by John Gribbin
In one sense, the discovery of the Lamb shift was an indication that the Dirac theory was incomplete. But physicists already knew that, because of the way infinities came into the theory of quantum electrodynamics (QED) when they tried to calculate the self-interaction of an electron in an electromagnetic field. Indeed, the infinite term resulting from the self-interaction would, if it were real, have corresponded to an infinite Lamb shift, whatever that might mean. So in another sense, Lamb’s work showed that the Dirac theory might not be so bad after all, because the disagreement with the experiments, far from being infinite, was a tiny number corresponding to a very small shift in the energy levels. If Lamb had found zero shift, that would have meant Dirac was right, which would have flown in the face of what was already known, and would in that sense have been bad news. But the Lamb shift told the physicists at Shelter Island that what they had to try to find was not zero or infinity but a finite, very small, and now precisely known, quantity. That, they thought, they ought to be able to handle; with real numbers on the table in front of them, perhaps there was, at last, a chance to make sense out of QED.
Like other participants, Feynman also contributed to the conference, a talk about his spacetime approach to quantum mechanics, and path integrals; but, like most of the other presentations, this contribution (essentially a summary of his thesis work) made very little impact alongside the sensational news about the Lamb shift. The big question was, could quantum theory be tweaked up to predict the right amount of change in the energy levels?
At that time, Hans Bethe had a summer job as a consultant for the General Electric Company’s research laboratory in Schenectady, New York, and it was on the train from New York to Schenectady, immediately after the Shelter Island Conference, that he made the first, imperfect but suggestive, calculation of the Lamb shift. Bethe seemed to like working on trains – in similar circumstances, back in 1938, he had solved the puzzle of how nuclear fusion reactions keep the Sun hot (the work for which he won his own Nobel Prize) on a train ride back to Cornell after a conference in Washington D.C. Now, he had worked out a trick to get rid of the infinities in QED, and leave behind a small, finite amount of interaction, corresponding to the Lamb shift. There was one snag; in this first stab at the problem, he had not taken account of the effects required by relativity theory, and only had a non-relativistic calculation of the shift. But still, it was a big step in the right direction.
What Bethe did, in effect, was to calculate the energy for an electron in a hydrogen atom, which came out as the usual infinity plus a correction caused by the presence of the nearby atomic nucleus (in this case, a single proton). Then he subtracted from this the energy of a free electron, which is infinity, leaving behind the correction – the energy shift required. This approach, called ‘renormalization’, came originally from work by the Dutch physicist Hendrik Kramers (another of the Shelter Island participants) on another puzzling infinity that arises in quantum theory, and it has no right to work. Infinity is a funny thing. Infinity plus a little bit is also infinity, and at one level you might think that subtracting the two quantities Bethe was playing with (infinity plus a little bit, minus infinity) ought to leave zero. On the other hand, you could imagine ‘making’ infinity by adding up all of the integer numbers there are, and making another infinity by doubling each integer and adding up the doubled numbers. Bizarrely, the second infinity is smaller than the first one, because it contains only even numbers, whereas the first infinity contains all the even numbers and all the odd ones. If you subtract the second infinity from the first one, now you will be left with infinity again, the sum of all the odd numbers alone! In fact, a mathematician can arrange to get almost any answer you want by subtracting infinity from infinity; the fact that, as Bethe found, the infinities really could be cancelled out of the quantum equations in this way to give the right answer for the Lamb shift seemed to some people a miracle, to others a fraud, while to most physicists it meant that he had made a fundamental discovery about the way the world works, although they weren’t quite sure what that discovery was (this final position is still roughly where physics stands today).
The discovery highlights one of the great features of Bethe’s work. Given a number, a link with experiment, he would take the appropriate theory and shake it by the scruff of its neck until either it fell apart or it was forced to agree with the experiment. Feynman’s great weakness, up to this point, was that he had developed a whole new way of looking at quantum theory, but had never tried to use it to calculate numbers that could be compared with experiment in this way. He still had not learned the lesson of his encounter with Jehle. And yet, one of the great features of Feynman’s version of quantum theory was that it had relativity built into it – it was, in the jargon, relativistically invariant. As news of Bethe’s work spread, many physicists tried to find a way to develop a relativistic version of the appropriate equations. Feynman first heard the news in an excited phone call from Bethe in Schenectady, but didn’t immediately take in its importance.16 It was only when Bethe returned to Cornell and gave a formal lecture on his discovery, ending by pointing out the need for a relativistically invariant version of the calculation, that the penny dropped. Feynman went up to Bethe after the lecture and said, ‘I can do that for you. I’ll bring it in for you tomorrow.’17
Up to that point, though, Feynman hadn’t even used his beautiful new machinery to calculate the self-energy of the electron. For the first time,18 he applied the path integral approach to ordinary electrodynamics, instead of using the half advanced and half retarded formulation. The theory was clear enough, but Feynman had never tried to do anything like this with it before. Probably as a result, when he did try to work through the Lamb shift problem with Bethe the next day, somehow he made an error, and when they tried to apply renormalization the infinities refused to disappear (in other words, the equations diverged). He had to go back to his room and worry away at the problem, learning how to calculate the self-energy and all the other things he had ignored, over the next couple of months. Then he tackled the problem again. The calculation worked out right, and the infinities disappeared – in the jargon, the equations converged in just the right way, using the renormalization trick. It was now early in the autumn of 1947. Having realized, at last, the power of his new tool, Feynman set out to calculate everything in sight. By the time of the next of the three big meetings, the Pocono Conference held in April 1948, he had done just about all of the work for which he would win the Nobel Prize, including an updated discussion of positrons as electrons going backwards in time; but the material was not yet in a form that could be immediately understood by other physicists, brought up on the old ways of the Hamiltonian approach and the Schrödinger equation.
