Finally we come to the strongest of all forces, those that hold the atomic nucleus together. A nucleus is composed of electrically neutral neutrons and positively charged protons. No negative charges are found in a nucleus. Why doesn’t it blow up? Because the individual protons and neutrons attract with a nonelectric force about fifty times stronger than the electrical repulsion. The quarks that make up a single proton have even stronger forces binding them together. How is it that our protons and neutrons aren’t attracted to the protons and neutrons of the earth by such powerful forces? The answer is that although the nuclear force is powerful, it is also very short range. It is easily strong enough to overcome the electric repulsion of protons, but only when particles are close together. Once they separate by more than a couple of proton diameters, the force becomes negligible. Underlying the strong inter-actions are the powerful forces between quarks, the elementary particles that make up hadrons.
I often feel a discomfort, a kind of embarrassment, when I explain elementary-particle physics to laypeople. It all seems so arbitrary—the ridiculous collection of fundamental particles, the lack of pattern to their masses, and especially the four forces, so different from each other, with no apparent rhyme or reason. Is the universe “elegant” as Brian Greene tells us? Not as far as I can tell, not the usual laws of particle physics anyway. But in the context of a megaverse of wild diversity, there is a pattern. All of the forces and most of the elementary particles are absolutely essential. Change any of it more than a bit, and life as we know it becomes impossible.
Origins of String Theory
A peculiar ideology insinuated itself into the high-energy theoretical physics of the 1960s. It paralleled almost exactly a fad that had taken hold in psychology. B. F. Skinner was the guru of the behaviorists, who insisted that only the external behavior of a human being was the proper material of mind science. According to Skinner, psychologists had no business inquiring into the inner mental states of their subjects. He even went so far as to declare that no such thing existed. The business of psychology was to watch, measure, and record the external behavior of subjects without ever inquiring about internal feelings, thoughts, or emotions. To the behaviorists a human was a black box that converted sensory input into behavioral output. While it is probably true that Freudians went too far in the other direction, the behaviorists carried their ideology to extremes.
The behaviorism of physics was called S-matrix theory. Sometime in the early sixties, while I was a graduate student, some very influential theoretical physicists, centered in Berkeley, decided that physicists had no business trying to explain the inner workings of hadrons. Instead, they should think of the Laws of Physics as a black box—a black box called the Scattering Matrix, or S-matrix for short. Like the behaviorists the S-matrix advocates wanted theoretical physics to stay close to experimental data and not wander off into speculation about unobservable events taking place inside the (what was then considered) absurdly small dimensions characteristic of particles like the proton.
The input to the black box is some specified set of particles coming toward one another, about to collide. They could be protons, neutrons, mesons, or even nuclei of atoms. Each particle has a specified momentum as well as a host of other properties like spin, electric charge, and so on. Into the metaphorical black box they disappear. And what comes out of the black box is also a group of particles—the products of collision, again with specified properties. The Berkeley dogma forbade looking into the box to unravel the underlying mechanisms. The initial and final particles are everything. This is very close to what experimental physicists do with accelerators to produce the incoming particles and with detectors to detect what emerges from the collision.
The S-matrix is basically a table of quantum-mechanical probabilities. You plug in the input, and the S-matrix tells you the probability for a given output. The table of probabilities depends on the direction and energy of both the incoming and outgoing particles, and according to the prevailing ideology of the mid-1960s, the theory of elementary particles should be confined to studying the way the S-matrix depends on these variables. Everything else was forbidden. The ideologues had decided that they knew what constituted good science and became the guardians of scientific purity. S-matrix theory was a healthy reminder that physics is an empirical subject, but like behaviorism, the S-matrix philosophy went too far. For me it turned all of the wonder of the world into the gray sterility of an accountant’s actuarial tables. I was a rebel, but a rebel without a theory.
In 1968 Gabriele Veneziano was a young Italian physicist living and working at the Weizmann Institute, in Israel. He was not especially ideological about S-matrix theory, but the mathematical challenge of figuring out the S-matrix appealed to him. The S-matrix was supposed to satisfy certain technical requirements, but no one at that time could point to a specific mathematical expression that satisfied the rules. So Veneziano tried to find one. The attack was brilliant. The result, famous today as the “Veneziano amplitude,” was extremely neat. But it was not a picture of what particles were made of or of how the processes of collision could be visualized. The Veneziano amplitude was an elegant mathematical expression—an elegant mathematical table of probabilities.
The discovery of String Theory, which in a sense is still ongoing, was full of twists of fate, reversals of fortune, and serendipity. My own involvement with it began sometime in 1968 or early 1969. I was beginning to tire of the problems of elementary particles, especially hadrons, which seemed to have little to offer in the way of deep, new principles. I found the S-matrix approach boring and was beginning to think about the relation between quantum mechanics and gravity. Putting the General Theory of Relativity together with the principles of quantum mechanics seemed far more exciting, even if all the experimental data were about hadrons. But just at that time, a friend from Israel visited me in New York. The friend, Hector Rubinstein, was extremely excited about Veneziano’s work. At first I was not very interested. Hadrons were exactly what I wanted to forget about. Mainly out of politeness I decided to hear Hector out.
