Part 13 (1/2)
Lucas numbers are like Fibonacci numbers but begin with 2: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, and so on. Like the Fibonacci series, the Lucas series also produces the Golden Ratio.
Fibonacci and Lucas numbers pop up all over the place, from how rabbits reproduce to the shape of mollusk sh.e.l.ls, to leaf arrangements that sometimes spiral at angles derivable from the Golden Ratio.
Because all these numbers are related, any formula for 137 in terms of the Golden Ratio can be rewritten in terms of Fibonacci and Lucas numbers, though whether this is anything more than merely abstruse relations.h.i.+ps between certain numbers is not clear.
If one tries hard enough, 137 can be deduced from devilishly complicated combinations of ”magic numbers” such as 22 (the first 22 human chromosomes are numbered), 23 (the number of chromosomes from each parent), 28 (the length of a woman's menstrual cycle), 46 (the number of pairs of chromosomes a person has), 64 (the number of possible values for the 20 amino acids in DNA), and 92 (the number of naturally occurring elements in the periodic table).
Sadly, all these are pure coincidences with no scientific basis. And still 137 continues to tantalize. In fact 137 has become something of a cult. According to one Web site, ”The Fine Structure Constant holds a special place among cult numbers. Unlike its more mundane cousins, 17 and 666, the Fine Structure Constant seduces otherwise sane engineers and scientists into seeking mystical truths and developing farfetched theories.”
Enrico Fermi with his incorrect equation for the fine structure constant.
Even Heisenberg had a go, fired by Eddington's number speculations. In a letter to Bohr in 1935 he reported ”playing around” with the fine structure constant, which he expressed as . He was quick to add, ”but the other research on it is more serious,” referring to his and Pauli's attempts to derive it from quantum electrodynamics.
Many years later, Enrico Fermi, the physicist who christened Pauli's newfound weakly interacting particle the neutrino, was asked to pose for a photograph. He took his place in front of a blackboard on which he had written the fine structure constant-but incorrectly. Instead of he put . ( is shorthand for Planck's constant h divided by 2: h/2.) It was an excellent joke-and a joke comprehensible only to scientists. However, the joke backfired when the photograph was used for a stamp to commemorate him after his death. So he is caught forever standing next to this iconic equation-incorrectly written.
Surprisingly, 137 also crops up in an entirely different context. In the 1950s, Pauli developed a close friends.h.i.+p with Gershom Scholem, a prominent scholar of Jewish mysticism. When his former a.s.sistant Victor Weisskopf went to Jerusalem, Pauli urged him to meet Scholem, though he gave no hint of his own interests in Jewish mysticism. Scholem asked Weisskopf what the deep unsolved problems of physics were. Weisskopf replied, ”Well, there's this number, 137.”
Scholem's eyes lit up. ”Did you know that 137 is the number a.s.sociated with the Kabbalah?” he asked.
In ancient Hebrew, numbers were written with letters, and each letter of the Hebrew alphabet has a number a.s.sociated with it. Adepts of the philosophical system known as the Gematria add the numbers in Hebrew words and thus find hidden meanings in them. The word Kabbalah is written in Hebrew: is 5, is 30, is 2, and is 100. The four letters add up to...137!
It is an extraordinary link between mysticism and physics. Two key words in the Kabbalah are ”wisdom,” which has a numerical value of 73, and ”prophesy” (64): 73 + 64 = 137. G.o.d himself is One-1-which can also be written 10 (1 + 0 = 10). Take 10's const.i.tuent prime numbers, 3 and 7, and add the original 1: together they can be written 137.
In the bible the key phrase ”The G.o.d of Truth” (Isaiah 65) adds up to 137. So does ”The Surrounding Brightness” (Ezekiel 1) , and the Hebrew word for ”crucifix”, .
It turns out that, according to the Gematria, the number content of the letters in the Hebrew word for 137 add up to 1664. This happens to be the numerical value for a portion of the well-known pa.s.sage from Revelations 13:18: ”Here is wisdom. Let him that hath understanding count the number of the beast.” The rest of the pa.s.sage reads: ”for the number is that of man; and his number is six hundred and sixty-six.”
