally by workmen, but when he returned, someone had put it back. His suspicions aroused, the guard informed the police, who caught five men breaking into the offices of the Democratic Party’s national committee. It turned out that the men were associated with President Nixon’s reelection committee.

  The discovery had little effect on the election itself; Nixon won in a landslide. But the story wouldn’t go away, and slowly the tentacles of the Watergate affair reached higher and higher in the Nixon administration. Two reporters from the Washington Post, Bob Woodward and Carl Bernstein, pursued the story with dogged persistence, assisted by the clandestine revelations of “Deep Throat.” No one knew who he was, but it was clear that he had to be a very senior official. In 2005, Deep Throat was revealed as Mark Felt, the second-in-command in the Federal Bureau of Investigation.

  The information that Deep Throat leaked to the press was dynamite. By April 1974, Nixon had been forced to ask for the resignations of two senior aides. Then it turned out that the president had bugged his own office, and there were recorded tapes of sensitive conversations. After a legal battle to secure access to the tapes, gaps were found in some of the recordings, apparently the result of deliberate erasure.

  The attempt to cover up the relation between the burglary and the White House was almost universally perceived as a worse crime than the burglary itself. The House of Representatives began a formal process that could lead to the president being impeached—tried for “high crimes and misdemeanors” before the U.S. Senate, and if found guilty, removed from office. When impeachment and conviction became inevitable, Nixon resigned.

  Nixon’s opponent in the election was Senator George McGovern. Announcing his candidacy for the Democratic nomination in Sioux Falls, South Dakota, McGovern made some prophetic remarks:

  Today, our citizens no longer feel that they can shape their own lives in concert with their fellow citizens. Beyond that is the loss of confidence in the truthfulness and common sense of our leaders. The most painful new phrase in the American political vocabulary is “credibility gap”—the gap between rhetoric and reality. Put bluntly, it means that people no longer believe what their leaders tell them.

  Among the minor figures in McGovern’s campaign was a would-be political journalist whose career would probably have taken off had Mc-Govern been elected. In that variant of history, politics might have been richer, but fundamental physics and advanced mathematics would have been much the poorer. In the year 2004 of the history that actually happened, the journalist was listed by Time magazine as one of the year’s one hundred most influential people—but not for his journalism.

  Instead, he was listed for his groundbreaking contributions to mathematical physics. He is responsible for some of the most original mathematics in the world—for which he won the Fields Medal, the top honor in mathematics, comparable in prestige to a Nobel Prize—but he is not a mathematician. He is one of the world’s leading theoretical physicists, and was awarded the National Medal of Science, but his first degree was in history. And he is the prime mover, though not quite the original creator, of the current front-runner in the effort to unify the whole of physics. He is the Charles Simonyi professor of mathematical physics at the Institute for Advanced Study in Princeton, where Einstein used to work, and his name is Edward Witten.

  Like the great German quantum theorists but unlike poor Dirac, Witten grew up in an intellectual environment. His father, Louis Witten, is also a physicist, working on general relativity and gravitation. Edward was born in Baltimore, Maryland, and studied for his first degree at Brandeis University. After Nixon’s reelection, he went back to academic life, taking a PhD at Princeton University, and embarked on a career of research and teaching at various American universities. In 1987 he was appointed to the Institute for Advanced Study, where all academic positions focus purely on research, and this is where he currently works.

  Witten started research in quantum field theory, the first fruits of efforts to reconcile quantum theory with relativity. Here relativistic effects of motion are taken into account, but only in flat space-time. (Gravity, which requires curved space-time, is not considered.) In 1998, in a Gibbs lecture, Witten said that quantum field theory “encompasses most of what we know of the laws of physics, except gravity. In its seventy years there have been many milestones, ranging from the theory of ‘antimatter’ . . . to a more precise description of atoms . . . to the ‘standard model of particle physics.’” He pointed out that having been developed largely by physicists, much of it lacked mathematical rigor and so had had little impact on mathematics as such.

  The time was ripe, Witten said, to remedy that shortcoming. Several major areas of pure mathematics were effectively quantum field theory in disguise. Witten’s own contribution, the discovery and analysis of “topological quantum field theories,” had a direct interpretation in terms of concepts that various pure mathematicians had invented in quite different settings. These included the English mathematician Simon Donaldson’s epic discovery that four-dimensional space is unique in supporting many different “differentiable structures”—coordinate systems in which calculus can be carried out. Other aspects are a recent breakthrough in knot theory, known as the Jones polynomial, a phenomenon called “mirror symmetry” in multidimensional complex surfaces, and several areas of modern Lie theory.

