Hyperspace (30 page)

With the help of friends, Ramanujan managed to become a low-level clerk in the Port Trust of Madras. It was a menial job, paying a paltry £20 a year, but it freed Ramanujan, like Einstein before him at the Swiss patent office, to follow his dreams in his spare time. Ramanujan then mailed some of the results of his “dreams” to three well-known British mathematicians, hoping for contact with other mathematical minds. Two of the mathematicians, receiving this letter written by an unknown Indian clerk with no formal education, promptly threw it away. The third one was the brilliant Cambridge mathematician Godfrey H. Hardy. Because of his stature in England, Hardy was accustomed to receiving crank mail and thought dimly of the letter. Amid the dense scribbling he noticed many theorems of mathematics that were already well known.
Thinking it the obvious work of a plagiarist, he also threw it away. But something wasn’t quite right. Something nagged at Hardy; he couldn’t help wondering about this strange letter.

At dinner that night, January 16, 1913, Hardy and his colleague John Littlewood discussed this odd letter and decided to take a second look at its contents. It began, innocently enough, with “I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office of Madras on a salary of only 20 pounds per annum.”
¹⁵
But the letter from the poor Madras clerk contained theorems that were totally unknown to Western mathematicians. In all, it contained 120 theorems. Hardy was stunned. He recalled that proving some of these theorems “defeated me completely.” He recalled, “I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class.”
¹⁶

Littlewood and Hardy reached the identical astounding conclusion: This was obviously the work of a genius engaged in rederiving 100 years of European mathematics. “He had been carrying an impossible handicap, a poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe,” recalled Hardy.
¹⁷

Hardy sent for Ramanujan and, after much difficulty, arranged for his stay in Cambridge in 1914. For the first time, Ramanujan could communicate regularly with his peers, the community of European mathematicians. Then began a burst of activity: 3 short, intense years of collaboration with Hardy at Trinity College in Cambridge.

Hardy later tried to estimate the mathematical skill that Ramanujan possessed. He rated David Hilbert, universally recognized as one of the greatest Western mathematicians of the nineteenth century, an 80. To Ramanujan, he assigned a 100. (Hardy rated himself a 25.)

Unfortunately, neither Hardy nor Ramanujan seemed interested in the psychology or thinking process by which Ramanujan discovered these incredible theorems, especially when this flood of material came pouring out of his “dreams” with such frequency. Hardy noted, “It seemed ridiculous to worry him about how he had found this or that known theorem, when he was showing me half a dozen new ones almost every day.”
¹⁸

Hardy vividly recalled,

I remember going to see him once when he was lying ill in Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be
rather a dull one, and that I hoped that it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.”
¹⁹

(It is the sum of 1 × 1 × 1 and 12 × 12 × 12, and also the sum of 9 × 9 × 9 and 10 × 10 × 10.) On the spot, he could recite complex theorems in arithmetic that would require a modern computer to prove.

Always in poor health, the austerity of the war-torn British economy prevented Ramanujan from maintaining his strict vegetarian diet, and he was constantly in and out of sanitariums. After collaborating with Hardy for 3 years, Ramanujan fell ill and never recovered. World War I interrupted travel between England and India, and in 1919 he finally managed to return home, where he died a year later.

Ramanujan’s legacy is his work, which consists of 4,000 formulas on 400 pages filling three volumes of notes, all densely packed with theorems of incredible power but without any commentary or, which is more frustrating, any proof. In 1976, however, a new discovery was made. One hundred and thirty pages of scrap paper, containing the output of the last year of his life, was discovered by accident in a box at Trinity College. This is now called Ramanujan’s “Lost Notebook.” Commenting on the Lost Notebook, mathematician Richard Askey says, “The work of that one year, while he was dying, was the equivalent of a lifetime of work for a very great mathematician. What he accomplished was unbelievable. If it were a novel, nobody would believe it.” To underscore the difficulty of their arduous task of deciphering the “notebooks,” mathematicians Jonathan Borwein and Peter Borwein have commented, “To our knowledge no mathematical redaction of this scope or difficulty has ever been attempted.”
²⁰

Looking at the progression of Ramanujan’s equations, it’s as though we have been trained for years to listen to the Western music of Beethoven, and then suddenly we are exposed to another type of music, an eerily beautiful Eastern music blending harmonies and rhythms never heard before in Western music. Jonathan Borwein says, “He seems to have functioned in a way unlike anybody else we know of. He had such a feel for things that they just flowed out of his brain. Perhaps he didn’t see them in any way that’s translatable. It’s like watching somebody at a feast you haven’t been invited to.”

As physicists know, “accidents” do not appear without a reason. When performing a long and difficult calculation, and then suddenly having thousands of unwanted terms miraculously add up to zero, physicists know that this does not happen without a deeper, underlying reason. Today, physicists know that these “accidents” are an indication that a symmetry is at work. For strings, the symmetry is called conformal symmetry, the symmetry of stretching and deforming the string’s world sheet.

This is precisely where Ramanujan’s work comes in. In order to protect the original conformal symmetry from being destroyed by quantum theory, a number of mathematical identities must be miraculously satisfied. These identities are precisely the identities of Ramanujan’s modular function.

In summary, we have said that our fundamental premise is that the laws of nature simplify when expressed in higher dimensions. However, in light of quantum theory, we must how amend this basic theme. The correct statement should now read: The laws of nature simplify when
self-consistently
expressed in higher dimensions. The addition of the word
self-consistently
is crucial. This constraint forces us to use Ramanujan’s modular functions, which fixes the dimension of space-time to be ten. This, in turn, may give us the decisive clue to explain the origin of the universe.

