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Authors: Leonard Susskind

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BOOK: The Cosmic Landscape
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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 leptons
1
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.

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.

BOOK: The Cosmic Landscape
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