  Witten made a bold prediction: a major theme in twenty-first-century mathematics would be the integration of ideas from quantum field theory into the mathematical mainstream:

  One has here a vast mountain range, most of which is still covered with fog. Only the loftiest peaks, which reach above the clouds, are seen in the mathematical theories of today, and these splendid peaks are studied in isolation . . . Still lost in the mist is the body of the range, with its quantum field theory bedrock and the great bulk of the mathematical treasures.

  Witten’s Fields Medal celebrated his uncovering of a few of those hidden t
reasures. Among them was a new and improved proof of the “positive mass conjecture,” to the effect that a gravitational system with positive local mass density must have positive total mass. It may sound obvious, but in the quantum world mass is a subtle concept. The proof of this long-sought result, published by Richard Schoen and Shing-Tung Yau in 1979, had earned Yau a Fields Medal in 1982. Witten’s new, improved proof exploited “supersymmetry,” the first application of that concept to a significant problem in mathematics.

  We can understand supersymmetry in terms of an old puzzle, which asks for a cork that can fit into a bottle whose opening may be circular, square, or triangular. Amazingly, such shapes do exist, and the traditional answer is a cork with a circular base that tapers like a wedge. Viewed from below, it looks like a circle; viewed from the front it is a square; viewed from the side it is a triangle. A single shape can perform all three tasks because a three-dimensional object can have several different “shadows,” or projections, in different directions.

  Now, imagine a Flatlander living on the “floor” of my picture, able to observe the projection of the cork onto the floor but unaware of the other projections. One day he discovers to his amazement that the circular shape has somehow morphed into a square. How can that be? It’s certainly not a symmetry.

  Not in Flatland. But while the Flatlander’s back was turned, someone living in three dimensions rotated the cork so that its projection onto the floor changed to a square. And rotation is a symmetry transformation in three dimensions. So a symmetry in a higher dimension can sometimes explain a rather baffling transformation in a lower dimension.

  How supersymmetry works. Left: A cork to fit three shapes of hole. Right: Effect of rotating the cork.

  Something very similar happens in supersymmetry, but instead of changing circles into squares, it changes fermions into bosons. This is amazing. It means that you can do calculations with fermions, hit everything with a supersymmetry operation, and deduce results for bosons with no extra effort. Or the other way round.

  We expect this kind of thing to happen with genuine symmetries. If you stand in front of a mirror and juggle several balls, then whatever happens on your side of the mirror completely determines what happens on the other side. There, an image of you juggles images of the balls. If it takes 3.79 seconds to complete one sequence of juggles on the real side of the mirror, you know without doing the measurements that it will also take 3.79 seconds to complete the corresponding sequence of juggles on the other side. The two situations are related by a reflectional symmetry; whatever happens in one also happens, reflected, in the other.

  Supersymmetries are not as obvious as this, but they have a similar effect. They let us deduce features of one type of particle from features of an entirely different type of particle. It is almost as if you could reach into some higher-dimensional region of the universe and twist a fermion into a boson. Particles come in supersymmetric pairs: an ordinary particle is matched with its twisted version, called a sparticle. Electrons are paired with selectrons, quarks with squarks. For historical reasons the photon is twinned not with the sphoton but with the photino. There is a kind of “shadow world” of sparticles that interacts only weakly with the ordinary world.

  This idea makes for elegant mathematics, but the masses of these predicted shadow particles are too great for them to be observed in experiments. Supersymmetry is beautiful, but it may not be true. But even though direct confirmation is out of the question, indirect confirmation is still possible. Science mainly checks theories through their implications.

  Witten pursued supersymmetry vigorously, and in 1984 he wrote an article titled “Supersymmetry and Morse theory.” Morse theory is an area of topology, named for the pioneer Marston Morse, that relates the overall shape of a space to its peaks and valleys. Sir Michael Atiyah, probably Britain’s most distinguished living mathematician, described Witten’s paper as “obligatory reading for geometers interested in understanding modern quantum field theory. It also contains a brilliant proof of the classic Morse inequalities . . . The real aim of the paper is to prepare the ground for supersymmetric quantum field theory [in terms of] infinite-dimensional manifolds.” Subsequently, Witten applied these techniques to other hot topics at the frontiers of topology and algebraic geometry.

  It should be obvious that when I said Witten is not a mathematician, I did not mean he lacks mathematical ability—quite the reverse. Arguably no one on the planet has more mathematical ability. But in Witten’s case it is complemented by an amazing physical intuition.

