Why Beauty is Truth

by Ian Stewart

  Strings poke out of ordinary space-time into new dimensions.

  This, in turn, hinted that quantum particles, with their discrete quantum numbers like charge, might be topological features of a smooth space-time. Mathematicians had already observed the tendency of basic topological properties—such as the number of holes in a surface—to be discrete. It all seemed to fit. But as always, the devil was in the detail, and the detail was devilish. String theory was the first attempt to get the detail in agreement with the real world.

  String theory did not start out as a possible route to a Theory of Everything but as a proposal to explain the particles collectively known as hadrons. These include most of the common particles found in the atomic nucleus, such as the proton and neutron, together with a host of more exotic particles. However, the theory had a flaw: it predicted the existence of a particle with zero mass and spin 2, which had never been observed (and still hasn’t). Additionally, it failed to predict any particles with spin ½—and rather a lot of hadrons, including the proton and neutron, have spin ½. It was like a midsummer weather forecast that predicts hailstones a foot across but has nothing to say about whether the temperature will be warm. Physicists were unimpressed. In 1974, when quantum chromodynamics came along and explained all known hadrons, and even successfully predicted a new one, the omega-minus, the fate of string theory seemed sealed.

  At that point, however, John Schwarz and Joel Scherk noticed that string theory’s unwanted zero-mass spin-2 particle might be the long-sought graviton, the hypothetical particle believed to carry the force of gravity. Might string theory be a quantum theory of gravity, rather than hadrons? If so, it would be an attractive contender for a Theory of Everything—well, for a Theory of Many Things, because there are many particles that are not hadrons.

  At this point, supersymmetry came into play, because it converts fermions into bosons. Hadrons include particles of both kinds, but other particles, such as the electron, are not hadrons. If supersymmetry could be incorporated into string theory, then a number of new particles would automatically come within the theory’s grasp—carried along by super-symmetric partners that were already part of the theory.

  The combined theory, developed by Pierre Ramond, André Neveu, and Schwarz, was superstring theory. This theory did include spin-½ particles, and it eliminated a nasty feature of ordinary string theory, a particle that goes faster than light. The presence of such a particle in a theory is now seen as evidence that it is unstable, which rules it out.

  From 1980 onward, Michael Green, a British theoretical physicist, worked out more and more of the mathematics of superstrings, using techniques from Lie group theory and topology, and it quickly became clear that whatever its physical credentials, superstring theory possessed extraordinary mathematical beauty. The physics remained obstinate: in 1983, Luis Alvarez-Gaume and Witten discovered a new snag with string theories, including superstrings and even good old quantum field theory. Namely, these theories normally possess anomalies. An anomaly occurs when the process of converting a classical system to its quantum analogue changes an important symmetry.

  Green and Schwarz had discovered that very occasionally, the anomalies miraculously disappear, but only if space-time has 26 dimensions (in the first version of the theory, called bosonic string theory) or 10 dimensions (in later modifications). Why? In their calculations for bosonic string theory, the mathematical terms that would create an anomaly are multiplied by d – 26, where d is the dimension of space-time. So these terms vanish precisely when d = 26. Similarly, in the modified version, the factor becomes d –10. Time always remains one-dimensional, but space somehow acquires an extra 6 or 22 dimensions. Schwarz put it this way:

  In 1984 Michael Green and I did a calculation for one of these superstring theories to see whether, in fact, this anomaly occurred or not. What we discovered was quite surprising to us. We found that, in general, there was indeed an anomaly that rendered the theory unsatisfactory. Now there was freedom to choose the particular symmetry structure that one used in defining a theory in the first place. In fact, there were an infinite number of possibilities for these symmetry structures. However, for just one of them the anomaly magically cancelled out of the formulae, whereas for all the others it didn’t. So amid this infinity of possibilities, just one unique one was being picked out as being potentially consistent.

  If you were prepared to ignore the weird numbers 10 or 26, this discovery was very exciting. It suggested that there might be a mathematical reason for space-time to have a particular number of dimensions. It was disappointing that the number was not four, but it was a start. Physicists had always wondered why space-time has the dimensions it does; now it looked as though there might be a better answer to that question than, “well, it could be anything, but in our universe it’s four.”

  Perhaps other theories would lead to a four-dimensional space-time. It would have been ideal, but nothing along those lines seemed to work, and the funny dimensions refused to go away. So maybe they were there. This was an old idea of Kaluza’s: space-time might have extra dimensions that we are unable to observe. If so, the strings would remain one-dimensional loops, but those loops would vibrate in an otherwise invisible higher-dimensional space. The quantum numbers associated with the particles, like charge or charm, would be determined by the form of the vibrations.

  A basic question was, what do the hidden dimensions look like? What shape is space-time?

  At first, physicists hoped the extra dimensions would form some simple shape like the 6-dimensional analogue of a torus. But in 1985, Philip Candelas, Gary Horowitz, Andrew Strominger, and Witten reasoned that the most suitable shape would be a so-called Calabi–Yau manifold. There are tens of thousands of these shapes; here is a typical one:

  A Calabi–Yau manifold (schematic).

