Unfortunately, however, the same type of procedure that Feynman used to change the theory on small distance scales when considering the self-energy of the electron did not work when considering the infinite impact of vacuum polarization diagrams. Feynman could find no alteration for small electron-positron loops that maintained the nice mathematical properties of the theory without such an alteration. This is probably another reason, in addition to his sense that these diagrams might not be physically appropriate to the problem at hand, why he ignored them in his original Lamb shift calculation.
Feynman wrestled with this problem on and off during 1948 and 1949. He was able to derive quantitatively accurate results by intuitively supposing that various extra terms in his equations, induced by altering the form of the electron-positron loop, were likely to be unphysical, because they did not respect the mathematical niceties of QED, and therefore could be safely ignored. This unsatisfactory situation was resolved when late in 1949 Hans Bethe informed Feynman of a trick Wolfgang Pauli discovered that allowed a mathematically consistent alteration to be introduced in vacuum polarization diagrams.
Once Feynman incorporated this scheme in his calculations, all of the infinities of QED could be tamed, and every physical quantity could be calculated to arbitrarily high precision and compared with experimental results. From the practical perspective of having a theory of electrons and photons with which one could calculate finite (and accurate) predictions for all processes, Feynman had achieved the goal that had first motivated him in his graduate work with Wheeler.
But Feynman didn’t easily let go of his intuitive doubts, and in his famous 1949 paper, “A Space-Time Approach to QED,” which outlined his diagrammatic techniques and results, he added a footnote suggesting that one needed Lamb shift experiments that were more precise, to see if the small contribution that he and others had by then calculated to be due to vacuum polarization effects was real.
It was.
CHAPTER 10
Through a Glass Darkly
My machines came from too far away.
—RICHARD FEYNMAN, TO
SYLVAN SCHWEBER, 1984
Feynman’s tentative reaction to his own results may seem surprising, but it is not. Nor is it unique. The right answers in science are not always obvious at the time they are developed. When working tentatively, at the edge of knowledge, with many wrong twists, blind alleys, and dead ends, it is easy to be skeptical when nature seems to obey the mathematics envisaged at one’s desk. Hence, this is not the end of this part of Feynman’s story, at least if we are to understand his true scientific legacy. Instead, we need to examine some associated events, personalities, and twists of fate that actually govern history, the kind of things that one often avoids when attempting to present an after-the-fact logical exposition of scientific concepts.
Two personalities dominated the immediate environment in which Feynman’s discoveries were developed and perceived: Julian Schwinger and Freeman Dyson. We have already encountered Julian Schwinger, the boy wonder. Like Feynman, he was drawn to the most important fundamental question in theoretical physics at the time: how to turn QED into a consistent theory of nature. Like Feynman, Schwinger had contributed to the war effort, and his work then would also have a profound influence on his approach. Schwinger, who worked at the MIT Radiation Laboratory, began to follow an engineering approach based on classical electrodynamics, focusing on sources and responses. And like Feynman he was fiercely driven by a competition primarily with himself and what he thought he should be able to do.
But the similarities stop there. While also heralding from New York, Schwinger grew up in Manhattan, a world away from Long Island, and perhaps no physicist could have projected a more different aura. Brilliant and polished, he was recruited to Columbia by Rabi at age seventeen after he helped resolve a debate Rabi was having with a colleague in the hallway about a subtle point of quantum mechanics. At age twenty-one, he received his PhD, and eight years later became the youngest tenured professor in Harvard’s history. He projected an air of supreme confidence and organization. Even though Schwinger always lectured without notes, everything seemed planned, from the place where the chalk would first hit the blackboard to the place where it would last leave it, or rather the places, as he sometimes wrote with both hands. The flow of ideas might be complicated, many would say more complicated than it needed to be, but it was precise, logical, and nothing if not elegant.
Feynman’s brilliance manifested itself instead in a kind of intellectual impatience. If he was interested in solving something he would thrust forward to get to the answer and then work backward to understand and fill in the steps. At times he had little patience for those who couldn’t keep up, and few could. As he said, in responding to a later job offer from Caltech, “I do not like to suggest a problem and suggest a method for its solution and feel responsible after the student is unable to work out the problem by the suggested method by the time his wife is going to have a baby so that he cannot get a job. What happens is that I find that I do not suggest what I do not know will work and the only way I know it works is by having tried it at home previously, so I find the old saying that ‘A Ph.D. thesis is research done by a professor under particularly trying circumstances’ is for me the dead truth.”
For this reason, perhaps, Feynman had very few successful students in physics. Schwinger, on the other hand, advised over 150 doctoral students during his career, three of whom later won the Nobel Prize, two in physics and one in biology. It is therefore not surprising that the physics community flocked to listen to Schwinger, not least when he was trying to solve the greatest outstanding puzzle at the forefront of fundamental physics.
