It occurred to Dyson that he was rushing into print with accounts of theories not yet published by their inventors and that the inventors themselves might take offense. He visited Bethe, temporarily in New York visiting Columbia, and they took a long walk in Riverside Park as the sun set over the Hudson River. Bethe warned him that there could be problems. Dyson said it was Schwinger’s and Feynman’s own fault that they had not published “any moderately intelligible account”: Schwinger, he suspected, was polishing obsessively, while Feynman simply couldn’t be bothered with paperwork. It was irresponsible. They were retarding the development of science. By publicizing their work Dyson was performing a service to humanity, he argued. He and Bethe ended up agreeing that Feynman would not mind but that Schwinger might, and that it would be poor tactics for an ambitious young physicist to irritate Schwinger. “So the result of all this,” Dyson wrote his parents,
is that I am reversing the tactics of Mark Antony, and saying very loud at various points in my paper, “I come to praise Schwinger, not to bury him.” I only hope he won’t see through it.
Still, he made his judgment clear. The distinctions he drew and the characterizations he set down soon became the community’s conventional wisdom: that Schwinger’s and Tomonaga’s approach was the same, while Feynman’s differed profoundly; and that Feynman’s method was original and intuitive, while Schwinger’s was formal and laborious.
Dyson well understood that he was reaching out to an audience that wanted tools. When he showed a Schwinger formula with commutators threatening to subdivide like branches on a tree and remarked that “their evaluation gives rise to long and rather difficult analysis,” he knew that his readers would not suspect him of overstating the difficulty. Ease of use was the Feynman virtue he stressed. To “write down the matrix elements” for a certain event, he explained, one need only take a certain set of products, replace them by sums of matrix elements from another equation, reassemble the various terms in a certain form, and undertake a certain type of substitution. Or, he said, one could simply draw a graph.
The simplest Dyson graph.
Graph was the mathematician’s word for a network of points joined by lines. Dyson showed that there was a graph for every matrix and a matrix for every graph—the graphs provided a means of cataloging these otherwise-misplaceable arrays of probabilities. So alien did this conceit seem that Dyson left it to his readers to draw the graphs in their minds. The journal editors made room for just one figure. Dyson called the solid lines, with an implicit direction, electron lines. The directionless dotted lines were photon lines. Feynman, he mentioned, had something more in mind than the mere bookkeeping of matrices: “a picture of the physical process.” For Feynman the points represented the actual creation or annihilation of particles; the lines represented paths of electrons and photons, not through a measurable real space but through the history from one quantum event to another.
Oppenheimer depressed Dyson with a coolness bordering on animosity. It was the last response he had expected: a defeatist Oppenheimer, a lethargic Oppenheimer, an Oppenheimer hostile to new ideas and unwilling to listen. He had been in Europe, where he had summarized the present state of the theory at two international conferences. It was “Schwinger’s theory” and “Schwinger’s program.” There were developments “the first largely, the second almost wholly, due to Schwinger.” In passing, there were “Feynman’s algorithms”—an exotically disdainful phrase.
Dyson decided that there would be no prize for timidity and—still in his first weeks at the institute—sent Oppenheimer by interoffice mail an aggressive manifesto. He argued that the new quantum electrodynamics promised to be more powerful, more self-consistent, and more broadly applicable than Oppenheimer seemed to think. He did not mince words.
From Mr. F. J. Dyson.
Dear Dr. Oppenheimer:
As I disagree rather strongly with the point of view expressed in your Solvay Report (not so much with what you say as with what you do not say) …
I… . I am convinced that the Feynman theory is considerably easier to use, understand, and teach.
II. Therefore I believe that a correct theory, even if radically different from our present ideas, will contain more of Feynman than of Heisenberg-Pauli. …
V. I do not see any reason for supposing the Feynman method to be less applicable to meson theory than to electrodynamics… .
VI. Whatever the truth of the foregoing assertions may be, we have now a theory of nuclear fields which can be developed to the point where it can be compared with experiment, and this is a challenge to be accepted with enthusiasm.
