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percent probability of an error … In the quantum world probabilities were expressed as complex numbers, numbers with both a quantity and a phase, and these so-called amplitudes were squared to produce a probability.

This was the mathematical procedure necessary to capture the wavelike aspects of particle behavior. Waves interfered with one another. They could enhance one another or cancel one another, depending on whether they were in or out of phase. Light could combine with light to produce darkness, alternating with bands of brightness, just as water waves combining in a lake could produce doubly deep troughs and high crests.

Feynman described for his readers what they already knew as the canonical thought experiment of quantum mechanics, the so-cal ed two-slit experiment. For Niels

mechanics, the so-cal ed two-slit experiment. For Niels Bohr it had il ustrated the inescapable paradox of the wave-particle duality. A beam of electrons (for example) passes through two slits in a screen. A detector on the far side records their arrival. If the detector is sensitive enough, it wil record individual events, like bul ets striking; it might be designed to click as a Geiger counter clicks. But a peculiar spatial pattern emerges: the probabilities of electrons arriving at different places vary in the distinct manner of diffraction, precisely as though waves were passing through the slit and interfering with one another. Particles or waves? Sealing the paradox, quantum mechanical y, is a conclusion that one cannot escape: that each electron

“sees,” or “knows about,” or somehow goes through both slits. Classical y a particle would have to go through one slit or the other. Yet in this experiment, if the slits are alternately closed, so that one electron must go through A and the next through B, the interference pattern vanishes. If one tries to glimpse the particle as it passes through one slit or the other, perhaps by placing a detector at a slit, again one finds that the mere presence of the detector destroys the pattern.

Probability amplitudes were normal y associated with the likelihood of a particle’s arriving at a certain place at a certain time. Feynman said he would associate the probability amplitude “with an entire motion of a particle”—

with a path. He stated the central principle of his quantum mechanics: The probability of an event which can happen in several different ways is the absolute square of a sum

of complex contributions, one from each alternative way.

These complex numbers, these amplitudes, were written in terms of the classical action; he showed how to calculate the action for each path as a certain integral. And he established that this peculiar approach was mathematical y equivalent to the standard Schrödinger wave function, so different in spirit.

The central mystery of quantum mechanics—the one to which al others could ultimately be reduced.

A gun (obeying the classical laws)

sprays bul ets toward a target. First they must pass through a screen with two slits. The pattern they make shows how their probability of arrival varies from place to place. They are likeliest to strike directly behind one of the slits. The pattern happens to be simply the sum of the patterns for each slit considered separately: if half the bul ets were fired with only the left slit open and then half were fired with just the right slit open, the result would be the same.

With waves, however, the result is very different, because of interference . If the slits were opened one at a time, the pattern would resemble the pattern for bul ets: two distinct peaks. But when the slits are open at the same time, the waves pass through both slits at once and interfere with each other: where they are in phase they reinforce each other; where they are out of phase they cancel each other out.

Now the quantum paradox: Particles, like bul ets, strike the target one at a time. Yet, like waves, they create an interference pattern. If each particle passes individual y through one slit, with what does it “interfere”? Although each electron arrives at the target at a single place and a single time, it seems that each has passed through—or somehow felt the presence of—both slits at once.

The Physical Review had printed nothing by Feynman since his undergraduate thesis almost a decade before. To his dismay, the editors now rejected this paper. Bethe helped him rewrite it, showing him how to spel out for the reader what was old and what was new, and he tried the more retrospective journal Reviews of Modern Physics , where final y it appeared the next spring under the title

“Space-Time Approach to Non-Relativistic Quantum

Mechanics.” He plainly admitted that his reformulation of quantum mechanics contained nothing new in the way of results, and he stated even more plainly where he thought the merit lay: “There is a pleasure in recognizing old things from a new point of view. Also, there are problems for which the new point of view offers a distinct advantage.”

(For example, when two particles interacted, it became possible to avoid the laborious bookkeeping of two different coordinate systems.) His readers—and at first they were few—found no fancy mathematics, just this shift of vision, a bit of physical intuition laid atop a foundation of clean, classical mechanics.

Few immediately recognized the power of Feynman’s vision. One who did was the Polish mathematician Mark Kac, who heard Feynman describe his path integrals at Cornel and immediately recognized a kinship with a problem in probability theory. He had been trying to extend the work of Norbert Wiener on Brownian motion, the herky-jerky random motion in the diffusion processes that so dominated Feynman’s theoretical work at Los Alamos.

Wiener, too, had created integrals that summed many possible paths a particle could take, but with a crucial difference in the handling of time. Within days of Feynman’s talk, Kac had created a new formula, the Feynman-Kac Formula, that became one of the most ubiquitous of mathematical tools, linking the applications of probability and quantum mechanics. He later felt that he was better known as the K in F-K than for anything else in his career.

Even to physicists wel accustomed to theoretical

constructions with awkward philosophical implications, Feynman’s summings of paths—path integrals—seemed bizarre. They conjured a universe where no potential goes uncounted; where nothing is latent, everything alive; where every possibility makes itself felt in the outcome. He had expressed his conception to Dyson:

The electron does anything it likes. It just goes in any direction at any speed, forward or backward in time, however it likes, and then you add up the amplitudes and it gives you the wave function.

Dyson gleeful y retorted that he was crazy. Stil , Feynman had caught the intuitive essence of the two-slit experiment, where an electron seems aware of every possibility.

Feynman’s path-integral view of nature, his vision of a

“sum over histories,” was also the principle of least action, the principle of least time, reborn. Feynman felt that he had uncovered the deep laws that gave rise to the centuries-old principles of mechanics and optics discovered by Christiaan Huygens, Pierre de Fermat, and Joseph-Louis Lagrange. How does a thrown bal know to find the particular arc whose path minimizes action? How does a ray of light know to find the path that minimizes time?

Feynman answered these questions with images that served not only for the novel mysteries of quantum mechanics but for the treacherously innocent exercises posed for any beginning physics student. Light seems to angle neatly as it passes from air to water. It seems to