by Lee Smolin
This was a much more challenging problem than applying quantum theory to particles, because a field has a value at every point of space. If we assume that space is continuous—which is what special relativity asserts—then there are a continuous infinity of variables. In quantum theory, each variable is subject to an uncertainty principle. One implication is that the more precisely you try to measure a variable, the more it fluctuates uncontrollably. An infinite number of variables fluctuating uncontrollably can easily get out of hand. When you ask the theory questions, you have to be very careful not to get infinite and inconsistent answers.
The quantum theorists already knew that for each electromagnetic wave there is a quantum particle, the photon. It took only a few years to work this out in detail, but the result was just a theory of photons moving freely; the next step would be to incorporate charged particles, such as electrons or protons, and describe how they interact with photons. The goal was a fully consistent theory of quantum electrodynamics, or QED. This was much more challenging. QED was first solved by the Japanese physicist Sin-Itiro Tomonaga during World War II, but the news did not reach the rest of the world until 1948 or so. By then, QED had been constructed twice more, independently, by the young Americans Richard Feynman and Julian Schwinger.
Once QED was understood, the task was to extend quantum field theory to the strong and weak nuclear forces. This would take another quarter century, and the key would be the discovery of two new principles: The first defined what electromagnetism and these nuclear interactions have in common. It is called the gauge principle, and as I will describe, it leads to a unification of all three forces. The second principle explains why, although unified, the three forces are so different. It is called spontaneous symmetry breaking. These two principles together form the cornerstone of the standard model of particle physics. Their precise application had to await the discovery that particles like the proton and neutron are not elementary after all; instead, they are made of quarks.
The proton and neutron each have three quarks, while other particles, called mesons, have two (more properly, a quark and an antiquark). This discovery was made in the early 1960s, independently, by Murray Gell-Mann at Caltech and George Zweig at CERN (the European Organization for Nuclear Research) in Geneva. Shortly afterward, James Bjorken, of the Stanford Linear Accelerator Center (SLAC), and Richard Feynman at Caltech proposed experiments that, when they were later carried out at SLAC, confirmed that the proton and neutron are indeed each made up of three quarks.
The discovery of the quarks was an essential step toward unification because the interaction of protons, neutrons, and other particles was exceedingly complicated. But there was a hope that the force between the quarks might itself be simple and that the observed complexities arise because protons and neutrons are composite objects. This kind of notion had proved true before: Whereas the forces between molecules are complicated, the forces between the atoms that make them up can be easily understood in terms of electromagnetism. The idea was for theorists to give up trying to understand the force between protons and neutrons in fundamental terms and ask instead how that force affected the quarks. This is reductionism at work—the old trick that the laws governing the parts are often simpler than those governing the whole—and it eventually paid off in the discovery of the deep commonality that connects the two nuclear forces, strong and weak, with electromagnetism. All three are consequences of the simple but powerful gauge principle.
The gauge principle is best understood in terms of something physicists refer to as a symmetry. Put simply, a symmetry is an operation that doesn’t change how something behaves relative to the outside world. For example, if you rotate a ball, you don’t change it; it’s still a sphere. So when physicists talk of a symmetry, they can be referring to an operation in space, like rotation, that doesn’t change the result of an experiment. But they can also be talking about any kind of change we make to an experiment that doesn’t alter the outcome. For example, suppose you take two groups of cats—say, east-side cats and west-side cats—and you test their abilities in jumping. If there is no difference in the average jump a cat can make, then we say that cat-jumping is symmetric under the operation of trading all your east-side cats for west-side cats.
Here is another example, simplified and idealized to make the point. Consider an experiment in which a beam of protons is accelerated and then aimed at a target consisting of certain kinds of nuclei. You, as the experimenter, observe the pattern the protons make as they scatter off the nuclei in the target. Now, without changing the energy or the target, you substitute neutrons for protons. In certain cases, the pattern of scattering hardly changes. The experiment is said to reveal that the forces involved act the same on protons and neutrons. In other words, the act of replacing protons with neutrons is a symmetry of the forces between them and the nuclei in the target.
Knowing the symmetries is a good thing, because they tell you something about the forces involved. In the first example, we learn that the force of gravity on cats doesn’t depend on where they come from; in the second, that certain nuclear forces can’t tell the difference between a proton and a neutron. Sometimes all we get from a symmetry is such partial information about the forces. But there are special situations in which the symmetries completely determine the forces. This turns out to be the case for a class of forces called gauge forces. I won’t bother you with exactly how this works, as we won’t need it.1 But the fact that all the properties of a force can be determined by knowing the symmetries is one of the most important discoveries of twentieth-century physics. This idea is what is meant by the gauge principle.2
There are two things we do need to know about the gauge principle. One is that the forces it leads to are conveyed by particles called gauge bosons. The other thing we need to know is that the electromagnetic, strong, and weak forces each turned out to be forces of this kind. The gauge boson that corresponds to the electromagnetic force is called the photon. Those that correspond to the strong force holding the quarks together are called gluons. Those that correspond to the weak force have a less interesting name—they are called, simply, weak bosons.