Enthusiasm was not immediately forthcoming, but Oppenheimer did set up a series of forums to let Dyson make his case. They became an occasion. Bethe came down from New York to listen and lend moral support. As the seminars went on, Oppenheimer was a dramatically nerve-tightening presence. He interrupted continually, criticizing, jabbing, pouncing on errors. To Dyson he seemed uncontrollably nervous—always chain-smoking and fidgeting in his chair. Feynman himself was following Dyson’s progress by long-distance as he continued his own work. Dyson visited him at Cornell one weekend and watched, amazed, as he rattled off two new fundamental calculations in a matter of hours. Then Feynman fired off a hasty letter: “Dear Freeman: I hope you did not go bragging about how fast I could compute the scattering of light by a potential because on looking over the calculations last night I discovered the entire effect is zero. I am sure some smart fellow like Oppenheimer would know such a thing right off.”
In the end Bethe turned Oppenheimer around. He cast his vote explicitly with the Feynman theory and let the audience know that he felt Dyson had more to say. He took Oppenheimer aside privately, and the mood shifted. By January, the war had been won. At the American Physical Society meeting Dyson found himself almost as much a hero as Schwinger had been the year before. Sitting in the audience with Feynman beside him, he listened as a speaker talked admiringly of “the beautiful theory of Feynman-Dyson.” Feynman said loudly, “Well, Doc, you’re in.” Dyson had not even got a doctoral degree. He went on an excited lecture tour and told his parents that he was a certified big shot. The reward that lasted, however, was a handwritten note that had appeared in his mailbox in the dying days of the fall, saying simply, “Nolo contendere. R. O.”
Dyson Graphs, Feynman Diagrams
It was the affair of Case and Slotnick at the same January meeting that brought home to Feynman the full power of his machinery. He heard a buzz in the corridor after an early session. Apparently Oppenheimer had devastated a physicist named Murray Slotnick, who had presented a paper on meson dynamics. A new set of particles, a new set of fields: would the new renormalization methods apply? With physicists looking inward to the higher-energy particles implicated in the forces binding the nucleus, meson theories were now rising to the fore. The flora and fauna of meson theories did seem to resemble quantum electrodynamics, but there were important differences—chief among them: the counterpart of the photon was the meson, but mesons had mass. Feynman had not learned any of the language or the special techniques of this fast-growing field. Experiments were delivering data on the scattering of electrons by neutrons. Infinities again seemed to plague many plausible theories. Slotnick investigated two species of theory, one with “pseudoscalar coupling” and one with “pseudovector coupling.” The first gave finite answers; the second diverged to infinity.
So Slotnick reported. When he finished Oppenheimer rose and asked, “What about Case’s theorem?”
Slotnick had never heard of Case’s theorem—and could not have, since Kenneth Case, a postdoctoral fellow at Oppenheimer’s institute, had not yet publicized it. As Oppenheimer now revealed, Case’s theorem proved that the two types of coupling would have to give the same result. Case was going to demonstrate this the next day. For Slotnick, the assault was unanswerable.
Feynman had not studied meson theories, but he scrambled for a briefing and went back to his hotel room t
o begin calculating. No, the two couplings were not the same. The next morning he buttonholed Slotnick to check his answer. Slotnick was nonplussed. He had just spent six intensive months on this calculation; what was Feynman talking about? Feynman took out a piece of paper with a formula written on it.
“What’s that Q in there?” Slotnick asked.
Feynman said that was the momentum transfer, a quantity that varied according to how widely the electron was deflected.
Another shock for Slotnick: here was a complication that he had not dared to confront in a half-year of work. The special case of no deflection had been challenge enough.
This was no problem, Feynman said. He set Q equal to zero, simplified his equation, and found that indeed his night’s work agreed with Slotnick. He tried not to gloat, but he was afire. He had completed in hours a superior version of a calculation on which another physicist had staked a major piece of his career. He knew he now had to publish. He possessed a crossbow in a world of sticks and clubs.
