Hyperspace (29 page)

Research in quantum gravity went into other direction. From 1974 to 1984, when string theory was in eclipse, a large number of alternative theories of quantum gravity were successively studied. During this period, the original Kaluza-Klein theory and then the supergravity theory enjoyed great popularity, but each time the failures of these models also became apparent. For example, both Kaluza-Klein and supergravity theories were shown to be nonrenormalizable.

Then something strange happened during that decade. On the one hand, physicists became frustrated by the growing list of models that were tried and then discarded during this period. Everything failed. The realization came slowly that Kaluza-Klein theory and supergravity theory were probably on the right track, but they weren’t sophisticated enough to solve the problem of nonrenormalizability. But the only theory complex enough to contain both Kaluza-Klein theory and the supergravity theory was superstring theory. On the other hand, physicists slowly became accustomed to working in hyperspace. Because of the Kaluza-Klein renaissance, the idea of hyperspace didn’t seem that farfetched or forbidding anymore. Over time, even a theory defined in 26 dimensions didn’t seem that outlandish. The original resistance to 26 dimensions began to slowly melt away with time.

Finally, in 1984, Green and Schwarz proved that superstring theory was the
only
self-consistent theory of quantum gravity, and the stampede began. In 1985, Edward Witten made a significant advance in the field theory of strings, which many people think is one of the most beautiful achievements of the theory. He showed that our old field theory could be derived using powerful mathematical and geometric theorems (coming from something called
cohomology theory
) with a fully relativistic form.

With Witten’s new field theory, the true mathematical elegance of string field theory, which was concealed in our formalism, was revealed. Soon, almost a hundred scientific papers were written to explore the fascinating mathematical properties of Witten’s field theory.
¹²

Assuming that string field theory is correct, in principle we should be able to calculate the mass of the proton from first principles and make contact with known data, such as the masses of the various particles. If the numerical answers are wrong, then we will have to throw the theory out the window. However, if the theory is correct, it will rank among the most significant advances in physics in 2,000 years.

After the intense, euphoric fanfare of the late 1980s (when it appeared that the theory would be completely solved within a few years and the Nobel Prizes handed out by the dozen), a certain degree of cold realism has set in. Although the theory is well defined mathematically, no one has been able to solve the theory. No one.

The problem is that
no one is smart enough to solve the field theory of strings
or any other nonperturbative approach to string theory. This is a well-defined problem, but the irony is that solving field theory requires techniques that are currently beyond the skill of any physicist. This is frustrating. Sitting before us is a perfectly well-defined theory of strings. Within it is the possibility of settling all the controversy surrounding higher-dimensional space. The dream of calculating everything from first principles is staring us in the face. The problem is how to solve it. One is reminded of Julius Caesar’s famous remark in Shakespeare’s play: “The fault, dear Brutus, is not in our stars, but in ourselves.” For a string theorist, the fault is not in the theory, but in our primitive mathematics.

The reason for this pessimism is that our main calculational tool, perturbation theory, fails. Perturbation theory begins with a Veneziano-like formula and then calculates quantum corrections to it (which have the shape of loops). It was the hope of string theorists that they could write down a more advanced Veneziano-like formula defined in four dimensions that would uniquely describe the known spectrum of particles. In retrospect, they were too successful. The problem is that millions upon millions of Veneziano-like formulas have now been discovered. Embarrassingly, string theorists are literally drowning in these perturbative solutions.

The fundamental problem that has stalled progress in superstring theory in the past few years is that no one knows how to select the correct solution out of the millions that have been discovered. Some of these solutions come remarkably close to describing the real world. With a few modest assumptions, it is easy to extract the Standard Model as one vibration of the string. Several groups have announced, in fact, that they can find solutions that agree with the known data about subatomic particles.

The problem, we see, is that there are also millions upon millions of other solutions describing universes that do not appear anything like our universe. In some of these solutions, the universe has no quarks or too many quarks. In most of them, life as we know it cannot exist. Our universe may be lost somewhere among the millions of possible universes that have been found in string theory. To find the correct solution, we
must use nonperturbative techniques, which are notoriously difficult. Since 99% of what we know about high-energy physics is based on perturbation theory, this means that we are at a total loss to find the one true solution to the theory.

There is some room for optimism, however. Nonperturbative solutions that have been found for much simpler theories show that many of the solutions are actually unstable. After a time, these incorrect, unstable solutions will make a quantum leap to the correct, stable solution. If this is true for string theory, then perhaps the millions of solutions that have been found are actually unstable and will decay over time to the correct solution.

To understand the frustration that we physicists feel, think, for a moment, of how nineteenth-century physicists might react if a portable computer were given to them. They could easily learn to turn the dials and press the buttons. They could learn to master video games or watch educational programs on the monitor. Being a century behind in technology, they would marvel at the fantastic calculational ability of the computer. Within its memory could easily be stored all known scientific knowledge of that century. In a short period of time, they could learn to perform mathematical feats that would amaze any of their colleagues. However, once they decide to open up the monitor to see what is inside, they would be horrified. The transistors and microprocessors would be totally alien to anything they could understand. There would be really nothing in their experience to compare with the electronic computer. It would be beyond their ken. They could only stare blankly at the complicated circuitry, not knowing in the slightest how it works or what it all means.

The source of their frustration would be that the computer exists and is sitting there in front of their noses, but they would have no reference frame from which to explain it. Analogously, string theory appears to be twenty-first-century physics that was discovered accidentally in our century. String field theory, too, seems to include all physical knowledge. With little effort, we are able to turn a few dials and press a few buttons with the theory, and out pops the supergravity theory, Kaluza-Klein theory, and the Standard Model. But we are at a total loss to explain why it works. String field theory exists, but it taunts us because we are not smart enough to solve it.