Hector became so excited while explaining the Italian’s idea that I really couldn’t follow the details. As far as I could make out, Veneziano had worked out a formula for describing what happens when two hadrons collide. He finally wrote down Veneziano’s formula on the blackboard in my office. It immediately struck a chord. It was extremely simple, and features of the formula looked familiar to me. I recall asking Hector, “Does this formula represent some kind of simple quantum-mechanical system? It looks like it has something to do with harmonic oscillators.” Hector didn’t know of a physical picture that went with the formula, so I wrote it down on a sheet of paper to remember.
I was intrigued enough to postpone thinking about quantum gravity and give hadrons another chance. As it turned out I didn’t seriously think about gravity again for more than a decade. I pondered the formula above for several months before I began to see what it really represented.
The term harmonic oscillator is physics language for anything that can vibrate or swing back and forth with a periodic (repeating) motion. A child on a playground swing or a weight hanging at the end of a spring are familiar harmonic oscillators. Vibrations of a violin string or even the oscillations of the air when a sound wave passes through it are also good examples. If the vibrating system is small enough—the vibrations of atoms in a molecule are an example—then quantum mechanics becomes important, and energy can be added to the oscillator only in discrete steps. I had mentioned the harmonic oscillator to Hector because certain features of Veneziano’s formula reminded me of the mathematical properties of quantum-mechanical harmonic oscillators. I imagined a hadron as two weights connected by a spring, vibrating in periodic oscillation—the weights first approaching and then receding from each other. I was clearly playing with forbidden fruit, trying to picture the internal machinery inside elementary particles, and I knew it.
Being tantalizingly close to the answer
but not quite being there is maddening. I tried all sorts of quantum-mechanically oscillating systems, attempting to match them with Veneziano’s formula. I was able to produce formulas that looked a lot like Veneziano’s from the simple weight and spring model, but they weren’t quite right. During that period I spent long hours by myself, working in the attic of my house. I hardly came out, and when I did I was irritable. I barked at my wife and ignored my kids. I couldn’t put the formula out of my mind, even long enough to eat dinner. But then for no good reason, one evening in the attic I suddenly had a “eureka moment.” I don’t know what provoked the thought. One minute I saw a spring, and the next I could visualize an elastic string, stretched between two quarks and vibrating in many different patterns of oscillation. I knew in an instant that replacing the mathematical spring with the continuous material of a vibrating string would do the trick. Actually, the word string is not what flashed into my mind. A rubber band is the way I thought of it: a rubber band cut open so that it became an elastic string with two ends. At each end I pictured a quark or, more precisely, a quark at one end and an antiquark at the other.
I quickly did a few calculations in my notebook to test the idea, but I already knew that it would work. The simplicity of it was stunning. Veneziano’s S-matrix formula precisely described two colliding “rubber bands.” I didn’t know why I hadn’t thought of it earlier.
Nothing is quite like the excitement of a new discovery. It doesn’t happen often, even for the greatest physicists. You say to yourself, “Here I am, the only one on the planet who knows this thing. Soon the rest of the world will know, but for the moment I am the only one.” I was young and unknown but with visions of glory.
But I wasn’t the only one. At just about the same time, a physicist in Chicago was doing the same calculations. Yoichiro Nambu was a good deal older than I and had long been one of the most eminent theoretical physicists in the world. Born in Japan, he came to the University of Chicago as a young physicist right after World War II. Nambu was a star who had the reputation of seeing things long in advance of anyone else. Later I was to find out that yet another physicist in Denmark, Holger Bech Nielsen, was thinking about very similar ideas. I won’t deny that I was disappointed when I found out that I wasn’t alone in thinking of the “rubber band theory,” but being in the same company as the great Nambu had its own satisfactions.
Today’s modern String Theory is all about the elusive unification of quantum mechanics and gravity, over which physicists banged their collective head for much of the twentieth century. That means that it is a theory of what the world is like at that fabulously tiny scale of the Planck length, 10–33 centimeters. As I have explained, it started out much more modestly as a theory of hadrons. We will see in the next chapter how it morphed into a much deeper fundamental theory, but let’s follow its earlier incarnation.
Hadrons are small objects, typically about 100,000 times smaller than an atom. This makes them 10–13 centimeters in diameter. It takes an enormous force to bind quarks at such small separation. Hadronic strings, the rubber bands of my imagination, although microscopically small, are prodigiously strong. If you could find a way to attach one end of a meson (one kind of hadron) to a car and the other end to a crane, you could easily lift the car. Hadronic strings are not particularly small on the scale of today’s experiments. Modern accelerators are probing nature at scales from a hundred to a thousand times smaller. Just for comparison let me get ahead of the story and tell you what the strength of a string is in the modern reincarnation. In order to hold particles together at the Planck distance, a string would have to be about 1040 times stronger than the hadronic strings; one of them could support a weight equal to the entire mass of our galaxy if we could somehow concentrate the galaxy at the surface of the earth.