Some of Feynman’s new work was presented in a talk he gave at the Institute for Advanced Study, in Princeton, on 12 November. Dirac was in the audience, and one of the other participants wrote to a colleague that ‘Dirac is very impressed by Feynman, and thinks he does some interesting things.’19 But Dirac was then in a minority, as far as appreciating Feynman’s new work was concerned.
For most physicists, the next really exciting development in QED came from Julian Schwinger, who presented his version of the Lamb shift calculation, in relativistically invariant form, to the annual meeting of the American Physical Society which took place in New York in January 1948. He had also calculated the important property called the magnetic moment of the electron, and the extent of its departure from the value predicted by the Dirac equation. So many people wanted to hear the talk that it had to be repeated in the afternoon. After this talk, Feynman, who was in the audience, stood up and mentioned that he, too, had got the same results (in one case, going a step further than Schwinger) by a different method. He later regretted this. Schwinger was, at the time, more well known than Feynman (not least because Feynman had hardly published anything since his undergraduate senior thesis; even the work in his PhD thesis would only be published in a journal,
Reviews of Modern Physics, in 1948), and Feynman felt that his comments came across with the air of a small boy saying ‘me too’, when he had really just been trying to say that the results must be right if two separate calculations gave the same answers.20 In Feynman’s own mind, though, it was an important moment, because it meant that he really was on the right track, if he was getting the same results as Schwinger. Of course, there was an element of rivalry, felt especially keenly by Feynman because he was the lesser known of the two. He wanted to catch up with Schwinger, and overtake him; but most of all, he wanted to solve the problems of QED, whether Schwinger solved them first or not, just as, long ago, he had solved mathematical problems for his own satisfaction, without worrying whether some Greek mathematician in ancient times had solved them first.
The trouble with Schwinger, though, was that his work was difficult to follow because it was so complicated. This was partly in the nature of the Hamiltonian approach, and partly, many physicists suspected, through Schwinger’s own love of mathematics. If there were two ways to prove a mathematical point, it seemed, Schwinger would always choose the more elegant but also more complicated way, showing off his erudition in the process. It meant that his version of quantum electrodynamics involved hundreds of equations, developed with great mathematical skill and precision, but with few signposts, in the form of links with physics of the kind Bethe so revelled in, to point the way. Schwinger was a virtuoso with equations, but to anyone lacking his virtuosity it was often hard to fathom where he got his answers from. Nevertheless, his great triumph – the last great fling of the old way of doing quantum mechanics – came at the conference held at the Pocono Manor Inn, in the Pocono Mountains of Pennsylvania, between 30 March and 2 April 1948.
This time, there were 28 physicists at the meeting. Schwinger offered them their first glimpse of a complete relativistically invariant theory of quantum electrodynamics, taking up almost a full day. There were few questions, because nobody there had enough mathematical skill to find any flaws in the argument, even if there had been any. But everyone agreed that it was a triumph. Then Feynman, seven weeks short of his thirtieth birthday, gave his talk, under the title Alternative Formulation of Quantum Electrodynamics’. Partly at the suggestion of Bethe, who had noticed how Schwinger’s equations stunned the audience into silence, he made the mistake of offering his version of BED from a mathematical perspective, instead of kicking off from the physics he knew and loved so well. Feynman’s approach was new and unfamiliar, and nobody understood it. When he talked of electrons going forwards and backwards in time, they were baffled. There was no communication. In the end, he gave up. He knew he was right, that his theory was as good as Schwinger’s, but somehow he couldn’t get the message across. He decided to go back to Cornell and write it all up for publication, so that they could study it in cold print.21
But the Pocono Conference was far from being a disaster for Feynman. In the intervals between formal lectures, over lunch and coffee and whenever they could get together, he and Schwinger compared notes. Neither of them really understood what the other was doing, but they trusted and respected each other. For every problem that they had both tackled, it turned out that they had got the same answer:
We came at things entirely differently, but we came to the same end. So there was no problem with my believing that I was right and everything was OK.22
To Feynman and Schwinger, being told the same thing twice by the equations meant it must be true. In Lewis Carroll’s The Hunting of the Snark, ‘what I tell you three times is true’. The third telling of the story of QED was about to happen in spectacular fashion.