Homing in on the yet more arcane, a group of numerologists noticed that the number 82943 appeared at key places in Aztec and Roman texts and went on to argue that it was a key to a ”new universal consciousness.” Their confidence in this a.s.sertion was bolstered when they discovered they could relate it to the fine structure constant and the number of the beast-666-as follows: And more way out still, 137 is the foundation stone of a fiendishly complex ”biblical mathematics” referred to by followers of ”The Bible Wheel” as a ”Holographic Generating Set.” It is based on three geometric forms: a cube, A, divided into 27 subcubes; a hexagon, B, divided into 37 subhexagons; and a star of David, C, divided into 73 circles Of course 27 + 37 + 73 = 137. From this ”generating set” with its Pythagorean-geometric aura, aficionados of the Bible Wheel claim to be able to generate biblical pa.s.sages and probe mystical numbers by multiplying A, B, and C in various ways.
Thus 137 continues to fire the imagination of everyone from scientists and mystics to occultists and people from the far-flung edges of society.
The last challenge.
With World War II behind them, Heisenberg and Pauli resumed their scientific correspondence. But the days of their collaboration and the frequent exchange of letters seemed to have ended. After all, they had been on opposite sides in the war. And Heisenberg's reputation was colored by the fact that he had remained in Germany and had ended up in charge of the German atomic bomb project. Most of his postwar colleagues, of course, had been involved in the Manhattan Project. Even Heisenberg's brilliance was not enough to overcome this stain.
Then in 1957 Heisenberg wrote to Pauli that he had the germ of a theory that could explain the ma.s.ses of elementary particles as well as most of the symmetries. In a preliminary test he was almost able to deduce the fine structure constant from his new theory. In his calculations it came to 1/250, which is not very far away from 1/137, in the same way as 1/3 is not far from 1/4, even though 3 is far from 4. This was extraordinary, given that Heisenberg's theory was still in its formative stages. Pauli immediately took up Heisenberg's suggestion to join him in his project.
”Never before or afterward have I seen [Pauli] so excited about physics,” Heisenberg later recalled. It seemed as if the old days had come back. The two giants of quantum physics were working together once again.
”The picture keeps s.h.i.+fting all the time. Everything in flux. Nothing for publication yet but it's all bound to turn out magnificently,” Pauli wrote exuberantly to Heisenberg at the start of 1958. ”This is powerful stuff.... The cat is out of the bag and has shown its claws: division of symmetry reduction. I have gone out to meet it with my antisymmetry-I gave it fair play-whereupon it made its quietus.... A very happy New Year. Let us march forward toward it. It's a long way to Tipperary, it's a long way to go.”
Heisenberg was equally excited. For him the theory was to be the culmination of his life's work. Over the years he had become entranced by the power of mathematics to probe and understand the physical world. And he knew how to use it. ”A wonderful combination of profound intuition and formal virtuosity inspired Heisenberg to conceptions of striking brilliance,” a colleague wrote. He had used his formidable insight and daring to apply mathematics to make his startling discoveries in quantum mechanics. These included the uncertainty principle; the first steps toward understanding the force that holds the nucleus together; and his attempts to produce a coherent theory of electrons and light, known as quantum electrodynamics.
The theory he was working on with Pauli was exactly what he was looking for. ”The last few weeks have been full of excitement for me,” he wrote to his wife's sister Edith in January 1958: I have attempted an as yet-unknown-ascent to the fundamental peak of atomic theory with great efforts during the last five years. And now, with the peak directly ahead of me, the whole terrain of interrelations.h.i.+ps in atomic theory is suddenly and clearly spread out before my eyes. That these interrelations.h.i.+ps display, in all their mathematical abstraction, an incredible degree of simplicity, is a gift we can only accept humbly. Not even Plato could have believed them to be so beautiful. For these interrelations.h.i.+ps cannot have been invented; they have been there since the creation of the world.
Perhaps he also saw it as his chance to vindicate himself, to put behind him the fact that he had been the leader of the German atomic bomb project.
Together Pauli and Heisenberg wrote a joint paper that Pauli planned to lecture on during his forthcoming visit to the United States that January. Before leaving he wrote to Aniela Jaffe. He had been entirely engrossed in work with Heisenberg on a ”new physical-mathematical theory of the smallest particles,” he said.
The pace of their work was so overwhelming that often letters were not fast enough. The phone line between Zurich and Munich, where Heisenberg was based, was continually buzzing. For Pauli their research was so fundamental that he saw Jungian significance in it. Although he and Heisenberg were very different, he wrote to Jaffe, they were ”gripped by the same archetype”-by reflection symmetry, in the fullest sense of the CPT reflection. ”Director Spiegler! [Reflector] dictates to me what I should write and calculate,” he declared.
