Suppose we further reduce the size of the compact direction all the way down to the Planck length. Then no existing microscope would be able to resolve the second dimension. For all practical purposes the space would be one-dimensional. This process of making some directions finite and leaving the rest infinite is called compactification.
Now let’s make things a little harder. Take three-dimensional space with three axes: x, y, and now z. Let’s leave the x- and y-directions infinite but roll up the z-axis. It’s harder to visualize, but the principle is the same. If you move in the x- or y-direction, you go on forever, but moving along z brings you back to the start after a certain distance. If that distance were microscopic, it would be hard to tell that the space was not two-dimensional.
We can go a little further and compactify both the z- and the y-direction. For the moment completely ignore the x-direction and concentrate on the other directions. One thing you can do with two directions is to roll them into a 2-sphere. In this case you can move forever along the x-direction, but moving in the y- and z-direction is like moving on the surface of a globe. Again, if the 2-sphere were microscopic, it would be hard to tell that the space was not one-dimensional. So you see we can choose to hide any number of dimensions by rolling them up into a small compact space.
The 2-sphere is only one way to compactify two dimensions. Another very simple way is to use a torus. Just as the 2-sphere is the surface of a ball, the torus is the surface of a bagel. There are lots of other shapes you could use, but the torus is the most common.
Let’s return to the cylinder and imagine a particle moving on it. The particle can move up and down the infinite x-axis exactly as if the space were only one-dimensional. It has a velocity along the x-direction. But moving in the x-direction is not the only thing the particle can do: it can also move along the compact y-direction, circling the y-axis endlessly. With this new motion the particle has velocity along the hidden microscopic direction. It can move in the x-direction, the y-direction, or even with both motions simultaneously, in a helical (corkscrew) motion, winding around y while traveling along x. To the observer who cannot resolve the y-direction, that additional motion represents some new peculiar property of the particle. A particle moving with velocity along the y-axis is different from a particle with no such motion, and yet the origin of this difference would be hidden by the smallness of y. What should we make of this new property of the particle?
The idea that there might be an extra, unobserved direction to space is not new. It goes all the way back to the early years of the twentieth century, shortly after Einstein completed the General Theory of Relativity. A contemporary of Einstein’s named Theodor Franz Eduard Kaluza began to think about exactly this question—how would physics be influenced if there were an extra direction of space? At that time the two important forces of nature were the electromagnetic force and the gravitational force. In some ways they were similar, but Einstein’s theory of gravity seemed to have a much deeper origin than Maxwell’s theory of electromagnetism. Geometry itself—the elastic, bendable properties of space—were what gravity was all about. Maxwell’s theory just seemed like an arbitrary “add-on” that had no fundamental reason in the scheme of things. The geometry of space was just right to describe the properties of the gravitational field and no more. If the electric and magnetic forces were somehow to be united with gravity, the basic geometric properties of space would have to be more complex than envisioned by Einstein.
What Kaluza discovered was amazing. If one additional direction of space were added to the usual 3+1 dimensions, the geometry of space would encompass not only Einstein’s gravitational field but also Maxwell’s electromagnetic field. Gravity and electricity and magnetism would be unified under a single all-encompassing theory. Kaluza’s idea was brilliant and caught the attention of Einstein, who liked it very much. According to Kaluza, particles could move not only in the usual three spatial dimensions but also in a fourth, hidden dimension. However, the theory had one obvious, enormous problem. If space has an extra dimension, why don’t we notice it? How is the extra fourth dimension of space hidden from our senses? Neither Kaluza nor Einstein had an answer. But in 1926 the Swedish physicist Oscar Klein did have an answer. He added the new element that made sense out of Kaluza’s idea: the extra dimension must be rolled up into a tiny compact space. Today theories with extra compact dimensions are known as Kaluza Klein theories.