The problem is that while twenty-first-century physics fell accidentally into the twentieth century, twenty-first-century mathematics hasn’t been invented yet. It seems that we may have to wait for twenty-first-century
mathematics before we can make any progress, or the current generation of physicists must invent twenty-first-century mathematics on their own.

One of the deepest secrets of string theory, which is still not well understood, is why it is defined in only ten and 26 dimensions. If the theory were three dimensional, it would not be able to unify the known laws of physics in any sensible manner. Thus it is the geometry of higher dimensions that is the central feature of the theory.

If we calculate how strings break and re-form in
N
-dimensional space, we constantly find meaningless terms cropping up that destroy the marvelous properties of the theory. Fortunately, these unwanted terms appear multiplied by (
N
− 10). Therefore, to make these anomalies vanish, we have no choice but to fix
N
to be ten. String theory, in fact, is the only known quantum theory that specifically demands that the dimension of space-time be fixed at a unique number.

Unfortunately, string theorists are, at present, at a loss to explain why ten dimensions are singled out. The answer lies deep within mathematics, in an area called
modular functions
. Whenever we manipulate the KSV loop diagrams created by interacting strings, we encounter these strange modular functions, where the number ten appears in the strangest places. These modular functions are as mysterious as the man who investigated them, the mystic from the East. Perhaps if we better understood the work of this Indian genius, we would understand why we live in our present universe.

Srinivasa Ramanujan was the strangest man in all of mathematics, probably in the entire history of science. He has been compared to a bursting supernova, illuminating the darkest, most profound corners of mathematics, before being tragically struck down by tuberculosis at the age of 33, like Riemann before him. Working in total isolation from the main currents of his field, he was able to rederive 100 years’ worth of Western mathematics on his own. The tragedy of his life is that much of his work was wasted rediscovering known mathematics. Scattered throughout the
obscure equations in his notebooks are these modular functions, which are among the strangest ever found in mathematics. They reappear in the most distant and unrelated branches of mathematics. One function, which appears again and again in the theory of modular functions, is today called the
Ramanujan function
in his honor. This bizarre function contains a term raised to the twenty-fourth power.

In the work of Ramanujan, the number 24 appears repeatedly. This is an example of what mathematicians call magic numbers, which continually appear, where we least expect them, for reasons that no one understands. Miraculously, Ramanujan’s function also appears in string theory. The number 24 appearing in Ramanujan’s function is also the origin of the miraculous cancellations occurring in string theory. In string theory, each of the 24 modes in the Ramanujan function corresponds to a physical vibration of the string. Whenever the string executes its complex motions in space-time by splitting and recombining, a large number of highly sophisticated mathematical identities must be satisfied. These are precisely the mathematical identities discovered by Ramanujan. (Since physicists add two more dimensions when they count the total number of vibrations appearing in a relativistic theory, this means that space-time must have 24 + 2 = 26 space-time dimensions.
¹³
)

When the Ramanujan function is generalized, the number 24 is replaced by the number 8. Thus the critical number for the superstring is 8 + 2, or 10. This is the origin of the tenth dimension. The string vibrates in ten dimensions because it requires these generalized Ramanujan functions in order to remain self-consistent.
In other words, physicists have not the slightest understanding of why ten and 26 dimensions are singled out as the dimension of the string
. It’s as though there is some kind of deep numerology being manifested in these functions that no one understands. It is precisely these magic numbers appearing in the elliptic modular function that determines the dimension of space-time to be ten.

In the final analysis, the origin of the ten-dimensional theory is as mysterious as Ramanujan himself. When asked by audiences why nature might exist in ten dimensions, physicists are forced to answer, “We don’t know.’’ We know, in vague terms, why some dimension of space-time must be selected (or else the string cannot vibrate in a self-consistent quantum fashion), but we don’t know why these particular numbers are selected. Perhaps the answer lies waiting to be discovered in Ramanujan’s lost notebooks.

Ramanujan was born in 1887 in Erode, India, near Madras. Although his family was Brahmin, the highest of the Hindu castes, they were destitute, living off the meager wages of Ramanujan’s father’s job as a clerk in a clothing merchant’s office.

By the age of 10, it was clear that Ramanujan was not like the other children. Like Riemann before him, he became well known in his village for his awesome calculational powers. As a child, he had already rederived Euler’s identity between trigonometric functions and exponentials.

In every young scientist’s life, there is a turning point, a singular event that helps to change the course of his or her life. For Einstein, it was the fascination of observing a compass needle. For Riemann, it was reading Legendre’s book on number theory. For Ramanujan, it was when he stumbled on an obscure, forgotten book on mathematics by George Carr. This book has since been immortalized by the fact that it marked Ramanujan’s only known exposure to modern Western mathematics. According to his sister, “It was this book which awakened his genius. He set himself to establish the formulae given therein. As he was without the aid of other books, each solution was a piece of research so far as he was concerned…. Ramanujan used to say that the goddess of Namakkal inspired him with the formulae in dreams.”
¹⁴

Because of his brilliance, he was able to win a scholarship to high school. But because he was bored with the tedium of classwork and intensely preoccupied with the equations that were constantly dancing in his head, he failed to enter his senior class, and his scholarship was canceled. Frustrated, he ran away from home. He did finally return, but only to fall ill and fail his examinations again.