This way to the universe, p.9
This Way to the Universe, page 9
Another important symmetry is time reversal. In Newton’s world, make a video of some event—a ball rolling down a hill, a planet in its orbit around the sun—and then run the video backward. What you see may be surprising, perhaps challenging to arrange, but it obeys the laws of nature. Once we’ve given up parity as a symmetry, symmetries like time reversal would seem in jeopardy. In 1964, a small violation of time reversal symmetry was discovered in the weak interactions, of a very subtle form.
The symmetries we have so far encountered—translations in space and time, rotations, and parity—are symmetries we encounter in our day-to-day experience and of which we have some intuitive understanding. With the discovery of quantum mechanics, symmetries took on an even more central place in the laws of nature. Energy, momentum, and angular momentum were still conserved and their role, if anything, elevated. But quantum mechanics also revealed symmetries of a less familiar sort. Physicists call these symmetries, which have no obvious connection to space and time, “internal symmetries.” Perhaps the first of these was suggested by Werner Heisenberg, one of the inventors of quantum mechanics, famous for his uncertainty principle. Heisenberg noted that the proton and the neutron have very nearly the same mass. In fact, their masses are equal to roughly a part in 1,000. This might be a coincidence, but it is pretty remarkable. On the other hand, the proton and neutron would seem dramatically different—one particle carries electric charge, one does not. Heisenberg reasoned that, if you turn off the electric charge, perhaps you couldn’t tell these two particles apart—there is a symmetry that relates them. Since the number of neutrons in a nucleus determines which isotope of that type of an element one has, he called this “isotopic spin,” or “isospin.” He and others checked that the properties of nuclei were consistent with such a symmetry, again if one ignored the repulsive electric force between protons and the tiny difference in the proton and neutron masses.
Yang and Mills: Following Einstein, a New Type of Symmetry
C. N. Yang’s role in the discovery of parity violation, the breakdown of what to many, until that time, seemed a self-evident symmetry of nature, was remarkable. But slightly earlier, he made another discovery of profound importance. To appreciate this idea, we need to return to Einstein. One way to think about Einstein’s general relativity is in terms of symmetries. We know the laws of nature remain the same if one rotates a system about some axis. But while this is a property of Newton’s laws, it is actually strictly necessary to rotate the whole universe this way if the symmetry is to hold. This sounds a bit crazy, and it is. Einstein’s general relativity liberates the laws from this requirement. Within Einstein’s theory, one can rotate just a small part of the universe—in a lab, in my classroom, in the solar system. But this works only if one includes the gravitational field. The force of gravity, in other words, is forced upon us by this symmetry principle.
Yang and Robert Mills, in 1954, asked the same question about Heisenberg’s isospin symmetry. Shouldn’t one be able to trade a neutron for a proton anywhere in the universe, without having to do it everywhere? They wrote down a theory that implemented this idea. The mathematics was very pretty. Just as Einstein’s insistence that one can implement a rotation locally in space-time implies the existence of the gravitational field, Yang and Mills’s insistence that one can make an isotopic spin transformation anywhere in space-time leads to the prediction of three new fields, more similar to those of electricity and magnetism, and three massless particles, similar to the photon. No such particles exist, though Yang and Mills suggested that perhaps three massive particles known at that time might, somehow, be these particular “vector mesons.” How these particles might grow mass was a mystery, and the theory, for more than a decade, remained dormant.
Yang went on to many profound accomplishments in the field of condensed matter physics and to general aspects of theoretical physics. For many years, he headed the C. N. Yang Institute for Physics at the State University of New York at Stony Brook. Despite his contributions to our understanding of particle physics and basic laws of nature, he eventually became a harsh critic of the particle physics enterprise. He is particularly unenthusiastic about “big science,” of which particle physics is perhaps the epitome. As of this writing, Yang is a prominent opponent of proposals in China to build an extremely high energy accelerator (the “Chinese Collider”), a project requiring an investment comparable to the cost of the Large Hadron Collider at CERN in Switzerland, and of much higher energy.