He went off to Case’s lecture. At the end he leapt up with the question he had ready: “What about Slotnick’s calculation?”
Schwinger, meanwhile, found the spotlight sliding away. Dyson’s paper carried a sting—Dyson, who had seemed such an eager student the summer before. Now this strange wave of Dyson-Feynman publicity. As Schwinger said later with his incomparably sardonic obliqueness, “There were visions at large, being proclaimed in a manner somewhat akin to that of the Apostles, who used Greek logic to bring the Hebrew god to the Gentiles.”
Feynman now presented his own logic in his own voice. He and Dyson appeared at a third and last small gathering of physicists, this time at Oldstone-on-the-Hudson, New York, the final panel of the triptych that had begun at Shelter Island two years earlier. He published an extended set of papers—they would stretch over three years and one hundred thousand words—that defined the start of the modern era for the next generation of physicists. After his path-integrals paper came, in the Physical Review, “A Relativistic Cut-Off for Classical Electrodynamics,” “Relativistic Cut-Off for Quantum Electrodynamics,” “The Theory of Positrons,” “Space-Time Approach to Quantum Electrodynamics,” “Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction,” and “An Operator Calculus Having Applications in Quantum Electrodynamics.” As they appeared, the younger theorists who devoured them realized that Dyson had given only a bare summary of Feynman’s vision. They felt invigorated by his images—beginning with the unforgettable bombardier metaphor in the positron paper—and by his way of insisting on the plainest statements of physical principles in physical language:
The rest mass particles have is simply the work done in separating them against their mutual attraction after they are created… .
How would such a path appear to someone whose future gradually became past through a moving present? He would first see …
No aspiring physicist could read these papers without thinking about what space was, what time was, what energy was. Feynman was helping physics live up to the special promise it made to its devotees: that this most fundamental of disciplines would bring them face to face with the primeval questions. Above all, however, to young physicists the diagrams spoke loudest.
Feynman had told Dyson, with a slight edge, that he had not bothered to read his papers. “Feynman and I really understand each other,” Dyson wrote home cheerily. “I know that he is the one person in the world who has nothing to learn from what I have written; and he doesn’t mind telling me so.” Feynman’s students, however, sometimes noticed what seemed to them an undercurrent of anger in the pointed comments he would make about Dyson. He had started hearing about Dyson’s graphs—irritating. Why graphs? he asked Dyson. Was that the mathematician in him, putting on airs?
Feynman’s space-time method had other antecedents besides Dyson’s graphs, as it happened. A 1943 German textbook by Gregor Wentzel contained a parallel depiction of a particle exchange process in beta decay. A Swiss student of Wentzel’s, Ernst Stückelberg, had developed a diagrammatic approach that even embraced the conception of time-reversed positrons; parts of this he published, in French, and parts were returned as unpublishable. (Wentzel himself was the unimpressed referee.) Their diagrams showed glimmerings of the style of visualization that Feynman now brought to fruition. His own full-dress version finally appeared in a paper he sent off in late spring 1949. “The fundamental interaction”—an image that would burn itself into the brains of the next generation of field theorists—showed two electrons interacting by exchanging a single photon.
A diagram from a little-known 1941 paper of Ernst Stuckelberg, showing aversion of time reversal in particle trajectories.
He drew electrons as solid lines with arrows. For photons he used wavy lines without arrows: no directionality needed because the photon’s anti-particle is itself. “The fundamental interaction” reinterpreted the basic textbook process of electromagnetic repulsion. Two negative charges, electrons, repel. A standard picture, showing lines of force or merely two balls pressing apart from each other, would beg the question of how an entity feels the force of another entity at a distance. It would imply that force can be transmitted instantly, when in truth, as Feynman’s diagrams automatically made explicit, whatever carries force can move only as fast as light. In the case of electromagnetism, it is light—in the form of fugitive “virtual” particles that flash into existence just long enough to help quantum theorists balance their books.