All hadrons belong to one of three families: baryons, mesons, and glueballs. Nucleons, the ordinary protons and neutrons of nuclear physics, are the most familiar hadrons. They belong to the first family, called baryons.2 All baryons are composed of three quarks. The quarks are connected to one another by means of three strings in the manner of a gaucho’s bola: three strings joined at the center, with three quarks at their ends. The only thing wrong with the bola picture is that the hadronic strings are elastic, much like ideal stretchable bungee cords. The ordinary proton and neutron are the lowest energy configurations of the bola, with the quarks at rest at the ends of very short, unstretched strings.
The quarks at the ends of the strings can be set into motion in a number of ways. The bola can be spun around its center, the centrifugal force stretching the strings and pushing the quarks out from the center. This spinning motion requires energy (remember E = mc2), and that makes spinning hadrons heavier. As noted earlier, the jargon for a particle with extra energy is that it is excited. The quarks can also be excited without rotating. One way is through oscillating motions, moving toward and away from the center, in and out, in and out. In addition the strings themselves can be bent into curved, vibrating patterns almost as if they were plucked with a guitar pick. All of these motions, or at least indirect evidence of them, are routinely seen in real experiments on nucleons. Baryons really do behave like elastic quantum bolas.
What does it mean that they are quantum bolas? Quantum mechanics implies that the energy (mass) of any vibrating system can be added only in indivisible, discrete steps. In the earliest days of experimental hadron physics, physicists didn’t realize that the discretely different quantum states of the vibrating system were really the same object. They gave each energy level a different name and considered them all to be different particles. The proton and the neutron were the baryons with the least energy. The more massive ones had odd names that would mean absolutely nothing to most young physicists today. These particles are nothing more than rotating or vibrating excited states of the proton and neutron. When this was realized, of course, it brought a lot of order and unity to what had been a very messy zoo of particles.
Next come the mesons, the particles that I studied in my attic in 1969. They are simpler than baryons. Each meson is made of a single string with a quark at one end and an antiquark at the other. Mesons, like baryons, can rotate and vibrate in discrete quantum steps. The calculation that I did in the attic represented a fundamental process of interaction between two meson strings.
When two mesons collide they can do a number of things. Because quantum mechanics is a theory of probabilities, it is impossible to predict with certainty how the history of the collision will unfold. One possibility, in fact the most likely one, is that the two mesons will go right past each other, even if it means that the strings pass through each other. But a second, more interesting, possibility is that they can fuse, joining together, to form a single, longer string.
Imagine each string to be a group of dancers holding hands to form a line. At each end the dancers have one hand free (a quark or antiquark); all the others have both hands occupied. Picture the two lines racing toward each other. The only way they are allowed to interact is by a dancer at the end of one line clasping a free hand of the other group. Once they are joined they form a single chain. In this configuration they swing around one another in a complicated dance until somewhere along the chain a dancer releases his neighbor’s hand. Then the chain splits into two independent chains, and off they go, separating in some new direction. More precisely but less colorfully, the quark from the end of one string comes together with the antiquark of the other string. They collide and annihilate, as always, when a particle and an antiparticle come together. What they leave over is a single, longer string with a single quark and a single antiquark.
The resulting single string is usually left in an excited state vibrating and rotating. But after a short time, like the chain of dancers, the string can break in two by reversing the process that joined the original strings. The net result is an operation in which a pair of strings comes together, forms a compound string, and then splits back into two strings.
&nb
sp; The problem that I had solved in the attic was this: suppose two mesons (strings) were originally moving with a given energy in opposite directions before they collided. What is the quantum-mechanical probability that, after they collide, the resulting pair of strings will be moving along some specified new direction? It sounds like a horribly complicated problem, and it was something of a mathematical miracle that it could be solved.
The mathematical problem of describing an idealized elastic string had already been solved by the early nineteenth century. A vibrating string can be viewed as a collection of harmonic oscillators, one for each separate type of oscillating motion. The harmonic oscillator is one of the few physical systems that can be completely analyzed with very simple high school mathematics.
Adding quantum mechanics to make the string a quantum object was also simple. All that was necessary was to remember that the energy levels of any oscillating system come in discrete units of energy (see chapter 1). This simple observation was sufficient to understand the properties of a single vibrating string; but describing two interacting strings was much more intricate. For that I had to work out my own rules from scratch. What made it possible was that the complexity lasts for only an infinitesimal instant of time when the ends touch and join. Once that happens the two strings become one string, and the simple math of a single string takes over. A little later the single string splits, but again, the complicated event takes only an instant. Thus, with great precision, I was able to follow the two strings as they merged and separated. The results of the mathematical calculation could be compared with Veneziano’s formula, and gratifyingly, they agreed exactly.