He hoped that the ”new year will see a beautiful theory that will light up the world.” He had even had a dream about it: Pauli enters a room and finds a boy and a girl there. He calls out, ”Franca, here are two children!” He had seen Heisenberg just three days earlier and interpreted the children as the new ideas which he was confident would emerge from their work. From the fact that there were two children, he drew an a.n.a.logy to his ”mirror complex.”
The work could be taken as a realization of the unconscious and ”more specifically: a realization of the 'Self' (in the Jungian sense).” It was what Pauli had sought for years-physics and the psychology of the unconscious as mirror images, a scenario that had been destroyed by the violation of mirror symmetry (parity violation), but restored by CPT symmetry, Pauli's 1955 discovery about which he was still exultant.
On February 1, 1958, the lecture theater at the physics department at Columbia University was packed with over three hundred people. They were eager to see the great Pauli who was about to lecture on a theory formulated by the two giants of quantum theory. Niels Bohr, J. Robert Oppenheimer, T. D. Lee, and C. N. Yang, who had proposed the overthrow of parity, and C. S. Wu, the physicist who had performed the crucial experiment to prove it, all attended. The air was electric. But the distinguished audience had nothing but criticism for the new theory, though offered in a friendly manner. A key point in the theory was how newly discovered elementary particles decayed, how they transformed themselves into other particles. As Pauli was scribbling the equations on the blackboard, Abraham Pais, an eminent physicist and friend of Pauli's, raised his hand and objected, ”But Professor Pauli, this particle does not decay like that.” Pauli stopped midflow. There was a long silence. Then Pauli muttered, ”I must get in touch with my friends in Gottingen about that,” by which he meant Heisenberg. At this point, T. D. Lee remembers, you ”could almost feel the silence.” Others too pointed out loopholes in his mathematical proofs. Pauli continued his lecture but it was clear that the pa.s.sion had gone.
At one point Bohr and Pauli chased each other around a long table at the front of the room. Whenever Bohr ended up at the front he declared, ”It is not crazy enough.” Each time Pauli appeared he replied, ”It is crazy enough.” This was repeated several times and the audience burst into applause.
”We were all polite, but Pauli was obviously discouraged...It was obvious that his heart was no longer in their work,” Yang recalled. He vividly remembered Pauli's gloom as they were on their way in Lee's car to a restaurant after the lecture. ”Pauli oscillated back and forth in his seat and murmured some thing which I thought was, 'As I talked more and more, I believed in it less and less.' I was greatly saddened.” The physicist Freeman J. Dyson, of the Inst.i.tute of Advanced Study, commented that it was ”like watching the death of a n.o.ble animal.”
So what had happened? How could Pauli have been so enthusiastic about this new approach and then so totally cast down? Presenting a lecture on a subject can shed an entirely new light on it. So perhaps that was what happened. Suddenly he realized that the theory was full of holes. He was beginning to have second thoughts about his work with Heisenberg.
The following day he lectured at the American Physical Society meeting in New York, at that time the biggest and most important annual gathering of physicists from around the world. There was standing room only. But the criticism meted out by a younger and brasher generation of American physicists was even harsher. Lee could not bring himself to attend.
From there Pauli went on to California-where Feynman, for one, had had no compunction about telling the great Bohr that he was an idiot. Audiences there too offered ruthless criticism. Pauli was beginning to conclude, as he wrote to Heisenberg later that year, that ”something entirely new, in other words very 'crazy,' [was] needed” if he and Heisenberg were to crack the mystery of the ma.s.ses of elementary particles, one of the key aims of their unified theory.
Disillusioned, Pauli attacked Heisenberg's calculation of the fine structure constant as 1/250, which had seemed so promising and had played a part in his decision to join Heisenberg's project. He wrote to Fierz bitterly, ”I have never considered it as correct. It's so totally stupid.”
Some time later, Heisenberg's co-worker on that calculation, Renato Ascoli, recalled that he had originally deduced the fine structure constant as 8 on the basis of Heisenberg's theory. ”Only after Heisenberg had doctored it up, was the value reduced to 1/250.”
Later that month Heisenberg gave a lecture on his and Pauli's work at Heisenberg's Inst.i.tute in Gottingen. The room was packed. The great Heisenberg was about to announce a theory that could explain the behavior of every elementary particle in the world with a single equation-a ”world formula”-that would surely prove to be an abstruse and highly technical piece of mathematics.
