Why Beauty is Truth

by Ian Stewart

  A phase shift of a full vibrational period is the same as no phase shift at all, and this implies that in the abstract, changing phase is a rotation. So the symmetry group involved here—the “gauge group”—is SO(2), the rotation group in two dimensions. However, physicists like their quantum coordinate transformations to be “unitary”—defined by complex numbers, not real ones. Fortunately, SO(2) has another incarnation as the unitary group U(1)—rotations in the complex plane.

  In short: quantum electrodynamics has U(1) gauge symmetry.

  Gauge symmetries were the clue to the next two unifications of physics, the electroweak theory and quantum chromodynamics. Together these constitute the “standard model,” the currently accepted theory of all fundamental particles. Before we can see how this goes, we must explain exactly what is being unified: not theories but forces.

  Today’s physics recognizes four distinct kinds of force in nature: gravity, electromagnetism, the weak nuclear force, and the strong nuclear force. They have very different characteristics: they operate on different scales of space and time, some cause particles to attract each other, some cause them to repel each other, some do both depending on the particles, and some do both depending on how far apart the particles are.

  At first sight, each force bears little resemblance to the others. But beneath the surface there are signs that these differences are less important than they seem. Physicists have teased out evidence of a deeper unity, suggesting that all four forces have a common explanation.

  We feel the consequences of gravity all the time. When we drop a plate and it shatters on the kitchen floor, we see gravity pulling it towards the Earth’s center and the floor getting in the way. The plastic pigs on the freezer door (well, that’s what you will find in our house) remain in place thanks to the magnetic force, which Maxwell showed was merely one aspect of the unified electromagnetic force. The electrical aspect runs the freezer. Less obviously, the shattering plate also reveals the consequences of the electromagnetic force, because this is the main force acting in chemical bonds to hold bulk matter together. When the stress on the plate becomes too great for the electromagnetic force to hold its molecules together, it breaks.

  The two remaining forces, which act on the level of the atomic nucleus, are not so readily apparent; but without them there would not be any matter at all, because they hold atoms together. They are why the plate, pigs, freezer, floor, and kitchen exist.

  Other types of force could in principle give rise to other types of universe, and our ignorance of such possibilities is almost total. It is often claimed that without the particular forces we have, life would be impossible, proving that our universe is amazingly finely tuned to make life possible. This argument is bogus, a wild exaggeration based on too limited a view of what constitutes life. Life like ours would be impossible—but it is the height of arrogance to assume that our kind of life is the only kind of organized complexity that could exist. The fallacy here is to confuse sufficient conditions for life (those aspects of our universe on which our kind of life depends) with necessary ones.

  The first of the four forces to be formulated scientifically was gravity. As Newton observed, this is an attractive force: any two particles in the universe, he said, attract each other gravitationally. The force of gravity is long-range: it falls off fairly slowly with distance. On the other hand, the gravitational force is much weaker than the other three: a tiny magnet can attach a plastic pig firmly to the fridge, even though the entire Earth is trying to pull it off through the force of gravity.

  The next fundamental force to be isolated was electromagnetism, under whose influence particles may either attract or repel each other. What distinguishes the two is whether the particles have the same electric charge or the same magnetic polarity. If they do, the force is repulsive; if not, it is attractive. Again, this force is long-range.

  The nucleus of an atom is assembled from smaller particles—protons and neutrons. Neutrons, as the name suggests, have no electric charge, but all protons have positive charge. The electromagnetic repulsion among protons should cause the nucleus to explode. What holds it together? Gravity is too weak—think of the plastic pigs. There must be some other force—which physicists labeled the strong nuclear force.

  But if the strong force can overcome electric repulsion, why don’t all of the protons in the universe get sucked together into one gigantic atomic nucleus? Clearly, the effect of the strong force must fall off rapidly at distances greater than the size of the nucleus. So the strong force is short-range.