Kaluza and Klein discovered that the gravitational force between two particles was modified if both particles moved in the additional direction. The astonishing thing was that the extra force was identical to the electric force between charged particles. Moreover, the electric charge of each particle was nothing but the component of momentum in the extra dimension. If the two particles cycled in the same direction around the compact space, they repelled each other. If they moved in opposite directions, they attracted. But if either of them did not cycle in the compact direction, then only the ordinary gravitational attraction affected them. This smells like the beginnings of an explanation of why some particles (the electron, for example) are electrically charged, while other, similar particles (neutrinos) are electrically neutral. Charged particles move in the compact direction of space, while those without charge have no motion in this direction. It even begins to explain the difference between the electron and its antiparticle, the positron. The electron cycles around the compact direction one way, say, clockwise, while the positron moves counterclockwise.
Another insight was added by quantum mechanics. Like all other cyclic or oscillating motions, the motion around the compact y-axis is quantized. The particle cannot cycle the y-axis with an arbitrary value of the y-momentum: it is quantized in discrete units just like the motion of a harmonic oscillator or the electron in Bohr’s theory of the atom. This means that the y-motion, and therefore, the electric charge, cannot be any old number. Electric charge in Kaluza’s theory is quantized: it comes in integer multiples of the electron charge. A particle with charge twice or three times the electron would be possible but not one with charge 1/2 or .067 times the electron charge. This is a highly desirable state of affairs. In the real world no object has ever been discovered carrying a fractional electric charge: all electric charges are measured in integer multiples of the electron’s charge.
This was a spectacular discovery, which largely lay dormant for the rest of Kaluza’s life. But it’s the heart of our story. The Kaluza theory is a model of how properties of particles can arise from extra dimensions of space. Indeed, when string theorists discovered that their theory required six extra dimensions of space, they seized on Kaluza’s idea. Just roll up the extra six directions in some manner and use the motion in the new directions to explain the internal machinery of elementary particles.
String Theory is richer in possibilities than theories of point particles. Returning to the cylinder, let’s suppose a small closed string is moving on a cylinder. Start with a cylinder of circumference large enough to be visible to the naked eye. A tiny closed string moves on the cylinder pretty much the same way a point particle would. It can move along the length of the cylinder or around it. In this respect it is no different from the point particle. But the string can do something else that the point cannot. The string can wind around the cylinder just like a real rubber band can be wrapped around a cardboard cylinder. The wound string is different from the unwound string. In fact the rubber band can be wound any number of times around a cardboard cylinder, at least if it doesn’t break. This gives us a new property of particles: a property that depends not only on the compactness of a dimension but also on the fact that particles are strings or rubber bands. The new property is called the winding number, and it represents the number of times the string is wound around the compact direction.
The winding number is a property of a particle that we could not understand if our microscope was not strong enough to resolve the tiny distance around the compact direction. So you see, the extra dimensions n
eeded by String Theory are a blessing, not a curse: they are essential for understanding the complex properties of elementary particles.
A two-dimensional cylinder is easy to visualize, but a nine-dimensional space with six dimensions wrapped up into some kind of six-dimensional tiny space is beyond anyone’s powers of visualization. But making pictures in your head or on paper is not the only way to understand the mind-boggling six-dimensional geometry of String Theory. Geometry can often be reduced to algebra just the way you learned in high school when you used an equation to represent a straight line or a circle. Still, even the most powerful methods of mathematics are barely enough to scratch the surface of six-dimensional geometry.
For example, the number of possible ways String Theory allows of rolling up six dimensions runs into the millions. I won’t try to describe them other than to give you the special mathematical name for these spaces; they are called Calabi Yau spaces, after the two mathematicians who first studied them. I don’t know why mathematicians were interested in these spaces, but they came in handy for string theorists. Fortunately for us the only thing we have to know is that they are extremely complicated, with hundreds of “donut holes” and other features.