A press release was circulated reading (most offensively to Pauli): ”Professor Heisenberg and his a.s.sistant, W. Pauli, have discovered the basic equation of the cosmos.” The story was picked up by newspapers around the world. Pauli vented his anger in a letter to George Gamow, the physicist and prankster who had translated and ill.u.s.trated the Mephistopheles spoof at Bohr's Inst.i.tute in 1932. Pauli lampooned it by drawing an empty box saying, ”This is to show that I can paint like t.i.tian: Only the technical details are missing”-a case of the emperor's new clothes. Pauli requested Gamow not to publish his comment but, miffed at Heisenberg's insulting press release, added, ”please show it to other physicists and make it popular among them.” Gamow certainly did. A week later Weisskopf wrote to Pauli about the press release and added that he read it with Pauli's comment about t.i.tian well in mind.
Combining Pauli's well-known obsession with 137 with Heisenberg's for his new theory, a colleague wrote jokingly to Pauli, ”Since the Heisenberg equation is supposed to describe everything (see, for instance, New-York Herald Tribune, volume 137, p. i/137), it has as one of its solutions Heisenberg himself.” Pauli replied, ”Regarding Heisenberg I have the feeling, that the situation is slowly growing over his head; certainly he needs vacations.” The problem of how to describe the properties of elementary particles remained open. Pauli summed up the situation thus, ”Many questions, no good answers.”
In fact, Pauli was furious. He wrote to C. S. Wu about Heisenberg's ”poor taste” as far as the press releases were concerned: In some of these I had been, unfortunately, mentioned...but fortunately only in a ”mild” form as a secondary (or tertiary) auxiliary person of the Super-Faust, Super-Einstein and Super-man Heisenberg. (He seems to have mentioned his dreams on gravitational fields-about which one has not worked at all in Gottingen recently-and his revival of the old idea of a ”world formula”-which was never successful-in a quantized form.) He seems to have been relieved that he was not a.s.sociated too closely with Heisenberg's mistaken schemes. To Wu he recalled his hopes, dreams, and aspirations of thirty years earlier when he and Heisenberg were young and Heisenberg depended on his friend's criticism and inspiration. Now Pauli had had enough: Heisenberg's desire for publicity and ”glory” seems to be insatiable, while I am in this respect completely saturated. I only need something in science which interests me sufficiently and with which I can play (without being a hero in the limelight of the ”world.”) Heisenberg's opposite att.i.tude, with which he certainly wishes to compensate earlier failures, may have many reasons lying in the whole history of his life.
No doubt the last words were an oblique reference to the role Heisenberg played in the war.
Soon afterward Pauli withdrew from the collaboration. Heisenberg persisted in making promise after promise as to the wonders his theory would produce. ”He believes that if he publishes with me, then it is 1930 again! I have found it embarra.s.sing how he runs after me!” Pauli wrote to Fierz in May.
That July Pauli chaired a session at a conference at CERN at which Heisenberg was scheduled to speak. Pauli introduced Heisenberg with the words, ”What you will hear today is only a subst.i.tute for fundamental ideas.” He went on to make a request of the audience: ”don't laugh in the wrong place, ha, ha, ha....” The audience was already in fits of laughter. Pauli let Heisenberg finish speaking, then mercilessly demolished his paper.
When Pauli and Heisenberg met again later that summer, Heisenberg noticed that Pauli looked dispirited. Pauli encouraged him to go on with his work and wished him well, but added, ”For me, I have to drop out, I just haven't the strength, and that's that. Things have changed too much.”
Could it have been that the great criticizer had met his match in the lambasting he encountered in America? To be on the receiving end must have been shattering. Heisenberg had been afraid this would happen when Pauli ”in his present mood of exultation [encountered] the sober American pragmatists.” Franca too had noticed this c.h.i.n.k in Pauli's armor which, up until then, he had concealed so successfully: ”He was very easily hurt and therefore would let down a curtain. He tried to live without admitting reality. And his unworldliness stemmed precisely from his belief that that was possible.”
But there was something else that brought Pauli to the point of spiritual exhaustion. He had grown attached to his work with Heisenberg. Their new theory had all the trappings that he thought a theory should have: a high degree of mathematical symmetry and Jungian meaning too, taking it one step closer to not merely a unified theory of elementary particles from which the fine structure constant could one day be deduced, but to a theory of the mind as well.
As he always did, Pauli interpreted the failure of the theory as personal failure. Genius though he was, he had failed yet again. Those who saw him in the autumn of 1958 recalled that he seemed beaten.
That year Pauli told an interviewer, ”When I was young I thought I was the best formalist of my time. I believed that I was a revolutionary. When the great problems would come, I would be the one to solve them and to write about them. Others solved them and wrote about them. I was but a cla.s.sicist and no revolutionary.” He began writing letters as if saying farewell.
A different side of Pauli.