Hyperspace (27 page)

The most remarkable feature of string theory, as we have emphasized, is that Einstein’s theory of gravity is automatically contained in it. In fact, the graviton (the quantum of gravity) emerges as the smallest vibration of the closed string. While GUTs strenuously avoided any mention of Einstein’s theory of gravity, the superstring theories demand that Einstein’s theory be included. For example, if we simply drop Einstein’s theory of gravity as one vibration of the string, then the theory becomes inconsistent and useless. This, in fact, is the reason why Witten was attracted to string theory in the first place. In 1982, he read a review article by John Schwarz and was stunned to realize that gravity emerges from superstring theory from self-consistency requirements alone. He recalls that it was “the greatest intellectual thrill of my life.” Witten says, “String theory is extremely attractive because gravity is forced upon us. All known consistent string theories include gravity, so while gravity is impossible in quantum field theory as we have known it, it’s obligatory in string theory.”
⁶

Gross takes satisfaction in believing that Einstein, if he were alive, would love superstring theory. He would love the fact that the beauty and simplicity of superstring theory ultimately come from a geometric principle, whose precise nature is still unknown. Gross claims, “Einstein would have been pleased with this, at least with the goal, if not the realization….
He would have liked the fact that there is an underlying geometrical principle—which, unfortunately, we don’t really understand.”
⁷

Witten even goes so far as to say that “all the really great ideas in physics” are “spinoffs” of superstring theory. By this, he means that all the great advances in theoretical physics are included within superstring theory. He even claims that Einstein’s general relativity theory being discovered before superstring theory was “a mere accident of the development on planet Earth.” He claims that, somewhere in outer space, “other civilizations in the universe” might have discovered superstring theory first, and derived general relativity as a by-product.
⁸

String theory is such a promising candidate for physics because it gives a simple origin of the symmetries found in particle physics as well as general relativity.

We saw in
Chapter 6
that supergravity was both nonrenormalizable and too small to accommodate the symmetry of the Standard Model. Hence, it was not self-consistent and did not begin to realistically describe the known particles. However, string theory does both. As we shall soon see, it banishes the infinities found in quantum gravity, yielding a finite theory of quantum gravity. That alone would guarantee that string theory should be taken as a serious candidate for a theory of the universe. However, there is an added bonus. When we compactify some of the dimensions of the string, we find that there is “enough room” to accommodate the symmetries of the Standard Model and even the GUTs.

The heterotic string consists of a closed string that has two types of vibrations, clockwise and counterclockwise, which are treated differently. The clockwise vibrations live in a ten-dimensional space. The counterclockwise live in a 26-dimensional space, of which 16 dimensions have been compactified. (We recall that in Kaluza’s original five-dimensional theory, the fifth dimension was compactified by being wrapped up into a circle.) The heterotic string owes its name to the fact that the clockwise and the counterclockwise vibrations live in two different dimensions but are combined to produce a single superstring theory. That is why it is named after the Greek word for
heterosis
, which means “hybrid vigor.”

The 16-dimensional compactified space is by far the most interesting. In Kaluza-Klein theory, we recall that the compactified
N
-dimensional
space can have a symmetry associated with it, much like a beach ball. Then all the vibrations (or fields) defined on the
N
-dimensional space automatically inherit these symmetries. If the symmetry is SU(
N
), then all the vibrations on the space must obey SU(
N
) symmetry (in the same way that clay inherits the symmetries of the mold). In this way, KaluzaKlein theory could accommodate the symmetries of the Standard Model. However, in this way it could also be determined that the supergravity was “too small” to contain all the particles of the symmetries found in the Standard Model. This was sufficient to kill the supergravity theory as a realistic theory of matter and space-time.

But when the Princeton string quartet analyzed the symmetries of the 16-dimensional space, they found that it is a monstrously large symmetry, called E(8) × E(8), which is much larger than any GUT symmetry that has ever been tried.
⁹
This was an unexpected bonus. It meant that that all the vibrations of the string would inherit the symmetry of the 16-dimensional space, which was more than enough to accommodate the symmetry of the Standard Model.

This, then, is the mathematical expression of the central theme of the book, that the laws of physics simplify in higher dimensions. In this case, the 26-dimensional space of the counterclockwise vibrations of the heterotic string has room enough to explain all the symmetries found in both Einstein’s theory and quantum theory. So, for the first time, pure geometry has given a simple explanation of why the subatomic world must necessarily exhibit certain symmetries that emerge from the curling up of higher-dimensional space:
The symmetries of the subatomic realm are but remnants of the symmetry of higher-dimensional space
.

This means that the beauty and symmetry found in nature can ultimately be traced back to higher-dimensional space. For example, snowflakes create beautiful, hexagonal patterns, none of which are precisely the same. These snowflakes and crystals, in turn, have inherited their structure from the way in which their molecules have been geometrically arranged. This arrangement is mainly determined by the electron shells of the molecule, which in turn take us back to the rotational symmetries of the quantum theory, given by 0(3). All the symmetries of the low-energy universe that we observe in chemical elements are due to the symmetries cataloged by the Standard Model, which in turn can be derived by compactifying the heterotic string.

In conclusion, the symmetries that we see around us, from rainbows to blossoming flowers to crystals, may ultimately be viewed as manifestations of fragments of the original ten-dimensional theory.
¹⁰
Riemann and Einstein had hoped to find a geometric understanding of why forces
can determine the motion and the nature of matter. But they were missing a key ingredient in showing the relationship between wood and marble. This missing link is most likely superstring theory. With the ten-dimensional string theory, we see that the geometry of the string may ultimately be responsible for both the forces and the structure of matter.