Back to the two-dimensional cylinder. The distance around the cylinder is called the compactification scale. For a cardboard cylinder it might be a few inches, but for String Theory it should most likely be a few Planck lengths. You might think that this scale is so small that it doesn’t matter for anything we care about, but that’s not so. Although we can’t actually see such small scales, they nonetheless have their effect on ordinary physics. The compactification scale in Kaluza’s theory fixes the magnitude of the electric charge of a particle like the electron. It also fixes the masses of many of the particles. In other words the scale of compactification determines various constants that appear in the ordinary Laws of Physics. Vary the size of the cylinder, and the Laws of Physics change. Vary the values of scalar fields as in chapter 1, and the Laws of Physics change. Is there a connection? Absolutely! And we will return to it.
To specify the cylinder you need to specify only one parameter, the scale of compactification, but other shapes require more. For example, a torus is determined by three parameters. See if you can visualize them. First there is the overall size of the torus. Keeping the shape fixed, the torus can be magnified or shrunk. In addition, the torus can be “thin” like a narrow ring or “fat” like an overstuffed bagel. The parameter that determines the fatness is a ratio: the ratio of the size of the hole to the overall size. For the thin ring, the overall size and the size of the hole are about the same, so the ratio is near one. For the fat torus the hole is much smaller than the overall size, and the ratio is correspondingly small. There is one more quantity, which is harder to picture. Imagine taking a knife and cutting the ring, not in half but just so that it can be opened up to a section of a cylinder. Now twist one end of the cylinder, keeping the other end fixed. Finally, reconnect the ends of the cylinder so that it becomes a ring but with a twist. The angle of the twist is another variable. If you can’t picture it, that’s okay. You won’t need to.
Mathematicians call these parameters that determine the size and shape of the torus moduli (plural of modulus). A torus has three moduli. The cylinder, or more accurately, the circular cross section of the cylinder, has only one modulus. But a typical Calabi Yau space has hundreds. Perhaps you can see where this is going, but if not, I will spell it out. It is leading us to a Landscape—and an incredibly complicated one at that.
One very important issue is whether the size and shape of the compact component of the space can vary from one point to another. Visualize a clumsily constructed cylinder. Suppose that as you move along the length of the cylinder, the circumference of the cross section varies: here the cylinder is narrow, there it’s wider.
Keep in mind that even if the cylinder is extremely thin, far too thin to detect its compact dimension, the size of that dimension determines various coupling constants and masses. Evidently we have made a world where the Laws of Physics can vary from point to point. What does an ordinary physicist who cannot see the small dimension make out of all this? She says: “Conditions are varying from point to point. It seems that some kind of scalar field controls the value of the electric charge and mass of particles, and it can vary from point to point.” In other words, the moduli form some kind of Landscape—a Landscape of hundreds of dimensions.
A Calabi Yau space is vastly more complicated than the circular cross section of the cylinder, but the principle is the same: the size and shape of the compact space can vary with position just as if there were hundreds of scalar fields controlling the Laws of Physics! Now we begin to see why the Landscape of String Theory is so complicated.
The Elegant Supersymmetric Universe?
The real underlying principles of String Theory are largely shrouded in mystery. Almost everything we know about the theory involves a very special portion of the Landscape where the mathematics was amazingly simplified by a property that was mentioned in chapter 2 called supersymmetry. The supersymmetric regions of the Landscape form a perfectly flat plain at exactly zero altitude, with properties so symmetric that many things can be deduced without having complete mastery of the entire Landscape. If one were looking for simplicity and elegance, the flat plain of supersymmetric String Theory, a.k.a. superString Theory, is the place to look. In fact until a couple of years ago it was the only place string theorists looked. But some theoretical physicists are finally waking up and trying to wean themselves off the elegant simplifications of the superworld. The reason is simple: the real world is not supersymmetric.
