All forces in nature—be they gravitational, electric, or nuclear—have the same origin. Think of an electron circling a central nucleus. From time to time it emits a photon, and where does the photon go? If the atom is excited, the photon can escape while the electron jumps to a lower-energy orbit. But if the atom is already in its lowest-energy state, the photon cannot carry off any energy. The only alternative for the photon is to be absorbed, either by another electron or by the charged nucleus. Thus, in a real atom, the electrons and nucleus are constantly tossing photons back and forth in a kind of atomic juggling act. This “exchange” of particles, in this case the photon, is the source of all forces in nature. Force—whether it is electric, magnetic, gravitational, or any other—ultimately traces back to Feynman “exchange diagrams,” in which quanta hop from one particle to another. For the electric and magnetic force, photons are the exchanged quanta; for the gravitational forces the graviton does the job. You and I are kept anchored to the earth by gravitons jumping between earth and our bodies. But for the forces binding protons and neutrons into nuclei, the exchanged objects are pions. If one goes deeper into the protons and neutrons, quarks are found tossing gluons between them. This connection between force and exchanged “messenger” particles was one of the great themes of twentieth-century physics.
If the origins of the nuclear, electromagnetic, and gravitational forces are so similar, how is it that the results are so different? The electromagnetic and gravitational forces are long range, long enough for gravity to keep the planets in orbit, while the nuclear force becomes negligible when the nuclear particles are separated by the diameter of a single proton. If you are thinking that the difference has to do with some property of the messengers—graviton, photon, pion, and gluon—you’re absolutely right. The determining factor for the range of a particular force is the mass of the messenger: the lighter the messenger, the longer the range. The reason that gravity and electric forces are long range is that the graviton and photon are massless. But the pion is not massless; it is almost three hundred times heavier than the electron. The result of this much mass carried by the messenger is that, like an overweight athlete, it can’t jump more than a small distance to bridge the gap between distant particles.
String Theory is also a theory of forces. Let’s go back to the dance of the strings. As before, two lines of dancers approach each other. This time, instead of temporarily joining to form a single string, they do a different dance. Before they meet, one of the strings splits off a few of its members to form a short third string. The third string then runs over to the other group and joins up with it. All in all the two initial groups of dancers exchange a short string and, in so doing, a force is produced between the two groups.
From a distance the world sheet describing this exchange dance would look like the letter H, but under the microscope the lines that form it would be revealed as plumbing. The crossbar of the H is the world sheet of an exchanged string that jumps across the space between the vertical legs and creates a force between them. In those first days of String Theory, those of us who hoped to explain everything about hadrons were delighted with the possibility of explaining the nuclear force that binds protons and neutrons to form nuclei.
Unfortunately our hopes were soon shattered. When the calculations were done, the force law between strings looked nothing like the real forces holding the nucleus together. Instead of the short-range force of nuclear physics, we found long-range forces that more closely resembled electric and gravitational forces, as I mentioned earlier. The reason was not hard to find. Among the particle-like vibrating strings, there were two particular objects with a very special property—one, an open string of the kind that describes mesons, and the other, a closed glueball. Both of them had the exceptional trait of having absolutely no mass, just like the photon and the graviton! When juggled between other particles, they create forces almost exactly like the electric force between charged particles and the gravitational forces between masses. The open string mimicked the photon, but to me the biggest surprise was the similarity between the closed glueball and the mysterious elusive graviton. This would have been a source of unbounded joy if we were trying to create a new theory of gravity and electricity, but that was far from our goal. We were in the business of describing nuclear forces and had unquestionably failed. An impasse had been reached.
String Theory had one more difficulty. It was either a “theory of everything” or a “theory of nothing.” The original goal of String Theory was to describe hadrons—nothing more. The electron, photon, and graviton were to remain point particles. Experiments over the years had taught us that electrons and photons, if they had any size at all, were vastly smaller than hadrons. They might as well be mere points as far as anyone could tell. On the other hand, hadrons are obviously not points. A point cannot be spun about an axis. When I think of spinning an object, I think of a pizza chef spinning a wad of dough or a basketball player spinning the ball on his finger. But you can’t spin an infinitely small point. Hadrons can be easily spun: the excited spinning hadrons were encountered regularly in many accelerator laboratories. Hadrons must be more like a wad of dough than a mathematical point. But no one has ever spun a photon or an electron.4
Real hadrons can and do interact with point particles. A proton can absorb and emit a photon in the same way that an electron does. But when we tried to develop the theory so that the stringlike hadrons could interact with photons, all hell broke loose. One mathematical contradiction after another obstructed our attempts.
The obvious idea occurred to many people. The vibrating strings were certainly not points, but we had always assumed that the string ends were pointlike quarks. Why not allow all the electric charge to reside on the quarks? After all, making a point charge interact with a pointlike photon was child’s work. But as we know, the best-laid plans sometimes go awry. The mathematics just wouldn’t hold together.