Einstein often asked himself whether God had any choice in creating the universe. According to superstring theorists, once we demand a unification of quantum theory and general relativity, God had no choice. Self-consistency alone, they claim, must have forced God to create the universe as he did.

Although the mathematical sophistication introduced by superstring theory has reached dizzying heights and has startled the mathematicians, the critics of the theory still pound it at its weakest point. Any theory, they claim, must be testable. Since any theory defined at the Planck energy of 10
¹⁹
billion electron volts is not testable, superstring theory is not really a theory at all!

The main problem, as we have pointed out, is theoretical rather than experimental. If we were smart enough, we could solve the theory exactly and find the true nonperturbative solution of the theory. However, this does not excuse us from finding some means by which to verify the theory experimentally. To test the theory, we must wait for signals from the tenth dimension.

How strange it would be if the final theory were to be discovered in our lifetimes! The discovery of the final laws of nature will mark a discontinuity in human intellectual history, the sharpest that has occurred since the beginning of modern science in the seventeenth century. Can we now imagine what that would be like?

Steven Weinberg

ALTHOUGH superstring theory gives us a compelling formulation of the theory of the universe, the fundamental problem is that an experimental test of the theory seems beyond our present-day technology. In fact, the theory predicts that the unification of all forces occurs at the Planck energy, or 10
¹⁹
billion electron volts, which is about 1 quadrillion times larger than energies currently available in our accelerators.

Physicist David Gross, commenting on the cost of generating this fantastic energy, says, “There is not enough money in the treasuries of all the countries in the world put together. It’s truly astronomical.”
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This is disappointing, because it means that experimental verification, the engine that drives progress in physics, is no longer possible with our current generation of machines or with any generation of machines
in the conceivable future. This, in turn, means that the ten-dimensional theory is not a theory in the usual sense, because it is untestable given the present technological state of our planet. We are then left with the question: Is beauty, by itself, a physical principle that can be substituted for the lack of experimental verification?

To some, the answer is a resounding no. They derisively call these theories “theatrical physics” or “recreational mathematics.” The most caustic of the critics is Nobel Prize winner Sheldon Glashow of Harvard University. He has assumed the role of gadfly in this debate, leading the charge against the claims of other physicists that higher dimensions may exist. Glashow rails against these physicists, comparing the current epidemic to the AIDS virus; that is, it’s incurable. He also compares the current bandwagon effect with former President Reagan’s Star Wars program:

Here’s a riddle: Name two grand designs that are incredibly complex, require decades of research to develop, and may never work in the real world? Stars Wars and string theory…. Neither ambition can be accomplished with existing technology, and neither may achieve its stated objectives. Both adventures are costly in terms of scarce human resources. And, in both cases, the Russians are trying desperately to catch up.
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To stir up more controversy, Glashow even penned a poem, which ends:

The Theory of Everything, if you dare to be bold,
Might be something more than a string orbifold.
While some of your leaders have got old and sclerotic,
Not to be trusted alone with things heterotic,
Please heed our advice that you are not smitten—
The Book is not finished, the last word is not Witten.
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Glashow has vowed (unsuccessfully) to keep these theories out of Harvard, where he teaches. But he does admit that he is often outnumbered on this question. He regrets, “I find myself a dinosaur in a world of upstart mammals.”
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(Glashow’s views are certainly not shared by other Nobel laureates, such as Murray Gell-Mann and Steven Weinberg. Physicist Weinberg, in fact, says, “String theory provides our only present source of candidates for a final theory—how could anyone expect that many of the brightest young theorists would
not
work on it?”
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)

To understand the implications of this debate concerning the unification
of all forces, and also the problems with its experimental verification, it is instructive to consider the following analogy, the “parable of the gemstone.”

In the beginning, let us say, was a gemstone of great beauty, which was perfectly symmetrical in three dimensions. However, this gemstone was unstable. One day, it burst apart and sent fragments in all directions; they eventually rained down on the two-dimensional world of Flatland. Curious, the residents of Flatland embarked on a quest to reassemble the pieces. They called the original explosion the Big Bang, but did not understand why these fragments were scattered throughout their world. Eventually, two kinds of fragments were identified. Some fragments were polished and smooth on one side, and Flatlanders compared them to “marble.” Other fragments were entirely jagged and ugly, with no regularity whatsoever, and Flatlanders compared these pieces to “wood.”

Over the years, the Flatlanders divided into two camps. The first camp began to piece together the polished fragments. Slowly, some of the polished pieces begin to fit together. Marveling at how these polished fragments were being assembled, these Flatlanders were convinced that somehow a powerful new geometry must be operating. These Flatlanders called their partially assembled piece “relativity.”

The second group devoted their efforts to assembling the jagged, irregular fragments. They, too, had limited success in finding patterns among these fragments. However, the jagged pieces produced only a larger but even more irregular clump, which they called the Standard Model. No one was inspired by the ugly mass called the Standard Model.

After years of painstaking work trying to fit these various pieces together, however, it appeared as though there was no way to put the polished pieces together with the jagged pieces.

Then one day an ingenious Flatlander hit upon a marvelous idea. He declared that the two sets of pieces could be reassembled into one piece if they were moved “up”—that is, in something he called the third dimension. Most Flatlanders were bewildered by this new approach, because no one could understand what “up” meant. However, he was able to show by computer that the “marble” fragments could be viewed as outer fragments of some object, and were hence polished, while the “wood” fragments were the inner fragments. When both sets of fragments were assembled in the third dimension, the Flatlanders gasped at what was revealed in the computer: a dazzling gemstone with perfect three-dimensional symmetry. In one stroke, the artificial distinction between the two sets of fragments was resolved by pure geometry.