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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 oscillations 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 mortally 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 allowed to act on themselves at all. The idea seemed circular and silly. As he recognized, however, eliminating self-action meant eliminating the field itself. It was the field, the totality of the charges of all 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 Maxwell, the nineteenth-century Britons who contrived the notion and made it into an implement no more dispensable than a surgeon’s scalpel, started out apologetically. They did not mean to be taken literally when they wrote of “lines of force”—Faraday could actually see these when he sprinkled iron filings near a magnet—or “idle wheels,” the pseudomechanical, invisible vortices that Maxwell imagined filling 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 Maxwell could feel the way the “vibrations” or “undulations” could twist and spin like tubes or wheels. Crucially, the field also obviated the unpleasantly magical idea of action at a distance,