by James Gleick
The absorber theory of Wheeler and Feynman had by then lost the interest of an increasingly single-minded particle physics, but it held center stage in this eclectic gathering. It had been born of their concern with reversible and irreversible processes, and now it served as common ground for three different approaches to understanding time’s flow, the arrow of time. As particle physicists had passed the absorber theory by, a new generation of cosmologists had taken it up. Their field had begun a transition from mere stargazing astronomy to an enterprise asking the grandest questions about the universe: whence and wherefore. It was beginning to stand out among the modern sciences as an enterprise not fully scientific, but an amalgam of philosophy, art, faith, and not a little hope. They had so few windows through the murky atmosphere—a few overworked glass contraptions on mountain tops, a few radio antennae—yet they believed they could peer far enough, or guess shrewdly enough, to uncover the origins of space and time. Already their space was not the flat, neutral stuff of their parents’ pre-Einsteinian intuition, but an eerily plastic medium that somehow embodied both time and gravity. Some of them, but not all, believed that space was expanding at high speed and dragging its contents farther and farther apart, on account of an explosive big bang ten or fifteen billion years before. It no longer seemed safe to assume that the universe was the same everywhere, infinite, static, Euclidean, ageless, and homogeneous: world without end, amen. The strongest evidence for an expanding universe was still, in 1963, Edwin Hubble’s 1929 discovery that other galaxies are streaming away from ours, and that the farther away they are, the faster they seem to be moving. Whether this expansion would continue forever or whether it would reverse itself was—and would remain—an open question. Perhaps the universe bloomed and collapsed again and again in a cycle that ran through eternity.
The issue seemed linked to the nature of time itself. Assumptions about time were built into the equations for the particle interactions that led to the creation and dissipation of light. If one thought about time as Wheeler and Feynman had, one could not escape a cosmic connection between these intimate interactions and the process of universal expansion. As Hermann Bondi said at the meeting’s outset, “This process leads to the dark night sky, to the disequilibrium between matter and radiation, and to the fact that radiated energy is effectively lost … we accept a very close connection between cosmology and the basic structure of our physics.” By their boldness in constructing a time-symmetrical theory of half advanced and half retarded waves, Wheeler and Feynman had been forced into boldness of a cosmological sort. If the equations were to balance properly, they had to make the mathematical assumption that all radiation was eventually absorbed somewhere. A beam of light heading forever into an eternal future, never to cross paths with a substance that would absorb it, would violate their assumption, so their theory mandated a certain kind of universe. If the universe were to expand forever, conceivably its matter might so thin out that light would not be absorbed.
Physicists had learned to distinguish three arrows of time. Feynman described them: the thermodynamic or “accidents of life” arrow; the radiation or “retarded or advanced” arrow; and the cosmological arrow. He suggested keeping in mind three physical pictures: a tank with blue water on one side and clear water on the other; an antenna with a charge moving toward it or away; and distant nebulas moving together or apart. The connections between these arrows were connections between the pictures. If a film showed the water getting more and more mixed, must it also show the radiation leaving the antenna and the nebulas drifting apart? Did one form of time govern the others? His listeners could only speculate, and speculate they did.
“It’s a very interesting thing in physics,” said Mr. X, “that the laws tell us about permissible universes, whereas we only have one universe to describe.”
Least Action in Quantum Mechanics
Omega oil did nothing for Arline’s lumps and fevers, and she was admitted to the hospital in Far Rockaway with what her doctor feared was typhoid. Feynman began to glimpse the special powerlessness that medical uncertainty can inflict on a scientific person. He had come to believe that the scientific way of thinking brought a measure of calmness and control in difficult situations—but not now. However remotely, medicine was a part of the domain of knowledge he considered his. It belonged to science. At one time his father had hopefully studied a kind of medicine. Lately Richard had been sitting in on a physiology course, learning some basic anatomy. He read up on typhoid fever in Princeton’s library, and when he visited Arline in the hospital he started questioning the doctor. Had a Widal test been administered? Yes. The results? Negative. Then how could it be typhoid? Why were all of Arline’s friends and relatives wearing gowns to protect against supposed bacteria that even a sensitive laboratory test could not detect? What did the mysterious lumps appearing and disappearing in her neck and armpit have to do with typhoid? The doctor resented his questions. Arline’s parents pointed out that his status as fiancé did not entitle him to interfere in her medical care. He backed down. Arline seemed to recover.