Oppenheimer was by now Director of the Institute for Advanced Study, and when he got back to Princeton after the Pocono Conference he found a letter and a package of scientific papers waiting for him. They came from a Japanese physicist, Sin‑Itiro Tomonaga, who had worked out essentially the same version of QED as Schwinger, largely cut off from contact with Western scientists, in the harsh conditions of battered wartime and postwar Tokyo. This incredible achievement has been described in detail by Silvan Schweber, in QED and the Men Who Made It. Tomonaga had not only come up with a slightly simpler version of QED than Schwinger (proof, if any were needed, of Schwinger’s love of sometimes unnecessary complications) but he had actually been the first of the three physicists to complete his theory.
The physics community had been told three times that QED was true, and it was. But how did Feynman’s version of QED come to be recognized, before long, as the simplest approach, a break with tradition that, instead of being the last flowering of an old glory, provided a seed from which great new ideas would grow? Feynman did indeed start to publish his work in a series of clear and impressive papers after the Pocono Conference. But the key to getting his message across to a wider audience owed much to the presence in Princeton of another mathematical prodigy, the Englishman Freeman Dyson. Where Schwinger had demonstrated his talent by finishing his PhD work before completing his BSc, Dyson would demonstrate his in equally impressive fashion by (eventually) becoming a member of the Institute for Advanced Study without finishing a PhD at all.
Dyson had been born in 1923, and after graduating from Cambridge worked for the British wartime Bomber Command on statistical studies of the effectiveness of the bombing campaign over Germany. This was a doubly futile exercise – it was a waste of Dyson’s mathematical talent, and he soon discovered (although he was never able to convince his superiors) that the bombing effort was largely misdirected and a waste of the lives of inexperienced aircrew sent on impossible missions. In September 1947, he enrolled as a graduate student in the physics department at Cornell, working under Bethe and in an ideal position to observe the dramatic development of QED over the next few months. He has often told the story, most notably in his book Disturbing the Universe,23 from which the following account is largely drawn.
The first task Bethe gave Dyson was to redo Bethe’s calculation of the Lamb shift for a spin-zero electron (a fictitious simplification), with the requirements of the Special Theory of Relativity (corresponding to taking note of the spin) bolted on in an ad hoc fashion. This provided no new insight into the quantum world, but after hundreds of pages of calculations Dyson ended up with what he describes as a ‘pastiche’, no real improvement on Bethe’s calculation, which more or less gave the right answer. A good analogy with Bethe’s and Dyson’s efforts to explain the Lamb shift would be the Bohr model of the atom, a patchwork of ideas put together on an ad hoc basis which worked after a fashion, but gave no deep insight into what was going on. The hours Dyson spent in this calculation were, though, a valuable familiarization with the cutting edge of what was going on in quantum physics at the time. Dyson was too junior a researcher to be present at the Pocono Conference, but he was well aware of Feynman as ‘the liveliest person in our department’, who ‘refused to take anybody’s word for anything’ and had set out ‘to reinvent quantum mechanics’.
Dyson soon realized that Feynman, with his new quantum mechanics, could solve every problem that Bethe could solve using the older version, getting the same answers. But Feynman could also solve a lot of problems that the old quantum mechanics couldn’t handle. ‘It was obvious to me that Dick’s theory must be fundamentally right. I decided that my main job, after I finished the calculation for Hans, must be to understand Dick and explain his ideas in a language that the rest of the world could understand.’
It seemed that Dyson might not get the opportunity to do this, because after a year at Cornell he was scheduled to spend a year doing research at the Institute for Advanced Study, working with Oppenheimer. This left him only a few months to try to get to grips with Feynman’s work. He made an effort to see Feynman as much as possible, and happily accepted the way in which Feynman dealt with visitors. If he didn’t want to be disturbed, he would just shout, ‘Go away; I’m busy.’ But if he let you into his office, it meant that he really did have time to talk. They talked for hours about F
eynman’s theory, until Dyson began to feel that he was beginning (only beginning) to get the hang of it – but his time in Cornell was nearly up.
The reason why ordinary physicists had trouble getting hold of Feynman’s ideas, Dyson realized, was that Feynman thought in pictorial terms. He had a physical picture of how the world worked, a picture which gave him an insight into the solution of complicated problems without having to write down a lot of equations. In an interview with Silvan Schweber,24 Feynman said:
Visualization in some form or other is a vital part of my thinking… half-assed kind of vague, mixed with symbols. It is very difficult to explain, because it is not clear. My atom, for example, when I think of an electron spin in an atom, I see an atom and I see a vector and a ψ written somewhere, sort of, or mixed with it somehow, and an amplitude all mixed up with xs … it is very visual … a mixture of a mathematical expression wrapped into and around, in a vague way, around the object. So I see all the time visual things associated with what I am trying to do.
In What Do You Care, Feynman tried once again to explain how he thought about physics:
When I see equations, I see the letters in colors – I don’t know why. As I’m talking, 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.
Another great physicist who also thought in visual terms was Albert Einstein, although his pictures – a person riding on a beam of light, or falling in an elevator with a broken cable – seem to have been more clear-cut and down to earth than Feynman’s.