The gauge principle is the “beautiful mathematical idea,” noted in chapter 3, that was discovered by Herman Weyl in his failed 1918 attempt at the unification of gravity and electromagnetism. Weyl is one of the deepest mathematicians ever to ponder the equations of physics, and it was he who understood that the structure of Maxwell’s theory was entirely explained by a gauge force. In the 1950s, some people asked whether other field theories could be constructed using the gauge principle. It turned out that this could be done by basing them on symmetries involving the various kinds of elementary particles. These theories are now called Yang-Mills theories, after the names of two of their inventors.3 At first no one knew what to do with these new theories. The new forces they described would have an infinite range, like electromagnetism. Physicists knew that the two nuclear forces each had a short range, so it did not seem that they could be described by a gauge theory.
What makes theoretical physics as much an art as a science is that the best theorists have a sixth sense about what results can be ignored. Thus, in the early 1960s, Sheldon Glashow, then a postdoc at the Niels Bohr Institute, suggested that the weak force was indeed described by a gauge theory. He simply posited that some unknown mechanism limited the range of the weak force. If this range problem could be solved, the weak force could then be unified with electromagnetism. But the overall problem would still have to be faced: How could you unify forces that manifest themselves as differently as electromagnetism and the strong and weak nuclear forces do?
This is an example of a general problem that plagues nearly every attempt at unification. The phenomena you hope to unify are different—otherwise there would be nothing surprising about their unification. So even if you discover some hidden unity, you still have to understand why and how it is that they appear to be different.
As we saw ear
lier, Einstein had a wonderful way of solving this problem for special and general relativity. He realized that the apparent differences between the phenomena were not intrinsic to the phenomena but were due entirely to the necessity of describing the phenomena from the viewpoint of an observer. Electricity and magnetism, motion and rest, gravity and acceleration were all unified by Einstein in this way. The differences that observers perceive between them are therefore contingent, because they reflect only the viewpoint of the observers.
In the 1960s, a different solution to this general problem was proposed: The differences between unified phenomena were contingent, but not because of the viewpoint of particular observers. Instead, physicists made what seems at first an elementary observation: The laws may have a symmetry that is not respected by all features of the world they apply to.
Let me illustrate this first with our social laws. Our laws apply equally to all people. We may regard this as a symmetry of the laws. Substitute any person for any other and you do not change the laws they must obey. All must pay taxes, all must not exceed the speed limit. But this equality or symmetry before the law need not and does not require that our circumstances be the same. Some of us are wealthier than others. Not all of us have cars, and in those that do, the tendencies to exceed the speed limit can differ quite a bit.
Moreover, in an ideal society we all start out with equal opportunities. This is unfortunately not actually the case, but if it were, we could speak of a symmetry in our initial opportunities. As life goes on, this initial symmetry goes away. By the time we turn twenty, we have very different opportunities. A few of us have the opportunity to be concert pianists, a few to be Olympic athletes.
We can describe this differentiation by saying that the initial equality is broken as time goes on. Physicists who speak of equality as a symmetry would say that the symmetry between us at birth is broken by the situations we encounter and the choices we make. In some cases, it would be hard to predict the way the symmetry will be broken. We know that it must break, but looking at a nursery full of infants we are hard-pressed to predict how. In cases like this, physicists say that the symmetry is spontaneously broken. By this we mean that it is necessary that the symmetry break, but exactly how it breaks is highly contingent. This spontaneous symmetry breaking is the second great principle that underlies the standard model of particle physics.
Here is another example from human life. As a faculty member, I’ve sometimes had occasion to go to receptions for new undergraduates. Watching them meet one another, it has occurred to me that over the next year some will become friends, others lovers, a few will even marry. At this first moment, when they encounter one another as strangers, there is a lot of symmetry in the room; many possible couples and bonds of friendship could be forged in this group. But of necessity the symmetry must be broken as the actual human relationships develop out of a much larger space of possible relationships. This, too, is an example of spontaneous symmetry breaking.
Much of the structure of the world, both social and physical, is a consequence of the requirement that the world, in its actuality, break symmetries present in the space of possibilities. An important feature of this requirement is the trade-off between symmetry and stability. The symmetric situation, in which we are all potentially friends and romantic partners, is unstable. In reality, we must make choices, and this leads to more stability. We trade the unstable freedom of potentiality for the stable experience of actuality.