The theory of Yang and Mills, as we will see, has emerged as the underpinning of the Standard Model. It has also had profound impact in mathematics. But making sense of these theories and seeing their possible role would take over a decade. Much happened in those intervening years.
Why Are the Pions Light?—the Nambu-Goldstone Phenomenon
With the discovery of the pion, Yukawa’s picture appeared to give at least a rough description of the strong nuclear force. But it was not satisfying in the way the quantum theory of electric and magnetic forces (QED) was. First, precisely because the interactions were so strong, one couldn’t apply the methods of Feynman, Schwinger, and Tomonaga to this problem. So it was hard to establish what the theory predicted. But it also became clear rather quickly that the theory could not be complete as a model of the strong interactions. The discovery of the pions was followed by the discovery of other, heavier particles, which also seemed to play some role in the nuclear force. As a group, the strongly interacting particles were called hadrons. The questions now became: What is the role of all these other particles? And why are the pions somehow special? In particular, why are they significantly lighter than any of the others and the most important in explaining the force between neutrons and protons?
A solution came in the work of the great Japanese-American physicist Yoichiro Nambu. He had come to the United States as a young man, taking on a position at the Institute for Advanced Study in Princeton. While always a modest and gentle person, Nambu was not shy or retiring. When he arrived at the Institute, J. Robert Oppenheimer, who was then the director, told the new members (the title of postdocs at the Institute) that they were not to disturb the great Einstein. Nambu acknowledged the instruction and then immediately made an appointment to meet with Einstein. He was eager to acquire whatever wisdom the legendary physicist would be willing to impart. At their meeting, Einstein complained that none of the younger people came to see him. Nambu, somewhat unusually for that time, and particularly for a young person coming from abroad, owned a car, and after that first meeting, he made a point of offering Einstein rides to work. He surreptitiously took a picture of Einstein one morning walking toward the car, not a small feat with the cameras of those days. In our day, this would have gone viral on social media.
In any case, after his time at the Institute, Nambu went on to the University of Chicago. There he did a range of important work, but he was particularly interested in the symmetries of the strong interactions and their connection to the pions. Nambu considered the possibility that the strong interactions had a symmetry of a then unfamiliar type. Unfamiliar in two ways. First, because it emerged from the combination of relativity and quantum mechanics. If the electron had no mass, he realized, it would obey a curious conservation law. The spin of the electron points in some direction. In quantum mechanics, if the electron were massless, it would, like the photon, move at the speed of light. The direction of its spin might be along the direction in which the electron of its motion, or in the opposite direction. If along the direction of motion, it will stay that way; similarly, if opposite. This is the conserved property. The symmetry connected with this conservation law is called “chiral symmetry” because it has to do with handedness (the word chiral comes from the Greek word for “hand”). The electron is not massless (though at very high energies, its mass can sometimes be ignored and the conservation law holds), and the proton and neutron are certainly not.
Here came the second, great leap in Nambu’s thinking. He conjectured that the strong interactions have such a symmetry, but it is a broken symmetry. At first, this idea may seem a bit weird, but the notion of a broken symmetry is actually quite familiar. The laws of nature are symmetric under rotations, but objects we encounter are typically not. The handle on a water jug, for example, breaks the rotational symmetry. This is not paradoxical; the underlying symmetry is reflected in the fact that we can orient the jug so that its handle lies in any direction. Objects other than perfect spheres have some intrinsic direction and, if sitting at rest, must point in some direction. Physicists say that such a symmetry is “spontaneously broken.” Nambu reasoned that chiral symmetry in the nuclear forces is of this type. He also realized that if a symmetry of the laws of nature is broken in this way, there is a consequence: There must be a massless particle. There are no massless particles in the strong interactions, but the pions are much lighter than the others, and he identified these as the candidate massless particles. If the chiral symmetry is not an exact symmetry of the underlying laws of the strong interactions but is broken “a little bit,” this would account for the small masses of the pions. Jeffrey Goldstone of MIT proved a general result that massless particles arise from spontaneous symmetry breaking, so the massless particles are known as Nambu-Goldstone bosons. Extensive experimental study verified that the pions behave in just the way expected of such objects.