With Wheeler, meanwhile, Feynman was trying to move their work a crucial step forward. So far, despite its modern, acausal flavor, it was a classical theory, not a quantum one. It treated objects as objects, not as probabilistic smudges. It treated energy as a continuous phenomenon, where quantum mechanics required discrete packets and indivisible jumps in well-defined circumstances. The problem of self-energy was as severe in classical electrodynamics as in quantum theory. Unwanted infinities predated the quantum. They appeared as soon as one faced the consequences of a pointlike electron. It was as simple as dividing by zero. Feynman had felt from the beginning that the natural route would be to start with the classical case and only then work toward a quantized electrodynamics. There were already standard recipes for translating classical models into their modern quantum cousins. One prescription was to take all the momentum variables and replace them with certain more complicated expressions. The problem was that in Wheeler and Feynman’s theory there were no momentum variables. Feynman had eliminated them in creating his simplified framework based on the principle of least action.
Sometimes Wheeler told Feynman not to bother—that he had already solved the problem. Later in the spring of 1941 he went so far as to schedule a presentation of the quantized theory at the Princeton physics colloquium. Pauli, still dubious, buttonholed Feynman on his way into Palmer Library one day. He asked what Wheeler was planning to say. Feynman said he didn’t know.
“Oh?” Pauli said. “The professor doesn’t tell his assistant how he has it worked out? Maybe the professor hasn’t got it worked out.”
Pauli was right. Wheeler canceled the lecture. He lost none of his enthusiasm, however, and made plans for not one but a grand series of five papers. Feynman, meanwhile, had a doctoral thesis to prepare. He decided to approach the quantizing of his theory just as he had approached complicated problems at MIT, by working out cases that were stripped to their bare essentials. He tried calculating the interaction of a pair of harmonic oscillators, coupled, with a time delay—just a pair of idealized springs. One spring would shake, sending out a pure sine wave. The other would bounce back, and out of their interaction new wave forms would evolve. Feynman made some progress but could not understand the quantum version. He had gone too far in the direction of simplicity.
Conventional quantum mechanics went from present to future by the solving of differential equations—the so-called Hamiltonian method. Physicists spoke of “finding a Hamiltonian” for a system: if they could find one, then they could go ahead and calculate; if not, they were helpless. In Wheeler and Feynman’s view of direct action at a distance, the Hamiltonian method had no place. That was because of the introduction of time delays. It was not enough merely to write down a complete description of the present: the positions, momentums, and other quantities. One never knew when some delayed effect would hurtle into the picture out of the past (or in the case of Wheeler and F
eynman, out of the future). Because past and future interacted, the customary differential-equation point of view broke down. The alternative least-action or Lagrangian approach was no luxury. It was a necessity.
With all this on his mind, Feynman went to a beer party at the Nassau Tavern. He sat with a physicist lately arrived from Europe, Herbert Jehle, a former student of Schrödinger in Berlin, a Quaker, and a survivor of prison camps in both Germany and France. The American scientific world was absorbing such refugees rapidly now, and the turmoil of Europe seemed more palpable and near. Jehle asked Feynman what he was working on. Feynman explained and asked in turn whether Jehle knew of any application of the least-action principle in quantum mechanics.
Jehle certainly did. He pointed out that Feynman’s own hero, Dirac, had published a paper on just that subject eight years before. The next day Jehle and Feynman looked at it together in the library. It was short. They found it, “The Lagrangian in Quantum Mechanics,” in the bound volumes of Physikalische Zeitschrift der Sowjetunion, not the best-read of journals. Dirac had worked out the beginnings of a least-action approach in just the style Feynman was seeking, a way of treating the probability of a particle’s entire path over time. Dirac considered only one detail, a piece of mathematics for carrying the wave function—the packet of quantum-mechanical knowledge—forward in time by an infinitesimal amount, a mere instant.
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 all 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 follow, 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 small 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 collaboration 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 Hall 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 casually 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 unwilling 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 all 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 especially 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. All 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 tally 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 really 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 still. 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 hell it must look like to the students.”
In the past eight years neither Dirac nor any other physicist had been able to follow 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 radically 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 called 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 sense of paths in space-time seemed somehow cleaner—more direct. There seemed something quaint now about the peculiarly constrained oscillations of the post-ethereal field, the wavy inheritance of the 1920s.