The same is true in physics. A common example from physics is of a pencil balanced on its point. It is symmetric, in that while it is balanced on its point, one direction is as good as another. But it is unstable. When the pencil falls, as it inevitably must, it will fall randomly, in one direction or another, breaking the symmetry. Once it has fallen, it is stable, but it no longer manifests the symmetry—although the symmetry is still there in the underlying laws. The laws describe only the space of what possibly may happen; the actual world governed by those laws involves a choice of one realization from many possibilities.
This mechanism of spontaneous symmetry breaking can happen to the symmetries between the particles in nature. When it occurs for the symmetries that, by the gauge principle, give rise to the forces of nature, it leads to the differences in their properties. The forces become distinguished; they can have different ranges and different strengths. Before the symmetry breaks, all four fundamental forces have an infinite range, like electromagnetism, but afterward some will have a finite range, like the two nuclear forces. As noted, this is one of the most important discoveries of twentieth-century physics, because together with the gauge principle it allows us to unify fundamental forces that appear disparate.
The idea of combining spontaneous symmetry breaking with the gauge theories was invented by François Englert and Robert Brout in Brussels, in 1962, and independently a few months later by Peter Higgs of Edinburgh University. It should be called the EBH phenomenon, but it is unfortunately usually just called the Higgs phenomenon. (This is one of many examples in which something in science is named after the last person who discovered it, rather than the first.) The three of them also showed that there is a particle whose existence is a consequence of spontaneous symmetry breaking. This is called the Higgs boson.
A few years later, in 1967, Steven Weinberg and the Pakistani physicist Abdus Salam independently discovered that the combination of the gauge principle and spontaneous symmetry breaking could be used to construct a concrete theory that unified the electromagnetic and weak nuclear forces. The theory bears their name: the Weinberg-Salam model of the electroweak force. This was certainly a unification with consequences to be celebrated; it quickly led to predictions of novel phenomena that were successfully verified. It implies, for example, that there must be particles—analogous to the photon, which carries the electromagnetic force—to carry the weak nuclear force. There are three of them, called the W+, W−, and Z. All three have been found and exhibit exactly the properties predicted.
The use of spontaneous symmetry breaking in a fundamental theory was to have profound consequences, not just for the laws of nature but for the larger question of what a law of nature is. Before this, it was thought that the properties of the elementary particles are determined directly by eternally given laws of nature. But in a theory with spontaneous symmetry breaking, a new element enters, which is that the properties of the elementary particles depend in part on history and environment. The symmetry may break in different ways, depending on conditions like density and temperature. More generally, the properties of the elementary particles depend not just on the equations of the theory but on which solution to those equations applies to our universe.
This signals a departure from the usual reductionism, according to which the properties of the elementary particles are eternal and set by absolute law. It opens up the possibility that many—or even all—properties of the elementary particles are contingent and depend on which solution of the laws is chosen in our region of the universe or in our particular era. They could be different in different regions.4 They could even change in time.
In spontaneous symmetry breaking, there is a physical quantity whose value signals that the symmetry is broken and how. This quantity is usually a field, called the Higgs field. The Weinberg-Salam model requires that the Higgs field exist and that it manifest itself as the new elementary particle called the Higgs boson, which carries the force associated with the Higgs field. Of all the predictions required by the unification of the electromagnetic and weak forces, only this one has not yet been verified. One difficulty is that the theory does not allow us to precisely predict the mass of the Higgs boson; it is one of the free constants that the theory asks us to set. There have been many experiments designed to find the Higgs boson, but all we know is that if it exists, its mass must be greater than about 120 times the mass of the proton. One of the main goals of future accelerator experiments is to find it.
In the early 1970s, the gauge principle w
as applied to the strong nuclear force, the force that binds the quarks, and it was found that a gauge field is responsible for that force, too. The resulting theory is called quantum chromodynamics, or QCD for short. (The word chromo, from the Greek for “color,” refers to a fanciful designation used to refer to the fact that quarks come in three versions, which, for fun, are called colors.) QCD, too, has survived rigorous experimental test. Together with the Weinberg-Salam model, it is the basis of the standard model of elementary-particle physics.
The discovery that all three forces are expressions of a single unifying principle—the gauge principle—is the deepest accomplishment of theoretical particle physics to date. The people who did this are true heroes of science. The standard model is the result of decades of hard, often frustrating experimental and theoretical work by hundreds of people. It was completed in 1973, and it has held up for thirty years against a wide array of experiments. We physicists are justly proud of it.
But consider what happened next. All three forces were now understood to be expressions of the same principle, and it was obvious that they should be unified. To unify all the particles, however, you need a big symmetry that includes them all. You then apply the gauge principle, giving rise to the three forces. To distinguish all the particles and forces, you set it up so that any configuration of the system in which the symmetry is realized is unstable, while the stable configurations are asymmetrical. This is not hard to do because, as I discussed, symmetrical situations are often unstable in nature. Thus, the symmetry including all the particles together will be spontaneously broken. This can be done so that the three forces end up with the very properties they are observed to have.