The problem was that the strings of String Theory have an exceptionally violent case of the quantum jitters. The very high-frequency quantum fluctuations are so wild and out of control that the quark ends are most likely found far away at the very edges of the universe. It sounds crazy but the bits of string jitter and jiggle so violently that if you looked very quickly, you would discover that they are infinitely far away!
Let me try to explain this extraordinary unintuitive behavior of strings. The easiest way is to imagine a guitar string. The guitar string is somewhat different from the strings that we have to deal with in String Theory. For one thing their ends are held in place by the rest of the guitar. But that is not an important consideration right now. The important point is that both kinds of strings can vibrate in a wide variety of patterns. The guitar string can vibrate as a whole, forming a long, bowlike shape that oscillates as one long string. In that configuration, the string sounds the fundamental note.
But as every guitar player knows, a string can be made to vibrate in higher-pitched harmonics or modes of oscillation. These are vibrations in which the string vibrates in sections as if it were multiple shorter strings. For example, in the first harmonic the two halves of the string move separately.
In principle an ideal infinitely thin string could oscillate in an infinite number of harmonics at higher and higher frequency, although in practice, friction and other contaminating influences damp these vibrations almost before they get started.
Now let’s recall the quantum mechanics lesson of chapter 1. Every oscillation has a bit of jittery zero-point motion that is unavoidable. This has a very dramatic consequence for a perfectly ideal string: all of the possible vibrations, all infinitely many modes of oscillation simultaneously vibrate in a mad symphony of pure noise. All the various oscillations mount up in their influence on the location of bits of the string and would cause it to vibrate to infinite distance.
Why doesn’t this insane oscillation happen to a real guitar string? The reason is that an ordinary string is made of atoms spaced out along the
string. There is no meaning to vibrations in which the string vibrates in more pieces than the number of atoms it contains. But a mathematically ideal string, not made of atoms but continuous along its length, would vibrate in this uncontrolled way.
Perhaps the most amazing mathematical miracle of String Theory is that if everything is set up just right, including the fact that the number of space-time dimensions is ten, the wild vibrations of different strings precisely match each other and cause no harm. Your strings and my strings might wildly oscillate to the ends of the universe, but if the world is ten-dimensional, these oscillations are miraculously undetectable.
But this miracle of the strings works only if everything in the world is made of strings. If the photon were a point particle and the proton a string, a horrible clash would occur. For this reason the only things that strings can interact with are other strings! This is what I meant when I said that String Theory is either a theory of everything or a theory of nothing.
Violently jittering strings, fluctuating out to the boundaries of the universe, seemed so dismal a prospect that I gave up thinking about the inflexible mathematics of String Theory for more than a decade. But in the end this berserk behavior became the basis for one of the most exciting and strange developments of modern theoretical physics. In chapter 10 we will meet the Holographic Principle, which states that the world is a kind of quantum hologram at the boundaries of space. In part it was inspired by the extreme jitters of strings. But the Holographic Principle is a feature of the quantum mechanics of gravity, not nuclear physics.
Some theories are so mathematically precise that they are inflexible. That’s a good thing if the theory is successful. But if it doesn’t quite work, the inflexibility may become a liability. The String Theory that existed in the seventies, eighties, and most of the nineties could not interact with any objects that were not strings. If the purpose was to describe hadrons, the theory was not promising: it involved too many dimensions, massless gravitons and photons, and the impossibility of inter-acting with smaller objects. String Theory was in trouble, at least in its guise as a theory of hadrons. Nevertheless, there was no denying that hadrons behaved like stretchable strings with quarks at the end. In the thirty-five years since the discovery of String Theory, the stringy nature of hadrons has become a well-established experimental fact. But in the meantime String Theory found another life for itself. How String Theory was reborn as a fundamental theory unifying quantum mechanics and gravity is the subject of the next chapter.
CHAPTER EIGHT
Reincarnation
Although the String Theory of hadrons had failed in its most precise mathematical form, some brave souls saw opportunity in the train wreck. “If the mountain won’t come to Mohammad, Mohammad will go to the mountain.” If you can’t make a String Theory of hadrons because the theory persists in behaving like gravity, then let gravity be described by String Theory. Why not use it to describe everything: gravity, electromagnetic forces, quarks, and all the rest? Problems two and three from the last chapter are gone; the predicted range of forces now fits the reality, and everything is made of strings. The inflexibility of the theory becomes an asset. A radically new vision of the world made up of one-dimensional threads of energy, fluctuating wildly out to the edges of the universe, would replace an older vision of matter made of point particles.
To give you a picture of what this transformation of String Theory meant, let’s talk about size scales. Hadrons have a size somewhere between 10–13 and 10–14 centimeters. There is some variation, but mesons, baryons, and glueballs are all in this range. While 10–13 centimeters may seem terribly small, 100,000 times smaller than an atom, by the standards of modern particle physics, it’s very large. Accelerators have long probed objects 1,000 times smaller and are beginning to get down to 10,000 times smaller.