  Unlike mathematicians, physicists are seldom shy about employing physical intuition to paper over any gaps in mathematical logic. Mathematicians have learned to regard leaps of faith with suspicion, however strong the supporting evidence may be: to them, proof is all. Witten is unusual in that he can relate his intuition to mathematics as mathematicians understand it. Atiyah puts it like this: “His ability to interpret physical ideas in mathematical form is quite unique. Time and again he has surprised the mathematical community by his brilliant application of physical insight leading to new and deep mathematical theorems.”

  But there is a flipside to this intuitive prowess. Many of Witten’s most important ideas, being derived from physical principles or analogies, were arrived at without proofs, and some still lack proofs today. It’s not that he can’t do proofs—as his Fields Medal demonstrates—but he can make leaps of logic that lead to deep and correct mathematics without seeming to need proofs.

  The big question is, does Witten’s wonderfully elegant mathematics have anything to do with fundamental physics? Or has the search for beauty headed down a mathematical dead end, losing any connection with physical truth?

  By 1980, physicists had unified three of the four forces of nature: electromagnetic, weak, and strong. But Grand Unified Theories said nothing about gravity. The force that we experience most directly in everyday life, which literally keeps our feet on the ground, was embarrassingly absent from the synthesis.

  It was easy enough to write down combined theories of gravity and quantum theory that looked sensible. But whenever anyone tried to solve the resulting equations, they got nonsense. Typically, numbers that ought to represent reasonable physical quantities were infinite. An infinity in a physical theory is a sign that something is wrong. It was an infinity in the radiation law that inspired Planck to quantize light.

  Some physicists became convinced that the main source of the infinities was the ingrained habit of treating particles as points. A point—location without size—is a mathematical fiction. Quantum particles were probabilistic fuzzed-out points, but that didn’t cure the disease; something more drastic was needed. Even in the 1970s a few pioneers had begun to think that particles might more sensibly be modeled as tiny vibrating loops—“strings.” In the 1980s, when supersymmetry got in on the act, these mutated into superstrings.

  One could write an entire book about superstrings, and several people have, but we can manage with a rough hand-waving description. I want to focus on four features: the way relativistic and quantum pictures are combined, the need for extra dimensions, the interpretation of quantum states as vibrations in those extra dimensions, and the symmetries of the extra dimensions—or, more accurately, of various fields that live in them.

  Our starting point is Einstein’s idea of representing the trajectory of a particle in space-time as a curve, which he called its world line. Essentially, this is the curve that the particle traces in space-time as it moves. In relativity, world lines are smooth curves, because of the form of Einstein’s field equations. They do not branch, because in relativity the future of any system is completely determined by its past, indeed by its present.

  There is an analogous concept in quantum field theory called a Feynman diagram. Feynman diagrams depict the interaction of particles in a rather schematic space-time. For example, the left-hand picture (next page) is the Feynman diagram for an electron that emits a photon that is then captured by a se
cond electron. It is traditional to use wiggly lines for photons.

  The Feynman diagram is a bit like a relativistic world line, but it has sharp corners and it branches. In 1970, it occurred to Yoichiro Nambu that if the assumption that particles are points is replaced by the assumption that they are tiny loops, then Feynman diagrams can be converted into smooth surfaces—worldsheets—as in the right-hand picture. A worldsheet can be interpreted as a world line in a modified space-time, with an extra dimension for the loops to live in.

  (Left) Feynman diagram for interacting particles. (Right) The corresponding worldsheet, sliced into strings.

  The great thing about loops—aside from not being points—is that they can vibrate. Perhaps each vibrational pattern corresponded to a quantum state. That would explain why quantum states come in whole-number multiples of some basic quantity—for example spin, which is always an integer multiple of ½. The number of waves that fit into the loop has to be a whole number. In a violin string, these different patterns are the fundamental note and its higher harmonics. So quantum theory becomes a kind of music, played with superstrings instead of violin strings.

  Nambu’s idea did not come out of the blue. It had its roots in a remarkable formula derived by Gabriele Veneziano in 1968, which showed that apparently distinct Feynman diagrams represent the same physical process, and that any failure to take that into account leads to wrong answers in quantum field theory calculations. Nambu noticed that when the Feynman diagram is surrounded by tubes, different diagrams yield networks of tubes with the same topology. That is, these networks can be deformed into each other. So Veneziano’s formula seemed to be related to the topological properties of the tubes.