  Credit: Andrew J. Hanson, Professor and Chair, Indiana University.

  The great advantage of Calabi–Yau manifolds is that the supersymmetry of 10-dimensional space-time is inherited by the ordinary four-dimensional space-time that underlies it.

  For the first time, the exceptional Lie groups were taking on a prominent role in frontier physics, and this trend accelerated. Around 1990, there seemed to be five possible types of superstring theory, all with space-time dimension equal to 10. The theories are called Type I, Types IIA and IIB, and “heterotic” types HO and HE. Interesting gauge symmetry groups turn up; for example, in Types I and HO we find SO(32), the rotation group in 32-dimensional space, and in Type HE the exceptional Lie group E8 turns up as E8 × E8, two distinct copies acting in two different ways.

  The exceptional group G2 also makes an appearance in the latest twist to the story, which Witten calls M-theory. The “M,” he says, stands for magic, mystery, or matrix. M-theory posits an 11-dimensional space-time, which unifies all five of the 10-dimensional string theories, in the sense that each can be obtained from M-theory by fixing some of its constants to particular values. In M-theory, Calabi–Yau manifolds are replaced by 7-dimensional spaces known as G2-manifolds, because their symmetries are closely related to Killing’s exceptional Lie group G2.

  At the moment there is a bit of a backlash against string theory; not on the grounds that it is known to be wrong, but on the grounds that it’s not yet known to be right. Several prominent physicists, especially experimentalists, have never had much truck with superstrings anyway—mostly because it didn’t give them anything to do. There were no new phenomena to observe, no new quantities to measure.

  I’m not wedded to superstrings as the key to the universe, but I think this criticism is unfair. String theorists are being asked to prove their innocence, whereas normally it would be up to the critics to prove their guilt. It takes a lot of time and effort to develop radically new ways of thinking about the physical world, and string theory is technically very difficult. In principle, it can make new predictions about our world; the big problem is that doing the necessary sums is extraordinarily hard. The same
complaint could have been made about quantum field theory 40 years ago, but eventually the sums got done, through a combination of better computers and better mathematics, and the agreement with experiment turned out to be better than we find anywhere else in science.

  Moreover, much the same charge can be leveled at almost any hopeful Theory of Everything, and paradoxically, the better it is, the harder it will be to prove correct. The reason is inherent in the nature of a Theory of Everything. In order to be successful, it must agree with quantum theory whenever it is applied to any experiment whose results are consistent with quantum theory. It must also agree with relativity whenever it is applied to any experiment whose results are consistent with relativity. So the Theory of Everything is obliged to pass every experimental test yet devised. Asking for a new prediction that distinguishes the Theory of Everything from conventional physics is rather like asking for something that yields results identical to those predicted by theories describing all known physical phenomena, yet is different.

  Of course, eventually string theory will have to make a new prediction, and be tested against observations, to make the transition from speculative theory to real physics. The need to agree with everything currently known does not rule out such predictions, it just explains why they don’t come easily. Some tentative proposals for critical experiments already exist. For instance, recent observations of distant galaxies indicate that the universe is not only expanding, but expanding increasingly fast. Superstring theory offers a simple explanation—gravity is leaking away into those extra dimensions. However, there are other ways to explain this particular effect. What is clear is that if the theorists all stop investigating superstring physics, we will never have a chance to find out whether the theory is correct. It takes time and effort to come up with the crucial experiments, even if they exist.

  I don’t want to leave the impression that when it comes to unifying quantum theory with relativity, superstrings are the only game in town. There are many competing proposals—though they all suffer from the same lack of experimental support.

  One idea, known as “noncommutative geometry,” is the brainchild of the French mathematician Alain Connes. It rests on a new concept of the geometry of space-time. Most unifications start with the idea that space-time is some extension of Einstein’s relativistic model, and try to make the fundamental particles of subatomic physics fit in somehow. Connes does the opposite. He starts from a mathematical structure known as a non-commutative space, which contains all of the symmetry groups that arise in the standard model, and then deduces features similar to relativity. The mathematics of such spaces traces back to Hamilton and his noncommutative quaternions, but is extensively generalized and modified. Once again, though, this alternative theory is firmly rooted in Lie group theory.

  Another intriguing idea is “loop quantum gravity.” In the 1980s, the physicist Abhay Ashtekar worked out how Einstein’s equations would look in a quantum setting where space is “grainy.” Lee Smolin and Carlo Ravelli developed his ideas, leading to a model of space that is rather like medieval chain mail—constructed from very tiny lumps about 10–35 meters across, joined by links. They noticed that the detailed structure of the chain mail can get very complex as the links become knotted or braided together. However, it was not clear what these possibilities meant.

  An electron represented as a braid.