The natural size of a graviton is vastly smaller. Gravitons, after all, are the result of mixing gravity together with the quantum nature of matter. And whenever you work on the quantum level, you will always discover exactly what Planck discovered in 1900: the natural unit of length is the minuscule Planck length—10–33 cm. Physicists expect that the size of a graviton is about that small.
Just how much smaller is a graviton than a proton? If the graviton were expanded until it was as big as the earth, the proton would become about 100 times bigger than the entire known universe! Using exactly the same String Theory that had failed as a theory of hadrons, string theorists such as John Schwarz and Joel Sherk were proposing to leapfrog completely over that vast range of scales. Like MacArthur’s frog leap across the Pacific, it was either a very bold, heroic move or a very foolish one.
If the range of forces was no longer a problem, the dimensionality of space was; mathematical consistency still required nine dimensions of space and one more for time. But in the new context, this turned out to be a blessing. The list of elementary particles of the Standard Model—the particles that are supposed to be points—is a long one. It includes thirty-six distinct kinds of quarks, eight gluons, electrons, muons, tau leptons1 and their antiparticles, two kinds of W-bosons, a Z-boson, the Higgs particle, photons, and three neutrinos. Each type of particle is distinctly different from all the others. Each has its own particular properties. You might say they have their own personalities. But if particles are mere points, how can they have personalities? How can we account for all their properties, their quantum numbers, such as spin, isospin, strangeness, charm, baryon number, lepton number, and color?2 Particles evidently have a lot of internal machinery that can’t be seen from a distance. Their pointlike, structureless appearance is surely a temporary consequence of the limited resolving power of our best microscopes, i.e., particle accelerators. Indeed the resolving power of an accelerator can be improved only by increasing the energy of the accelerated particles, and the only way to do that is to increase the size of the accelerator. If, as most physicists believe, the internal machinery of elementary particles were revealed only at the Planck scale, it would be necessary to build an accelerator at least as big as our entire galaxy! So we go on thinking of particles as points, despite the fact that they obviously have so many properties.
But String Theory is not a theory of point particles. From a theorist’s point of view, String Theory provides plenty of opportunity for particles to have properties. Among other things strings can vibrate in many different quantized patterns of vibration. Anyone who ever played the guitar knows that a guitar string can vibrate in many harmonics. The string can vibrate as a whole or it can vibrate in two pieces with a node in the middle. It can also vibrate in three or any number of separate sections, thus producing a series of harmonics. The same is true of the strings of String Theory. The different patterns of vibration do produce particles of different types, but this in itself is not enough to explain the difference between electrons and neutrinos, photons and gluons, or up-quarks and charmed-quarks.
Here’s where string theorists made brilliant use of what had previously been their greatest embarrassment. The sow’s ear—too many dimensions—was turned into a silk purse. The key to the unexplained diversity of elementary particles—their electric charge, color, strangeness, isospin, and more—is very likely the extra six dimensions that previously dogged our efforts to explain hadrons!
At first sight there doesn’t seem to be an obvious connection. How does moving around in six extra dimensions explain electric charge or the difference between quark types? The answer lies in the profound changes in the nature of space that Einstein explained with his General Theory of Relativity—the possibility that space, or some part of space, can be compact.
Compactification
The easiest examples of compactification are two-dimensional. Once again let’s imagine that space is a flat sheet of paper. The paper could be unbounded, an infinite sheet that stretches endlessly in every direction. But there are other possibilities. When discussing Einstein’s and Friedmann’s universes, it was necessary to conceive of a two-
dimensional space with the shape of a 2-sphere—a closed-and-bounded space. No matter what direction you travel in, you eventually come back to the starting point.
Einstein and Friedmann were imagining space to be a gigantic sphere, big enough to move around in for billions of years without encountering the same galaxy or star twice. But now imagine shrinking the sphere smaller and smaller until it is far too small to hold a human being or even a molecule, an atom or even a proton. If the 2-sphere is shrunken to microscopic proportions, it becomes hard to distinguish it from a point—a space with no dimensions to move in. This is the simplest example of hiding dimensions by compactifying, or shrinking, them.
Can we somehow choose the shape of a two-dimensional space so that it looks for most purposes like a one-dimensional space? Can we effectively hide one of the two dimensions of the sheet of paper? Indeed, we can. Here’s how you do it: Start with an imaginary infinite sheet of paper. Cut out an infinite strip a few inches wide. Let’s say the strip is along the x-axis. The tip of your pencil can move forever along the x-axis, but if you move it in the y-direction, you’ll soon come to one of the edges. Now take the strip and bend it into a cylinder so that the upper and lower edges are joined in a smooth seam. The result is an infinite cylinder that can be described as compact (finite) in the y-direction, but infinite in the x-direction.
Let’s imagine such a space, but instead of making the y-direction a few inches in circumference, let’s take it to be a micron (one ten-thousandth of a centimeter). If we looked at the cylinder without a microscope, it would look like a one-dimensional space, an infinitely thin “hair.” Only if we look through a microscope will it reveal itself to be two-dimensional. In this way a two-dimensional space is disguised as one-dimensional.