Lots and Lots of Strongly Interacting Particles
The late 1940s saw the beginning of the accelerator era in particle physics. While primitive accelerators had been built in the years before World War II, physics was now viewed as an important part of the nation’s defense, and generous funding became available from the federal government. At the same time, technological developments during the war, and the training of a cadre of skilled people, both scientists and technicians, facilitated the boom. Accelerators were built and operated at UC Berkeley, the Brookhaven National Laboratory on Long Island, and elsewhere. The energies of these accelerators were more than enough to produce pions and soon yielded a raft of other particles as well. The details, for our purposes, are not important. What is important is that there were hundreds of these particles. Like the pions, these new particles were all short-lived, decaying in many cases in about 10-20 seconds! But they were sufficiently distinctive that their properties could be measured with precision.
Yet, now there was a mystery. Initially the proton, the neutron, and the pions were thought to be fundamental, structureless objects, like the electron. But the proliferation of newly discovered particles called this picture into question. Careful measurements, in fact, showed that the proton has a size and shape similar to an atomic nucleus, about 10-13 cm. So perhaps, just as the elements of the periodic table are built of electrons and nuclei, these new particles were built of some other entities. The way out of this confusion was provided by Murray Gell-Mann, who established an analog of the periodic table for the hadrons. His table was based on what was then, for most theoretical physicists, an unfamiliar branch of mathematics called group theory. The number eight played a crucial role in the table. The lightest mesons came in eight types, as did the lightest baryons—strongly interacting particles of spin ½ like the proton and neutron. Gell-Mann (who passed away in 2019) was quite erudite—and liked to let people know it—so showing off his interest in languages and Eastern religions, he called this the Eightfold Way. In Buddhism, the Eightfold Path is the path to nirvana, comprising eight aspects in which an aspirant must become practiced: right views, intention, speech, action, livelihood, effort, mindfulness, and concentration. In Gell-Mann’s periodic table, the eight referred to various more prosaic properties, such as electric charges of the particles.
Dmitri Mendeleev had put forward his periodic table of the elements based on regularities in chemical properties. Only with the discovery of quantum mechanics were its features understood in terms of properties of electrons and atomic nuclei. Gell-Mann, simultaneously with George Zweig, implemented this second step for the hadrons: They proposed that the Eightfold Way could be understood in terms of particles analogous to the electron, protons, and neutron in atoms, called quarks. Gell-Mann chose the name from a passage in James Joyce’s Finnegan’s Wake (Zweig called these objects Aces, but the name never stuck). Initially, there were three types of quarks, rather playfully called up, down, and strange.
The quarks worked beautifully to account for the properties of the strongly interacting particles—the hadrons. But they had some peculiar properties. They carried electric charge. From a human perspective, the electron is the most important of charged particles. It is what makes up electric current and is used to store information in computers or our cell phones. Because it is so important, it is natural to take the charge of the electron as a basic unit. By a convention that traces back to Benjamin Franklin, we will say that the electron has charge minus one. The proton, then, has charge plus one, so that atoms are electrically neutral. The quarks hypothesized by Gell-Mann and Zweig would have charges that were fractions of this basic unit: ⅔ or minus ⅓. Particles like the pion, with spin zero, in the quark model consist of a quark bound to an antiquark. These particles are known as mesons. Particles like the proton and its excitations, which carry spin like that of the electron, are bound states of three quarks. The proton, with charge one, was composed of two up quarks and one down quark. The neutron, with charge zero, was composed of one up and two down quarks. Other hadrons were built of other combinations. The π+ meson, for example, consisted of an up quark and an anti-down quark (the antiparticle of the down quark).[*]
While this new periodic table worked well, there was a puzzle. Electrons are easily ejected from atoms. We’re used to this from seeing sparks when a strong electric field (in a circuit, for example) rips electrons from atoms. Chemists work all the time with ions, and scientists study electrons and nuclei individually in many controlled situations. But for quarks, nothing similar happens. When protons collide with each other, with neutrons and with pions, no objects with fractional charge—charges plus or minus ⅓ or ⅔ that of the electron or proton—are observed in the debris. Scientists searched for such fractional charges all over the place, even in moon rocks. Gell-Mann himself retreated, for a time, to the view that quarks were just some sort of useful mathematical plaything, with no physical reality.