Following the conference at Shelter Island, Schwinger also threw himself at the task of deriving a relativistically consistent calculation in QED of one of the quantities that differed from the lowest-order predictions of Dirac’s theory, the anomalous magnetic moment of the electron. Using a set of tools he developed to address the problems of QED, and the renormalization ideas that he, Victor Weisskopf, and H. A. Kramers had been promoting, he was the first to arrive at an answer, late in 1947.
Almost immediately word began to spread in the physics community of Schwinger’s achievement. At the January 1948 meeting of the American Physical Society—the big physics meeting of the year—Schwinger gave an invited lecture titled “Recent Developments in QED.” The interest was so great that he was asked to repeat the lecture later that day, and then again to an even larger audience, to satisfy the demand to hear his results.
Meanwhile, Feynman got up after the talk and reported that he had also computed the same quantities that Schwinger had, and in fact claimed some more generality in the anomalous magnetic moment calculation. But he had yet to explain his methods, so his announcement had significantly less impact.
Feynman’s ideas generated less interest in the scientific community at the time, not because he did not have the same forum at that meeting, but because his entire approach to addressing the problems he eventually solved was unique. He had long ignored the safety net of conventional quantum field theory, and while his diagrams allowed him to calculate remarkable results, to others they may have looked like doodles scrawled in an effort to guess the answer to the problems of the day.
This reticence among the physics community to understand his work was on particular display at the first opportunity Feynman had to lecture about his idea, a few months later, at another National Academy of Sciences–sponsored workshop, the so-called Pocono Conference. His talk, titled “Alternative Formulation of Quantum Electrodynamics,” came after Schwinger’s, which lasted almost a full day. As unflappable as Schwinger was, even he got flustered from time to time, with Bohr, Dirac, and other big shots in the audience interrupting on a constant basis.
Hans Bethe noticed that Schwinger was interrupted the least whenever his presentation became more formal, so he
suggested Feynman make his presentation more formal and mathematical as well. This was like asking Bono to perform Bach on the harpsichord. Feynman had planned to present the material in almost the same way that he had worked on it, emphasizing successful calculations and results that would then take him backward to motivate the ideas. However, in deference to Bethe’s suggestion, Feynman emphasized mathematics associated with his space-time sums rather than physics. The result was, in Feynman’s words, “a hopeless presentation.”
Dirac kept interrupting him to ask if his theory was unitary—a mathematical way of stating the fact, as I described earlier, that the calculated sum of the probabilities for all possible physical outcomes in any situation must equal unity (that is, there is a 100 percent probability that something happens)—but Feynman hadn’t really thought about this, and since his particles were moving forward and backward in time, he answered that he simply didn’t know.
Next, when Feynman introduced the idea of positrons as acting like electrons going backward in time, a participant asked whether this implied that in some of the space-time paths he was including in his calculations, several electrons might appear to be occupying the same state—a clear violation of the Pauli exclusion principle. Feynman answered affirmatively because in this case the different electrons were not really different particles, just the same particles going forward and backward in time. Feynman later recalled that chaos then ensued.
Ultimately this led Bohr to question the very physical basis of Feynman’s space-time concept. He argued that the picture of space-time paths violated the tenets of quantum mechanics, which said that particles do not travel on individual specific trajectories at all. At this point Feynman gave up trying to convince his audience of the correctness of his scheme.
Bohr was wrong, of course. Feynman’s sum over paths made it explicitly clear that many different trajectories must be considered simultaneously when calculating physical results, and in fact Bohr’s son later came over and apologized, saying his father had misunderstood Feynman. But the type of questioning reflected what was clearly a deep skepticism and growing doubt that Feynman had developed a completely consistent picture. He was asking too much of his audience, which, while it contained some of the brightest physical minds in the twentieth century, couldn’t have been expected to adjust in a single lecture to this totally new and still incomplete way of thinking about fundamental processes. After all, Feynman himself had taken years and thousands of pages of calculations to develop it.
One might think that Feynman would have been jealous of Schwinger, especially given the different early receptions to their work. There is no doubt that they were both competitive. But at least as Feynman recalled it, they were more like co-conspirators. Neither could fully understand what the other was doing, but both knew and trusted the other’s abilities, and both felt they had left everyone else in the dust. Feynman’s memory might have been self-serving, for he certainly was not content at the time with not being understood. Disconsolate after the Pocono Conference, he decided that he had to get his ideas into print so he could properly explain what he was doing. To Feynman, who hated writing up his work for publication, the motivation was clear.
Recall that Feynman’s approach to solving physical problems could be framed as “The ends justify the means.” By this I mean that one might strike out with a new half-baked idea or method, but its validity lay in the results. If the calculated results agreed with nature, via experimentation, then the method was probably on the right track and was worth exploring further.
Feynman sensed that his sum-over-paths approach was right. He had by then calculated almost every quantity one could calculate in QED and his results agreed with other methods when they were available. He built his methods as he went along, in order to address the specific questions at hand. But how could he present this in print to a physics audience used to not working backward, but forward to understand a theory?
