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By Root 2198 0

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.

The White Plague

Twentieth-century medicine was struggling for the scientific footing that physics began to achieve in the seventeenth century. Its practitioners wielded the authority granted to healers throughout human history; they spoke a specialized language and wore the mantle of professional schools and societies; but their knowledge was a pastiche of folk wisdom and quasi-scientific fads. Few medical researchers understood the rudiments of controlled statistical experimentation. Authorities argued for or against particular therapies roughly the way theologians argued for or against their theories, by employing a combination of personal experience, abstract reason, and aesthetic judgment. Mathematics played no role in a biologist’s education. The human body