  In 2004, Sundance Bilson-Thompson discovered that some of these braids exactly reproduce the rules for combining quarks. The electric charge of the quark is reinterpreted in terms of the topology of the associated braid, and the combination rules follow from simple geometric operations with braids. This idea, still in its infancy, produces most of the particles observed in the standard model. It is the latest in a series of speculative proposals that matter—here realized as particles—might be a consequence of “singularities” in space, such as knots, localized waves, or more complicated structures where space ceases to be smooth and regular. If Bilson-Thompson is right, matter is just twisted space-time.

  Mathematicians have been studying the topology of braids for many years and have long known that braids themselves form a group, the braid group. The operation of “multiplication” arises when two braids are joined end to end—much as we joined permutations end to end when discussing Ruffini’s approach to the quintic. Yet again, physics is building on preexisting mathematical discoveries, mostly made “for their own sake” because they looked interesting. And yet again, a key ingredient is symmetry.

  In the latest version of superstrings, the biggest problem is an embarrassment of riches. Instead of making no predictions, the theory makes too many. The “vacuum energy”—the energy content of empty space—can be almost anything, depending on how the strings wrap around inside the extra dimensions of space. The number of ways for this to happen is gigantic—around 10500. Different choices yield different values for the vacuum energy.

  As it happens, the observed value is very, very small, around 10–120, but it is not zero.

  According to the conventional “fine-tuning” story, this particular value is exactly right for life to exist. Anything larger than 10–118 makes local space-time explode; anything smaller than 10–120 and space-time contracts in a cosmic crunch and disappears. So the “window of opportunity” for life is very small. By a miracle, our universe sits neatly within it.

  The “weak anthropic principle” points out that if our universe were not constituted the way it is, we wouldn’t be here to notice, but that leaves open the question why there is a “here” for us to occupy. The “strong anthropic principle” says that we’re here because the universe was designed specially for life to exist—which is mystical nonsense. No one actually knows what the possibilities would be if the vacuum energy were markedly different from what it is. We know a few things that would go wrong—but we have no idea what might go right instead. Most of the fine-tuning arguments are bogus.

  In 2000, Raphael Bousso and Joseph Polchinski proposed a different answer, using string theory and taking advantage of those 10500 possible values for the vacuum energy. Although 10–120 is very small, the possible vacuum energy levels are spaced about 10–500 units apart, which is even smaller. So plenty of string theories give vacuum energies in the “right” range. The probability that a randomly chosen one will do that is still negligible, but Bousso and Polchinski pointed out that this is irrelevant. Eventually, the “right” vacuum energy will inevitably occur. The idea is that the universe explores all possible string theories, sticking with any given one until it causes that universe to come to bits, and then “tunneling” quantum-mechanically to some other string theory. If you wait long enough, then at some stage the universe acquires a vacuum energy that happens to be in the range suitable for life.

  In 2006, Paul Steinhardt and Neil Turok proposed a variation on the “tunneling” theory: a cyclic universe that expands in a Big Bang and contracts in a Big Crunch, repeating this behavior every trillion years or so. In their model, the vacuum energy decreases in each successive cycle, so that eventually the universe has a very small, but nonzero, vacuum energy.

  In either model, a universe whose vacuum energy is low enough will hang around for a very long time. Conditions are suitable for life to arise, and life has plenty of time to evolve intelligence, and to wonder why it’s there.

  15

  A MUDDLE OF MATHEMATICIANS

  A gaggle of geese, a pride of lions, a charm of finches, an exaltation of skylarks . . . what is the collective noun for mathematicians? A magnificence of mathematicians? Too smug. A mystification of mathematicians? Too close to the mark. Having had many opportunities to observe the behavior of the mathematical species when congregated in large herds, I think that the most apt word is “muddle.”

  One such muddle invented one of the most bizarre structures in the entire subject, and discovered a hidden unity behind its puzzling facade. Their discoveries, mainly obtained by pottering around and seeing what turned up, a
re beginning to infiltrate theoretical physics, and they may just hold the key to some of the more curious features of superstrings.

  The mathematics of superstrings is so new that most of it has not been invented yet. But ironically, mathematicians and physicist have just discovered that superstrings, at the frontiers of modern physics, seem to have a curious relationship to a bit of Victorian algebra so old-fashioned that it is seldom mentioned in university mathematics courses. This algebraic invention is known as the octonions, and it is the next structure in line after real numbers, complex numbers, and quaternions.

  Octonions were discovered in 1843, published by someone else in 1845, and ever afterward credited to the wrong person—but that didn’t matter, since nobody took any notice. By 1900 they had fallen into obscurity even in mathematics. They experienced a brief revival in 1925 when Wigner and von Neumann tried to make them the basis of quantum mechanics, but then fell back into obscurity when the attempt failed. In the 1980s they resurfaced as a potentially useful gadget in string theory. In 1999 they turned up as a crucial ingredient in 10- and 11-dimensional superstring theory.

  Octonions tell us that there is something very strange about the number 8, and something even stranger about the physics of space, time, and matter. A Victorian whimsy has been reborn as the key to deep mysteries on the common frontier of mathematics and physics—especially the belief that space-time may have more dimensions than the traditional four, and that this is how gravity and quantum theory fit together.