  The strong force does not explain the phenomenon of radioactive decay, in which atoms of certain elements “spit out” particles and radiation, and change to different elements. Uranium, for example, is radioactive and eventually turns into lead. So there must be yet another subatomic force. This is the weak force, and it is even shorter-range than the strong force: it acts only at a distance one-thousandth the size of a proton.

  Physics was a lot easier when the only building blocks of matter were protons, neutrons, and electrons. These “elementary particles” were the components of the atom—which, it transpired, did split, even though the name means “indivisible.” In Niels Bohr’s early model, the atom was visualized as a tight collection of protons and neutrons orbited by much lighter, distant electrons. The proton carried a fixed positive electric charge, the electron carried the same amount of charge but negative, and the neutron was electrically neutral.

  Later, as quantum theory developed, this solar-system image was replaced by a subtler one. The electrons didn’t orbit the nucleus as well-defined particles but kind of smeared themselves around it in interestingly shaped clouds. These clouds were best interpreted as clouds of probability. If you looked for an electron, you were more likely to find it in the cloud’s denser regions and less likely to find it in the sparse regions.

  Physicists invented new ways to probe the atom, break it into pieces, and probe the inner structure of those pieces. The main method, still in use, is to hit it with another atom or particle and watch what flies off. Over time—the story is too complicated to tell in detail—more and more different kinds of particle were found. There was the neutrino, which could pass through a million miles of lead unhindered and was therefore rather hard to detect. There was the positron, like an electron but with the opposite electrical charge, predicted by Dirac’s matter/antimatter symmetry.

  As the number of “elementary” particles grew to more than sixty, physicists began to seek deeper ordering principles. There were too many building blocks for them to be fundamental. Each type of particle could be characterized by a series of properties: mass, charge, something called “spin” because the particles behaved as though they were spinning around some axis (except that this was an outmoded image and whatever spin was, it wasn’t really that). The particles did not spin in space, like the Earth or a spinning top. They “spun”—whatever that meant—in more exotic dimensions.

  Like everything in the quantum world, most of these features came in integer multiples of basic, very tiny amounts—quanta. All electrical charges were integer multiples of the charge on a proton. All spins were integer multiples of the spin of an electron. It was not clear whether mass was similarly quantized; the masses of the fundamental particles were a structureless mess.

  Some family resemblances emerged. An important distinction had to be made between particles whose spin was an odd integer multiple of the spin of the electron, and those whose spin was an even integer multiple. The reason was based on symmetry properties; the spins (in those exotic dimensions) did different things if you made the particles rotate in space. Somehow the exotic dimensions of spin and the prosaic dimensions of space were related.

  The odd particles were named fermions and the even ones bosons, after two giants of particle physics, Enrico Fermi and Satyendranath Bose. For reasons that once seemed sensible, the electron spin is defined to have value ½. So bosons have integer spin (even multiples of ½ are inte
gers) and fermions have spins ½, , , and so on, along with their negatives – ½, –, – . Fermions obey the Pauli exclusion principle, which says that in any prescribed quantum system, two distinct particles cannot be in the same state at the same time. Bosons do not obey the Pauli principle.

  Fermions include all of the familiar particles: the proton, neutron, and electron are all fermions. So are more esoteric particles like the muon, tauon, lambda, sigma, xi, and omega, all names derived from the Greek alphabet. So are three types of neutrino, associated with the electron, muon, and tauon.

  Bosons are more mysterious, with names like pion, kaon, and eta.

  The particle physicists knew that all of these particles existed, and they could measure their physical properties. The problem was making sense of this apparent mishmash. Was the universe built from whatever happened to be to hand? Or was there a hidden plan?

  The upshot of these deliberations was that many supposedly elementary particles were in fact composite. They were all made from quarks. Quarks (the name comes from Finnegans Wake) come in six distinct flavors, arbitrarily named: up, down, strange, charm, top, and bottom. They are all fermions, with spin ½. Each has an associated antiquark.