To Feynman, getting an understanding of his ideas that was clear enough to present to others meant doing more calculations for himself. Over the summer of 1948 he both refined his computational methods and generalized them, and he developed even more powerful ones. By making the computational methods more succinct, and more general, he probably felt that he could communicate the results to the physics community more easily. Finally, by the spring of 1949 Feynman had struggled sufficiently that he had completed his epochal works, “The Theory of Positrons” and “Space-Time Approach to Quantum Electrodynamics,” which essentially laid the basis of all of his ideas and his successful calculations performed over the previous two years.
Two other key factors contributed to cementing Feynman’s resolve and his legacy. The first had to do with a remarkable young mathematician-turned-physicist we have already met, Freeman Dyson, who arrived in the United States in 1947 from the University of Cambridge as a student to work with Bethe, and who would eventually explain Feynman to the rest of the world.
Already famous in the United Kingdom for his mathematical accomplishments, at the age of twenty-three Freeman Dyson had decided that the truly interesting intellectual questions at that time lay in theoretical physics, particularly in the effort to understand the quantum theory of electromagnetism. So, while a fellow at Trinity College, Cambridge, he contacted several physicists and asked where he should go to catch up on the most exciting recent developments, and everyone pointed him to Bethe’s group at Cornell.
Within a year, Dyson had completed a paper calculating the quantum corrections to the Lamb shift in a relativistic “toy” theory with spinless particles. Like Feynman, he was heavily influenced by his profound respect and admiration for Bethe, and his impression of the man was remarkably similar to Feynman’s. As Dyson wrote, “His view was to understand anything meant to be able to calculate the number. That was for him the essence of doing physics.”
By the spring of 1948 Dyson was focusing in depth on the conceptual problems of QED, and along with the rest of the physics world, alerted by Robert Oppenheimer, he read the first issue of the new Japanese journal Progress in Theoretical Physics. He was amazed to find that even though they were completely isolated, Japanese theorists had made remarkable progress during the war. In particular Sin-Itiro Tomonaga had essentially independently developed an approach to resolving the problems of QED using techniques similar to those developed by Schwinger. The difference was that Tomonaga’s approach appeared to be far simpler to Dyson, who wrote, “Tomonaga expressed his in a simple, clear language so that anybody could understand it and Schwinger did not.”
All during this time, Dyson was interacting with Feynman, learning at the blackboard exactly what he had accomplished. This gave him an almost unique opportunity to understand Feynman’s approach at a time when Feynman had yet to publish his work, or even give a coherent seminar on the subject.
If there was anyone Dyson would grow to admire as much or more than Bethe, it was Feynman, whose brilliance, combined with energy, charisma, and fearlessness, was captivating to the young man. Dyson soon realized that not only was Feynman’s space-time approach powerful, but if it were correct, it must be possible to unravel a relationship between this approach and the techniques Schwinger and Tomonaga developed.
At the same time, Dyson had so impressed his mentor that Bethe suggested he spend the second year of his Commonwealth Graduate Fellowship at the Institute for Advanced Study, with Oppenheimer. During the summer, before moving to New Jersey, Dyson went on that fateful cross-country car trip with Feynman to Los Alamos, then attended a summer school in Michigan, and took another cross-country trip, this time by Greyhound bus, to Berkeley and back. During the return trip from California, after forty-eight hours of what for many people would be mind-numbing bus travel, Dyson focused his thoughts intensively on physics, and was able complete in his mind the basic features of a proof that Feynman’s approach and Schwinger’s approach to QED were in fact equivalent. He w
as also able to fuse them together, as he described in a letter, in a “new form of the Schwinger theory which combines the advantages of both.”
By October of 1948, before Feynman had completed his own epic paper on QED, Dyson submitted his famous paper “The Radiation Theories of Tomonaga, Schwinger, and Feynman,” proving their equivalence. The psychological impact of this work was profound. Physicists had trust in Schwinger, but his methods were so complex as to be daunting. By demonstrating that Feynman’s approach was equally trustworthy and consistent and provided a much easier and ultimately systematic method for calculating higher-order quantum corrections, Dyson exposed the rest of the physics community to an effective new tool that everyone could begin to use.
Dyson followed his “Radiation” paper with another seminal paper early in 1949. Having developed the methods to allow the adaptation of Schwinger’s formalism to Feynman’s methods in order to allow the calculation of arbitrarily complicated higher-order contributions to the theory, Dyson set himself the task of proving that it all made sense, in a rigorous fashion, or at least a fashion that was rigorous enough for physicists. He demonstrated that once the problems with infinities in the simplest self-energy and vacuum polarization calculations were resolved, then there were no other infinities that would result in the higher-order calculations. This completed a proof of what has now become known as the renormalizability of the theory, in which all infinities, once first controlled by mathematical tricks of the type that Feynman’s method so easily allowed, can be incorporated in the unmeasurable bare mass and charge terms in the theory. When everything is expressed in terms of the renormalized physically measured masses and charges, all predictions become finite and sensible.
Quantum Man: Richard Feynman's Life in Science Page 13