But a different kind of experiment eventually established that quarks are real. In the mid-1960s, Richard Feynman at the California Institute of Technology (Caltech) and James Bjorken (known as BJ) at the then newly built Stanford Linear Accelerator Center (SLAC) started to think about what might happen in very high energy experiments if protons and neutrons consisted of quarks. Feynman, whose competition with Gell-Mann was notorious, refused, at least for some time, to call the constituents of the hadrons quarks, instead calling them partons. In any case, Feynman and Bjorken made a sharp prediction. Experiments similar to Rutherford’s, where electrons scattered off of nuclei, at very high energies should reveal the internal structure of the proton and neutron. A sequence of experiments at SLAC uncovered just this phenomenon. The proton was seen to be built of particles with fractional charges. This work won the Nobel Prize for Jerome Friedman, Henry Kendall, and Richard Taylor. For many, Henry Kendall is a familiar name; he went on to found the Union of Concerned Scientists, which has done notable work on issues of nuclear power and on environmental and energy policy more generally.
So it would seem that physicists were poised to uncover new laws of nature, those which governed the nuclear force. But the very successes of the quark model posed serious challenges. One might have thought that relativity and quantum mechanics required that the nuclear forces should be described by a quantum field theory. But it seemed that no quantum field theory had the required properties—either to explain why quarks were not seen free, isolated by themselves, or why hadrons would behave as if they were composed of quarks when banged together hard enough.
The Breakthrough: Yang-Mills Theories and Their Remarkable Properties
The explanation of the first puzzle, that the hadrons look like collections of quarks when collided hard together, was provided by David Gross and Frank Wilczek at Princeton and David Politzer at Harvard in 1973. They all realized that success in explaining this first point required that the theory have a property called asymptotic freedom. This is a fancy and rather colorful way of saying that the force between quarks must grow weaker as the quarks come closer together. But this didn’t seem to happen in the quantum field theories then familiar to theorists. In quantum electrodynamics, for example, one finds the opposite behavior: The force gets stronger as electrons approach each other. There was even an argument, seemingly based on general principles of quantum mechanics, that this would always be the case.
Gross and his then graduate student Wilczek, in fact, set out to prove that no known quantum field theory had this property. Politzer pursued the same problem but with a different outlook, at the suggestion of his thesis adviser, the late Sidney Coleman (who we’ll meet again later). For theories like QED and Yukawa’s meson theory, the calculations were relatively easy and familiar. But for one class of theories the problem was more challenging. These are the theories of Yang and Mills, known today as non-Abelian gauge theories or Yang-Mills theories, which we encountered earlier. For a decade, these theories had languished. They were fascinating but hard to understand, and no one put forth a convincing role for them in understanding the laws of nature. While there had been some progress, they were not understood at the level at which QED was understood. Richard Feynman, much as he had in the late 1940s for QED, guessed a set of rules for performing calculations in these theories. The role of Freeman Dyson in making sense of Feynman’s guess was played, at first, by two Soviet physicists, Ludvig Faddeev and Victor Popov. In a real sense they outdid Feynman. In figuring out the quantum mechanics of Yang-Mills theories, they took one of Feynman’s more outlandish early ideas and turned it into a useful and powerful tool.