  There are two ways to combine quarks. One is to use three ordinary quarks, in which case you end up with a fermion. The proton consists of two up quarks plus one down quark. The neutron is two down and one up. A bizarre particle called the omega-minus is made from three strange quarks. The other is to use a quark and an antiquark, which yield a boson. They don’t annihilate each other because they are kept apart by nuclear forces.

  For the electrical charges to work out correctly, the charges on quarks cannot be integers. Some have charge ⅓, some ⅔. Quarks come in three distinct “colors.” That makes 18 types of quark, plus 18 antiquarks. Oh, yes, there’s more. We have to add some more particles to “carry” the weak nuclear force, which holds the quarks together. The resulting theory, which has great mathematical elegance despite the proliferation of particles, is known as quantum chromodynamics.

  Quantum theory explains all physical forces in terms of exchanges of particles. Just as the tennis ball holds the two players together at opposite ends of the court as long as the game continues, so various particles carry the electromagnetic, strong, and weak forces. The electromagnetic force is carried by photons. The strong force is carried by gluons, the weak force by intermediate vector bosons, otherwise known as “weakons.” (Don’t blame me—I didn’t invent these names, which are mostly historical accidents.) Finally, it is widely conjectured that gravity must be carried by a hypothetical particle called the graviton. No one has yet observed a graviton.

  The large-scale effect of all these carrier particles is to fill the universe with “fields.” Gravitational interactions create a gravitational field, electromagnetic ones create an electromagnetic field, and the two nuclear forces together create something called a Yang–Mills field, after the physicists Chen Ning Yang and Robert Mills.

  We can summarize the main characteristics of the fundamental forces in a kind of physicist’s shopping list:

  • Gravity: Strength 6 × 10–39, range infinite, carried by gravitons (not observed, should have mass 0, spin 2), forms the gravitational field.

  • Electromagnetism: Strength 10–2, range infinite, carried by photons (mass 0, spin 1), forms the electromagnetic field.

  • Strong force: Strength 1, range 10–15 meters, carried by gluons (mass 0, spin 1), forms one component of the Yang–Mills field.

  • Weak force: Strength 10–6, range 10–18 meters, carried by weakons (large mass, spin 1), forms the other component of the Yang–Mills field.

  You may feel that 36 fundamental particles, plus assorted gluons, is not a big improvement on sixty or more, but quarks form a highly structured family with a huge amount of symmetry. They are all variations on the same theme—unlike the wild zoo of particles that physicists had to deal with before quarks were discovered.

  The description of fundamental particles in terms of quarks and gluons is known as the standard model. It fits experimental data extremely well. Some of the masses of some of the particles have to be adjusted to fit observations, but once you’ve done that, all the other masses slot neatly into place. The logic is not circular.

  Quarks are bound together very tightly, and you never see an isolated quark. All you can observe are the combinations of twos and threes. Nevertheless, particle physicists have confirmed the existence of quarks indirectly. They’re not just a clever numerological variation on the zoo. And to those who believe that the universe is at heart beautiful, the symmetry properties of quarks clinch it.

  According to quantum chromodynamics, a proton is made from three quarks—two up, one down. If you took the quarks out of a proton, shuffled them, and put them back, you would still have a proton. So the laws for protons ought to be symmetric under permutations of their constituent quarks. More interestingly, the laws also turn out to be symmetric under changes to the type of quark. You could turn an up quark into a down one, say, and the laws would still work.

  This implies that the actual symmetry group here is not just the group of six permutations of three quarks, but a closely related continuous group, SU(3), one of the simple groups on Killing’s list. Transformations in SU(3) leave the equations for laws of nature unchanged, but they can change the solutions to those equations. Using SU(3) you can “rotate” a proton into a neutron, for instance. All you have to do is turn all of its quarks upside down, so that two up and one down become two down and one up. The world of fermions has SU(3) symmetry, and the symmetries act by changing one fermion into another.

