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Infinitely smal nails, infinitely energetic hammers.

In a sense the equations were measuring the effect of the electron’s charge on itself, its “self-energy.” That effect would increase with proximity, and how much nearer could the electron be to itself? If the distance were zero, the effect would be infinite—impossible. The wave equation of quantum mechanics only made the infinities more

complicated. Instead of the grade-school horror of a division by zero, physicists now contemplated equations that grew out of bounds because they summed infinitely many wavelengths, infinitely many oscil ations in the field—

although even now Feynman did not quite understand this formulation of the infinities problem. Temporarily, for simple problems, physicists could get reasonable answers by the embarrassing expedient of discarding the parts of the equations that diverged. As Dirac recognized, however, in concluding his Principles of Quantum Mechanics , the electron’s infinities meant that the theory was mortal y flawed. It seems that some essentially new physical ideas are here needed.

Feynman quietly nursed an attachment to a solution so radical and straightforward that it could only have appealed to someone ignorant of the literature. He proposed—to himself—that electrons not be al owed to act on themselves at al . The idea seemed circular and sil y. As he recognized, however, eliminating self-action meant eliminating the field itself. It was the field, the totality of the charges of al electrons, that served as the agent of self-action. An electron contributed its charge to the field and was influenced by the field in turn. Suppose there was no field.

Then perhaps the circularity could be broken. Each electron would act directly on another. Only the direct interaction between charges would be permitted. One would have to build a time delay into the equations, for whatever form this interaction took, it could hardly surpass the speed of light.

The interaction was light, in the form of radio waves, visible light, X rays, or any of the other manifestations of electromagnetic radiation. “Shake this one, that one

shakes later,” Feynman said later. “The sun atom shakes; my eye electron shakes eight minutes later because of a direct interaction across.”

No field; no self-action. Implicit in Feynman’s attitude was a sense that the laws of nature were not to be discovered so much as constructed. Although language blurred the distinction, Feynman was asking not whether an electron acted on itself but whether the theorist could plausibly discard the concept; not whether the field existed in nature but whether it had to exist in the physicist’s mind. When Einstein banished the ether, he was reporting the absence of something real—at least something that might have been

—like a surgeon who opened a chest and reported that the bloody, pulsing heart was not to be found. The field was different. It had begun as an artifice, not an entity. Michael Faraday and James Clerk Maxwel , the nineteenth-century Britons who contrived the notion and made it into an implement no more dispensable than a surgeon’s scalpel, started out apologetical y. They did not mean to be taken literal y when they wrote of “lines of force”—Faraday could actual y see these when he sprinkled iron filings near a magnet—or “idle wheels,” the pseudomechanical, invisible vortices that Maxwel imagined fil ing space. They assured their readers that these were analogies, though analogies with the newly formidable weight of mathematical rectitude.

The field had not been invented without reason. It had unified light and electromagnetism, establishing forever that the one was no more or less than a ripple in the other. As an abstract successor to the now-defunct ether the field was ideal for accommodating waves, and energy did seem to ripple wavelike from its sources. Anyone who played with

electrical circuits and magnets as intently as Faraday and Maxwel could feel the way the “vibrations” or “undulations”

could twist and spin like tubes or wheels. Crucial y, the field also obviated the unpleasantly magical idea of action at a distance, objects influencing one another from afar. In the field, forces propagated sensibly and continuously from one place to the next. There was no jumping about, no sorcerous obeying of faraway orders. As Percy Bridgman, an American experimental physicist and philosopher, said,

“It is felt to be more acceptable to rational thought to conceive of the gravitational action of the sun on the earth, for example, as propagated through the intermediate space by the handing on of some sort of influence from one point to its proximate neighbor, than to think of the action overleaping the intervening distance and finding its target by some sort of teleological clairvoyance.” By then scientists had efficiently forgotten that the field, too, was a piece of magic—a wave-bearing nul ity, or empty space that was not quite empty (and more than space). Or in the elegant phrase of a later theorist, Steven Weinberg: “the tension in the membrane, but without the membrane.” The field grew so dominant in physicists’ thinking that even matter itself sometimes withdrew to the status of mere appendage: a “knot” of the field, or a “blemish,” or as Einstein himself said, merely a place where the field was especial y intense.

Embrace the field or abhor it—either way, by the nineteen-thirties the choice seemed more one of method than reality. The events of 1926 and 1927 had made that clear. No one could be so naïve now as to ask whether Heisenberg’s matrices or Schrödinger’s wave functions

existed . They were alternative ways of viewing the same processes. Thus Feynman, looking for a new eyepiece himself, began drifting back to a classical notion of unfieldlike particle interaction. The wavelike transmission of energy and the hocus-pocus of action at a distance were issues that he would have to address. In the meantime, Wheeler, too, had reasons to be drawn toward this implausibly pure conception. Electrons might interact directly, without the mediation of the field.

Folds and Rhythms

Feynman tended to associate more with the mathematicians than the physicists at the Graduate Col ege. Students from the two groups joined each afternoon for tea in a common lounge—more English tradition transplanted—and Feynman would listen to an increasingly alien jargon. Pure mathematics had swerved away from the fields of direct use to contemporary physicists and toward such seeming esoterica as topology, the study of shapes in two, three, or many dimensions without regard to rigid lengths or angles. An effective divorce had occurred between mathematics and physics.

By the time practitioners reached the graduate level, they shared no courses and had nothing practical to say to one another. Feynman listened to the mathematicians standing in groups or sitting on the couch at tea, talking about their proofs. Rightly or wrongly he felt he had an intuition for what theorems could be derived from what lemmas, even without quite understanding the subject. He enjoyed the strange

rhetoric. He enjoyed trying to guess the counterintuitive answers to their nearly unvisualizable questions, and he enjoyed applying the physicist’s favorite needle, the claim that mathematicians spent their time proving the obvious.

Although he teased them, he thought they were an exciting group—happy and interested in a kind of science that was getting beyond him. One friend was Arthur Stone, a patient young man attending Princeton on a fel owship from England. Another was John Tukey, who later became one of the world’s leading statisticians. These men spent their leisure time in curious ways. Stone had brought with him English-standard loose-leaf notebooks. The American-standard paper he bought at Woolworth’s overhung the notebooks by an inch, so he presently found himself with a supply of inch-wide paper ribbons, suitable for folding and twisting in different configurations. He tried diagonal folds at the 60-degree angle that produced rows of equilateral triangles. Then, fol owing these folds, he wrapped a strip into a perfect hexagon.