Джеймс Глик - Genius - The Life and Science of Richard Feynman

Infinitesimal time did not amount to much, but it was the starting point of the calculus. That limitation was not what troubled Feynman. As he looked over the few bound pages, he kept stopping at a single word: analogue . “A very simple quantum analogue,” Dirac had written. “… They have their classical analogues… . It is now easy to see what the quantum analogue of al this must be.” What kind of word was that, Feynman wondered, in a paper on physics? If two expressions were analogous, did it mean they were equal?

No, Jehle, said—surely Dirac had not meant that they were equal. Feynman found a blackboard and started working through the formulas. Jehle was right: they were not equal. So he tried adding a multiplication constant.

Calculating more rapidly than Jehle could fol ow, he substituted terms, jumped from one equation to the next, and suddenly produced something extremely familiar: the

Schrödinger equation. There was the link between Feynman’s Lagrangian-style formulation and the standard wave function of quantum mechanics. A surprise—by analogous Dirac had simply meant proportional .

But now Jehle had produced a smal notebook. He was rapidly copying from Feynman’s blackboard work. He told Feynman that Dirac had meant no such thing. In his view Dirac’s idea had been strictly metaphorical; the Englishman had not meant to suggest that the approach was useful. Jehle told Feynman he had made an important discovery. He was struck by the unabashed pragmatism in Feynman’s handling of the mathematics, so different from Dirac’s more detached, more aesthetic tone. “You Americans!” he said. “Always trying to find a use for something.”

The Aura

This was Richard Feynman nearing the crest of his powers.

At twenty-three he was a few years shy of the time when his vision would sweep hawklike across the breadth of physics, but there may now have been no physicist on earth who could match his exuberant command over the native materials of theoretical science. It was not just a facility at mathematics (though it had become clear to the senior physicists at Princeton that the mathematical machinery emerging in the Wheeler-Feynman col aboration was beyond Wheeler’s own ability). Feynman seemed to possess a frightening ease with the substance behind the equations, like Einstein at the same age, like the Soviet

physicist Lev Landau—but few others. He was a sculptor who sleeps and dreams with the feeling of clay alive in his fingers.

Graduate

students

and

instructors

found

themselves wandering over to the afternoon tea at Fine Hal with Feynman on their minds. They anticipated his bantering with Tukey and the other mathematicians, his spinning of half-serious physical theories. Handed an idea, he always had a question that seemed to pierce toward the essence. Robert R. Wilson, an experimentalist who arrived at Princeton from the famous cauldron of Ernest Lawrence’s Berkeley laboratory, talked casual y with Feynman only a few times before making a mental note: Here is a great man.

The Feynman aura—as it had already become—was strictly local. Feynman had not yet finished his second year of graduate school. He remained ignorant of the basic literature and unwil ing even to read through the papers of Dirac or Bohr. This was now deliberate. In preparing for his oral qualifying examination, a rite of passage for every graduate student, he chose not to study the outlines of known physics. Instead he went up to MIT, where he could be alone, and opened a fresh notebook. On the title page he wrote: Notebook Of Things I Don’t Know About. For the first but not the last time he reorganized his knowledge. He worked for weeks at disassembling each branch of physics, oiling the parts, and putting them back together, looking al the while for the raw edges and inconsistencies.

He tried to find the essential kernels of each subject. When he was done he had a notebook of which he was especial y proud. It was not much use in preparing for the examination, as it turned out. Feynman was asked which color was at the

top of a rainbow; he almost got that wrong, reversing in his mind the curve of refraction index against wavelength. The mathematical physicist H. P. Robertson asked a clever question about relativity, involving the apparent path of the earth as viewed through a telescope from a distant star.

Feynman did get that wrong, he realized later, but in the meantime he persuaded the professor that his answer was correct. Wheeler read a statement from a standard text on optics, that the light from a hundred atoms, randomly phased, would have fifty times the intensity of one atom, and asked for the derivation. Feynman saw that this was a trick. He replied that the textbook must be wrong, because by the same logic a pair of atoms would glow with the same intensity as one. Al this was a formality. Princeton’s senior physicists understood what they had in Feynman. In writing up course notes on nuclear physics, Feynman had been frustrated by a complicated formula of Wigner’s for particles in the nucleus. He did not understand it. So he worked the problem out for himself, inventing a diagram—a harbinger of things to come—that enabled him to keep a tal y of particle interactions, counting the neutrons and protons and arranging them in a group-theoretical way according to pairs that were or were not symmetrical. The diagram bore an odd resemblance to the diagrams he invented for understanding the pathways of folded-paper flexagons. He did not real y understand why his scheme worked, but he was certain that it did, and it proved to be a considerable simplification of Wigner’s own approach.

In high school he had not solved Euclidean geometry problems by tracking proofs through a logical sequence, step by step. He had manipulated the diagrams in his mind:

he anchored some points and let others float, imagined some lines as stiff rods and others as stretchable bands, and let the shapes slide until he could see what the result must be. These mental constructs flowed more freely than any real apparatus could. Now, having assimilated a corpus of physical knowledge and mathematical technique, Feynman worked the same way. The lines and vertices floating in the space of his mind now stood for complex symbols and operators. They had a recursive depth; he could focus on them and expand them into more complex expressions, made up of more complex expressions stil .

He could slide them and rearrange them, anchor fixed points and stretch the space in which they were embedded.

Some mental operations required shifts in the frame of reference, reorientations in space and time. The perspective would change from motionlessness to steady motion to acceleration. It was said of Feynman that he had an extraordinary physical intuition, but that alone did not account for his analytic power. He melded together a sense of forces with his knowledge of the algebraic operations that represented them. The calculus, the symbols, the operators had for him almost as tangible a reality as the physical quantities on which they worked. Just as some people see numerals in color in their mind’s eye, Feynman associated colors with the abstract variables of the formulas he understood so intimately. “As I’m talking,” he once said, “I see vague pictures of Bessel functions from Jahnke and Emde’s book, with light tan j ’s, slightly violet-bluish n ’s, and dark brown x ’s flying around. And I wonder what the hel it must look like to the students.”

In the past eight years neither Dirac nor any other

physicist had been able to fol ow up on the notion of a Lagrangian in quantum mechanics—a way of expressing a particle’s history in terms of the quantity of action. Now Dirac’s idea served as an explosive release in Feynman’s imagination. The uneasy elements of quantum mechanics broke loose and rearranged themselves into a radical y new formulation. Where Dirac had pointed the way to calculating how the wave function would evolve in an infinitesimal slice of time, Feynman needed to carry the wave function farther, through finite time. A considerable barrier separated the infinitesimal from the finite. Making use of Dirac’s infinitesimal slice required a piling up of many steps—infinitely many of them. Each step required an integration, a summing of algebraic quantities. In Feynman’s mind a sequence of multiplications and compounded integrals took form. He considered the coordinates that specify a particle’s position. They churned through his compound integral. The quantity that emerged was, once again, a form of the action. To produce it, Feynman realized, he had to make a complex integral encompassing every possible coordinate through which a particle could move. The result was a kind of sum of probabilities—yet not quite probabilities, because quantum mechanics required a more abstract quantity cal ed the probability amplitude. Feynman summed the contributions of every conceivable path from the starting position to the final position—though at first he saw more a haystack of coordinate positions than a set of distinct paths. Even so, he realized that he had burrowed back to first principles and found a new formulation of quantum mechanics. He could not see where it would lead. Already, however, his