  Two further symmetry groups contribute to the standard model. The gauge symmetries of the weak force, SU(2), can change an electron into a neutrino. SU(2) is another group on Killing’s list. And our old friend the electromagnetic field has U(1) symmetry—not the Lorentz symmetries of Maxwell’s equations, but the gauge (i.e., local) symmetry of phase changes. This group just misses Killing’s list because it is not SU(1), but it is morally on the list, since it’s a very close relative.

  The electroweak theory unified electromagnetism and the weak force by combining their gauge groups. The standard model incorporates the strong force as well, providing a single theory for all fundamental particles. It does this in a very direct manner: it just lumps all three gauge groups together as SU(3) × SU(2) × U(1). This construction is simple and straightforward but not terribly elegant, and it makes the standard model resemble something built out of chewing gum and string.

  Suppose you own a golf ball, a button, and a toothpick. The golf ball has spherical symmetry SO(3), the button has circular symmetry SO(2), and the toothpick has (say) just a single reflectional symmetry O(1). Can you find some unified object that has all three types of symmetry? Yes, you can: put all three into a paper bag. Now you can apply SO(3) to the contents of the bag by rotating the golf ball, SO(2) by rotating the button, and O(1) by flipping the toothpick. The symmetry group of the bag’s contents is SO(3) × SO(2) × O(1). This is how the standard model combines symmetries, but instead of using rotations it uses the “unitary transformations” of quantum mechanics. And it suffers from the same defect: it just lumps three systems together and combines their symmetries in an obvious, and rather trivial, way.

  A much more interesting way to combine the three symmetry groups would be to build something that contained the same objects but was more elegant than a paper bag. Maybe you could balance the toothpick on the golf ball and stick a button on the end of it. You could even have a whole system of toothpicks, like the spokes of a wheel; put the button at the hub, spin the wheel on top of the golf ball. If you were really clever, maybe the combined object would have lots and lots of symmetry, say the group K(9). (There is no such group. I made it up for the sake of this discussion.) The separate symmetry groups SO(3), SO(2), and O(1) might with luck be subgroups of K(9). That would be a far more impressive way to unify the golf ball, button
, and toothpick.

  Physicists felt the same way about the standard model, and they wanted K(9) to be something on Killing’s list or very close to it, because Killing’s groups are the fundamental building blocks of symmetry. So they invented a whole series of Grand Unified Theories, or GUTs, based on groups like SU(5), O(10), and Killing’s mysterious exceptional group E6.

  The GUTs seemed to suffer from the same defect as Kaluza–Klein theory—a lack of testable predictions. But then a really interesting prediction appeared. It was certainly new, so new that it seemed unlikely to be true, but it was testable. All GUTs predict that the proton can be “rotated” into an electron or a neutrino. So protons are unstable, and in the long run all matter in the universe should decay into radiation. The calculations said that on average, the life of a proton should be around 1029 years, much longer than the age of the universe. But individual protons would sometimes decay much sooner, and if you had enough protons, you might spot one decaying.

  A big tank of water has more than enough protons for a few to decay each year. By the end of the 1980s there were six experiments running, all trying to spot a decaying proton. The biggest tank contained over 3000 tons of extremely pure water. No one saw a proton decay. Not one. Which meant that the average lifetime is at least 1032 years. Protons live at least a thousand times longer than GUTs predict. GUTs just don’t hack it. In retrospect, it would have been a bit embarrassing if proton decay had been detected, because something very important is missing from GUTs: gravity.

  Any Theory of Everything has to explain why there are four fundamental forces, and why they take the strange forms that they do. This is a bit like trying to find a family resemblance between an elephant, a wombat, a swan, and a gnat.

  It would be much easier to organize the four forces if they could all be shown to be different aspects of a single force. In biology, this has been achieved: elephants, wombats, swans, and gnats are all members of the Tree of Life, united by their DNA, distinguished by a lengthy series of historical changes to DNA. All four evolved, step by step, from a common ancestor, which lived a billion or two billion years ago.