Given the enormous power of its symmetries, it is not surprising that superstring theory is radically different from any other kind of physics. It was, in fact, discovered quite by accident. Many physicists have commented that if this fortuitous accident had never occurred, then the theory would not have been discovered until the twenty-first century. This is because it is such a sharp departure from all the ideas tried in this century. It is not a continuous extension of trends and theories popular in this century; it stands apart.

By contrast, the theory of general relativity had a “normal” and logical evolution. First, Einstein postulated the equivalence principle. Then he reformulated this physical principle in the mathematics of a field theory of gravitation based on Faraday’s fields and Riemann’s metric tensor. Later came the “classical solutions,” such as the black hole and the Big Bang. Finally, the last stage is the current attempt to formulate a quantum theory of gravity. Thus general relativity went through a logical progression, from a physical principle to a quantum theory:

Geometry → field theory → classical theory → quantum theory

By contrast, superstring theory has been evolving
backward
since its accidental discovery in 1968. That’s why superstring theory looks so strange and unfamiliar to most physicists. We are still searching for its underlying physical principle, the counterpart to Einstein’s equivalence principle.

The theory was born quite by accident in 1968 when two young theoretical physicists, Gabriel Veneziano and Mahiko Suzuki, were independently leafing through math books, looking for mathematical functions that would describe the interactions of strongly interacting particles. While studying at CERN, the European center for theoretical physics in Geneva, Switzerland, they independently stumbled on the Euler beta function, a mathematical function written down in the nineteenth century by the mathematician Leonhard Euler. They were astonished
to find that the Euler beta function fit almost all the properties required to describe the strong interactions of elementary particles.

Over lunch at the Lawrence Berkeley Laboratory in California, with a spectacular view of the sun blazing down over San Francisco harbor, Suzuki once explained to me the thrill of discovering, quite by accident, a potentially important result. Physics was not supposed to happen that way.

After finding the Euler beta function in a math book, he excitedly showed his result to a senior physicist at CERN. The senior physicist, after listening to Suzuki, was not impressed. In fact, he told Suzuki that another young physicist (Veneziano) had discovered the identical function a few weeks earlier. He discouraged Suzuki from publishing his result. Today, this beta function goes by the name of the Veneziano model, which has inspired several thousand research papers, spawned a major school of physics, and now makes the claim of unifying all physical laws. (In retrospect, Suzuki, of course, should have published his result. There is a lesson to all this, I suspect: Never take too seriously the advice of your superiors.)

In 1970, the mystery surrounding the Veneziano-Suzuki model was partly explained when Yoichiro Nambu at the University of Chicago and Tetsuo Goto at Nihon University discovered that a vibrating string lies behind its wondrous properties.

Because string theory was discovered backward and by accident, physicists still do not know the physical principle that underlies string theory. The last step in the evolution of the theory (and the first step in the evolution of general relativity) is still missing.

Witten adds that

human beings on planet Earth never had the conceptual framework that would lead them to invent string theory on purpose…. No one invented it on purpose, it was invented in a lucky accident. By rights, twentieth-century physicists shouldn’t have had the privilege of studying this theory. By rights, string theory shouldn’t have been invented until our knowledge of some of the ideas that are prerequisite for string theory had developed to the point that it was possible for us to have the right concept of what it was all about.
¹¹

The formula discovered by Veneziano and Suzuki, which they hoped would describe the properties of interacting subatomic particles, was still
incomplete. It violated one of the properties of physics:
unitarity
, or the conservation of probability. By itself, the Veneziano-Suzuki formula would give incorrect answers for particle interactions. So the next step in the theory’s evolution was to add small quantum correction terms that would restore this property. In 1969, even before the string interpretation of Nambu and Goto, three physicists (Keiji Kikkawa, Bunji Sakita, and Miguel A. Virasoro, then all at the University of Wisconsin) proposed the correct solution: adding increasingly smaller terms to the Veneziano-Suzuki formula in order to restore unitarity.

Although these physicists had to guess at how to construct the series from scratch, today it is most easily understood in the framework of the string picture of Nambu. For example, when a bumblebee flies in space, its path can be described as a wiggly line. When a piece of string drifting in the air moves in space, its path can be likened to an imaginary two-dimensional sheet. When a closed string floats in space, its path resembles a tube.

Strings interact by breaking into smaller strings and by joining with other strings. When these interacting strings move, they trace out the configurations shown in
Figure 7.1
. Notice that two tubes come in from the left, with one tube fissioning in half, exchange the middle tube, and then veer off to the right. This is how tubes interact with each other. This diagram, of course, is shorthand for a very complicated mathematical expression. When we calculate the numerical expression corresponding to these diagrams, we get back the Euler beta function.

In the string picture, the essential trick proposed by Kikkawa-Sakita-Virasoro (KSV) amounted to adding
all possible
diagrams where strings can collide and break apart. There are, of course, an infinite number of these diagrams. The process of adding an infinite number of “loop” diagrams, with each diagram coming closer to the final answer, is perturbation theory and is one of most important weapons in the arsenal of any quantum physicist. (These string diagrams possess a beautiful symmetry that has never been seen in physics before, which is known as
conformal symmetry
in two dimensions. This conformal symmetry allows us to treat these tubes and sheets as though they were made of rubber: We can pull, stretch, bend, and shrink these diagrams. Then, because of conformal symmetry, we can prove that all these mathematical expressions remain the same.)

KSV claimed that the sum total of all these loop diagrams would yield the precise mathematical formula explaining how subatomic particles interact. However, the KSV program consisted of a series of unproven conjectures. Someone had to construct these loops explicitly, or else these conjectures were useless.