The world of experience that includes the Standard Model and a small cosmological constant is not located on this plain of zero altitude. It is somewhere in the rough, textured regions of the Landscape with hills, valleys, high plateaus, and steep descents. But there is at least some reason to think that our valley is close to the supersymmetric part of the Landscape and that there may be remnants of the mathematical supermiracles that would help us understand features of the empirical world. One example that we will encounter in this section involves the mass of the Higgs boson. In a real sense the discoveries that made this book possible are all about the initial timid explorations away from the safety of the supersymmetric plain.
Supersymmetry is all about the distinctions and similarities of bosons and fermions. As with so much else in modern physics, the principles trace back to Einstein. The year 2005 marks the one hundredth anniversary of the anno mirabilis—the “miracle year”—of modern physics. Einstein set two revolutions in motion that year and completed a third.3 It was of course the year of the Special Theory of Relativity. But what many people don’t know is that 1905 was much more than the “relativity year.” It also marked the birth of the photon, the start of modern quantum mechanics.
Einstein received only one Nobel Prize in physics, although I think it’s fair to say that almost every prize given after 1905 was in one way or another a tribute to his discoveries. The prize was ultimately awarded not for relativity but for the photoelectric effect. This, the most radical of Einstein’s contributions, is where the idea that light is composed of discrete quanta of energy was first argued. Physics was ready for the Special Theory of Relativity. In fact it was overdue. But the photon theory of light was a bolt from the blue. As noted previously, Einstein argued that a beam of light, usually thought of as a pure wave phenomenon, had a discrete, or grainy, structure. If the light had a definite color (wavelength), then the photons would all be marching in lock step, each photon identical to every other photon. Particles that can all be in the same quantum state like photons are called bosons, after the Indian physicist Satyendra Nath Bose.
Almost twenty years later, building on Einstein’s work, Louis de Broglie closed the cycle by showing that electrons, the quintessential particles, have a wavelike side to them. Like other waves, electrons can reflect, refract, diffract, and interfere. But t
here is a fundamental difference between electrons and photons: unlike photons, no two electrons can ever occupy the same quantum state. The Pauli exclusion principle ensures that each electron in an atom must exist in its own quantum state and that no other electron can ever shoulder its way into an already occupied state. Even outside an atom, two otherwise identical electrons cannot have the same position or the same momentum. Particles of this kind are called fermions, after the Italian physicist Enrico Fermi, although they should be called paulions. Of all the particles of the Standard Model, about half are fermions (electrons, neutrinos, and quarks) and half are bosons (photons, Z- and W-bosons, gluons, and Higgs bosons).
Fermions and bosons play very different roles in the grand design. Ordinarily we think of matter as made up of atoms, and that means electrons and nuclei. To review, the nuclei at one level are bunches of protons and neutrons stuck together by means of the nuclear force, but at a deeper level, the protons and neutrons are made of even smaller building blocks—quarks. All of these particles—electrons, protons, neutrons, and quarks—are fermions. Matter is made of fermions. But without bosons the atoms and nuclei as well as the protons and neutrons would just fall apart. It’s the bosons, especially photons and gluons, hopping back and forth between fermions that create the attraction holding everything together. Although both fermions and bosons are critically important to making the world what it is, they were always thought of as very different kinds of creatures.
But sometime in the early 1970s, as a consequence of discoveries by string theorists, physicists began to play around with a new mathematical idea, the idea that fermions and bosons are not really so different. The idea is that all particles come in precisely matched pairs, identical twins that are the same in every way except that one is a fermion and one is a boson. This was a wild hypothesis. If true in the real world, it would mean that physicists had completely missed half the particles of nature, had failed to discover them in their labs. For example, this new principle would require that there is a particle exactly like the electron—same mass and charge—except that it would be a boson. How could we have failed to detect it in particle physics laboratories like SLAC and CERN? Supersymmetry implies that a fermion twin for the photon, massless and with no electric charge, would have to exist. Similarly, boson partners for electrons and quarks would be required. A complete world of “opposites” was mysteriously missing if this new idea were right. In fact the whole exercise was only a mathematical game, a pure theoretical exploration of a new kind of symmetry that a world—some world not our own—might possess.