These were space-time diagrams, of course, representing time as one direction on the page. Typically the past sat at the bottom and the future at the top; one way to read the diagram would be to cover it with a sheet of paper, pull the paper slowly upward, and watch the history unfold. An electron changes course as it emits a photon. Another electron changes course when it absorbs the photon. Yet even the idea that the earlier event is emission and that the later is absorption represented a prejudice about time. It was built into the language. Feynman stressed how free his approach was from customary intuitions: these events are interchangeable.
The Feynman diagram: “The fundam ental interaction.” It is a space-time diagram: the progress of time is shown upward on the page. If one covers it with a sheet of paper and then draws the paper slowly upward:
•A pair of electrons-their paths shown as solid lines-move toward each other.
•When (6) is reached , a virtual photon is emitted by the right-hand electron (wiggly line), and the electron is deflected outward.
•At (5) the photon is reabsorbed by the other electron , and it, too, is deflected outward.
Thus this diagram depicts the ordinary force of repulsion between two electrons as a force carried by a quantum of light. Because it is a virtual particle, coming into existence for a mere ghostly instant, it can temporarily violate the laws that govern the system as a whole—the exclusion principle or the conservation of energy, for example. And Feynman noted that it is arbitrary to think of the photon as being emitted in one place and absorbed in the other: one can say just as correctly that it is emitted at (5), travels backward in time, and is then (earlier) absorbed at (6).
The diagram is an aid to visualization. But it serves physicists mainly as a bookkeeping device. Each diagram is associated with a complex number, an amplitude that is squared to produce a probability for the process shown.
In fact each diagram represented not a particular path, with specified times and places, but a sum of all such paths. There were other simple diagrams. He represented the self-energy of an electron—its interaction with itself—by showing a photon line returning to the same electron that spawned it. There was a grammar of permissible diagrams, corresponding, as Dyson had emphasized, to the permissible mathematical operations. Still, the diagrams could grow arbitrarily complicated, virtual particles appearing and disappearing in an intricate, recursive mesh. Feynman’s first H-shaped diagram for interacting electrons was the only such diagram with one virtual photon. Drawing a
ll the possible diagrams with two virtual photons showed how quickly the permutations grew. Each made a contribution to the final computation, and more complicated diagrams became enormously difficult to calculate. Fortunately the greater the complication the less the probability and the smaller, therefore, the effect on the answer. Even so, physicists would shortly find themselves agonizing over pages of diagrams resembling catalogs of knots. They found it was worth the effort; each diagram could replace an effective lifetime of Schwingerian algebra.
Self-interaction. It is necessary to sum the amplitudes corresponding tomany Feynman diagrams to add the contributions for every way an event can occur. The continual possibility of virtual particles materializing and vanishing causes increasing complexity. Here an electron interacts with itself, in effect- the self-energy problem that first troubled Feynman in his work with Wheeler. It emits and absorbs its own virtual photon.
Feynman diagrams seemed to depict particles, and they had sprung from a mind focused on a particle-centered style of visualization, but the theory they anchored—quantum field theory—gave center stage to the field. In a sense the paths of the diagrams, and the paths summed in the path integrals, were the paths of the field itself. Feynman read the Physical Review more avidly than ever in the past, watching for citations. For a while it was all Schwinger—a paper would be pages of glyphs and would culminate in a neat expression that Feynman felt he could simply have written down as a starting point. He was sure this could not last. It did not. Feynman’s method, Feynman’s rules, began to take over. In the summer of 1950 a paper appeared with small “Feynman diagrams” on the first page—“following the simplified methods introduced by Feynman.” A month later came another: “a technique due to Feynman… . The calculation of matrix elements can be simplified greatly by use of the Feynman-Dyson methods.” The unreasonable power of the diagrams in the hands of students frustrated some of the elders, who felt that physicists were waving a sword that they did not understand. As the flood of papers began to cite Feynman, Schwinger went into what he described as his retreat. “Like the silicon chip of more recent years, the Feynman diagram was bringing computation to the masses,” he said. Later, people who overlooked the note of hoi polloi quoted this remark as though Schwinger had intended a tribute. He had not. This was “pedagogy, not physics.”