Quantum Man: Richard Feynman's Life in Science (11 page)

The notion that somehow spin might produce additional phases when one is calculating probability amplitudes was prescient. However, when Feynman tried to move beyond one spatial dimension and associate more complicated phase factors as electrons turned corners and went off at different angles, he got nonsensical answers and couldn’t get results that corresponded to Dirac’s theory.

Feynman kept trying different, diffuse alternatives to reformulate the theory, but he made little progress. However, there is one area where his sum-over-paths methodology was particularly useful. Special relativity tells us that one person’s “now” may not be another person’s “now”—namely, observers in relative motion have different notions of simultaneity. Special relativity explains how this local notion of simultaneity is myopic, and how the underlying physical laws are independent of different observers’ individual preferences for “now.”

The problem with the conventional picture of quantum mechanics was that it depended explicitly on defining a “now,” in which an initial quantum configuration was established, and then determining how this configuration would evolve to a later time. In the process, the relativistic invariance of physical laws gets buried, because the minute we choose a particular spatial frame to define the initial wave function and an instant of time to call
t
=
0, we lose explicit contact with the underlying relativistic, frame-independent beauty of the theory.

Feynman’s space-time picture, however, was precisely tuned to make the relativistic invariance of the theory manifest. In the first place, it was defined in terms of quantities—Lagrangians—which can be written in an explicit relativistically invariant form. And secondly, since the sum-over-paths approach inevitably deals with all of space and time together, we do not have to restrict ourselves to defining specific instants in time or space. Thus, Feynman had trained himself to combine quantities in QED—that other-wise might be considered separately—together into combinations that behaved in a way in which the properties of relativity remained manifest. While he had made no real headway in explicitly reformulating the Dirac theory from first principles in any way that resolved the issues he was concerned with, the tricks he had developed would prove of crucial importance later in the ultimate solution.

T
HE SOLUTION CAME
into view, as it usually does, with an experiment. Indeed, while theorists normally take their guidance from experimental results, it is hard to overstress how literally important they were to driving progress in this case. Up to this point, the infinities were frustrating to theorists, but that is about all. As long as the zero-order predictions of Dirac’s equation were sufficient to explain, within the achievable experimental accuracy, all the results of atomic physics, theorists could worry about the fact that higher-order corrections, which should have been small, were in fact infinite, but the infinities were not yet a real practical impediment to using the theory in a physical context.

Theorists love to speculate, but I have found that until experimentalists actually produce concrete results that probe a theory at a new level, it is hard for theorists to take even their own ideas seriously enough to rigorously explore all of their ramifications, or to come up with practical solutions to existing problems. The preeminent U.S. experimental physicist at the time, I. I. Rabi, who had made Columbia University the experimental capital of the world for atomic physicists, mused on this inability of theorists to rise to the challenges of QED in the absence of experimental guidance. In the spring of 1947 he is reported to have said to a colleague over lunch, “The last eighteen years have been the most sterile of the century.”

All that changed within a few months. As I have already described, until that time the lowest-order calculations performed with Dirac’s relativistic theory produced results for the spectrum of energy levels of electrons bound to protons in hydrogen atoms which were sufficient not only to understand the general features of the spectrum, but also to strike quantitative agreement with observation, aside from a few possible inconsistencies that emerged at the very limit of experimental sensitivities and were thus largely ignored. That was, however, until a courageous attempt by the American physicist Willis Lamb—who worked in Rabi’s Columbia group, and who was one of the last of a breed of physicists who were equally adept in the laboratory and performing calculations—changed everything.

Recall that the initial great success of quantum mechanics in the early decades of the twentieth century lay in explaining the spectrum of light emitted by hydrogen. Neils Bohr was the first to propose a rather ad hoc quantum mechanical explanation for the energy levels in hydrogen: that electrons were able to jump between only fixed levels as they absorbed or emitted radiation. Later, Schrödinger, with his famous wave equation, showed that the electron energy levels of hydrogen could be derived precisely from using his “wave mechanics,” instead of by fiat as in the case of the Bohr atom.

Once Dirac derived his relativistic version of QED, physicists could attempt to replace the Schrödinger equation with the Dirac equation in order to predict energy levels. They did this and discovered that the energy levels of different states were “split” by small amounts, owing to relativistic effects (for example, more energetic electrons in atoms would be more massive, according to relativity) and to the nonzero spin of electrons incorporated into Dirac’s equation. Lo and behold, the predictions of Dirac agreed with observations of the more finely resolved spectra from hydrogen, where what were otherwise seen as single frequencies of emission and absorption were now shown to be split into two different, very finely separated frequencies of light. This
fine structure
of the spectrum, as it became known, was yet another vindication of the Dirac theory.

In 1946 Willis Lamb decided to measure the fine structure of hydrogen more accurately than it had ever been measured before, in order to test the Dirac theory. His proposal for this experiment explained his motivation: “The hydrogen atom is the simplest one in existence, and the only one for which essential exact theoretical calculations can be made. . . . Nevertheless, the experimental situation at present is such that the observed spectrum of the hydrogen atom does not provide a very critical test. . . . A critical test would be obtained from a measurement of . . . fine structure.”

On April 26, 1947 (back in the days when the time between proposing an experiment and completing it in particle physics was on the order of months, not decades), Lamb and his student Robert Retherford successfully completed a remarkable measurement that had previously been unthinkable. The result was equally astounding.

The lowest-order Dirac theory, like the Schrödinger theory before it, predicted that the same energy would be ascribed to two different states with the same total angular momentum of the electron in hydrogen—arising from the sum of the spin angular momentum and the orbital angular momentum—even if the separate pieces of the sum were different in the two states. However, Lamb’s experiment conclusively proved that the energy of electrons in one state differed from that of electrons in the other. Specifically he observed that the transitions of electrons between one state and a fixed higher state in hydrogen resulted in the emission or absorption of light whose frequency differed by about a billion cycles per second compared to transitions of electrons between the other state and the fixed higher state. This may seem like a lot, but the characteristic frequencies of light emitted and absorbed between energy levels in hydrogen were about ten million times bigger than even this frequency difference. Lamb was therefore required to measure frequencies with an accuracy of better than one part in ten million.

The state of theoretical physics following Dirac’s coup was such that the impact of this almost imperceptibly small, yet clearly nonzero, difference with the predictions of Dirac’s theory was profound. Suddenly, the problem with Dirac’s theory was concrete. It did not revolve around some obscure and ill-defined set of infinite results, but it now came down to real and finite experimental data that could be computed. Feynman later described the impact in his typically colorful fashion: “Thinking I understand geometry, and wanting to fit the diagonal of five-foot square I try to figure out how long it must be. Not being very expert I get infinity—useless. . . . It is not philosophy we are after, but the behavior of real things. So in despair, I measure it directly—lo, it is near to seven feet—neither infinity, nor zero. So, we have measured these things for which our theory gives such absurd answers.”

In June of 1947 the National Academy of Sciences convened a small conference of the greatest theoretical minds working on the quantum theory of electrodynamics (fortunately Feynman’s former supervisor, John Wheeler, was one of the organizers so Feynman was invited) in a small inn on Shelter Island, off Long Island, New York. The purpose of the “Conference on the Foundations of Quantum Theory” was to explore the outstanding problems in quantum theory that had been set aside during the war, when Feynman and his colleagues were laboring on producing the atomic bomb. In addition to Feynman, all the leading lights from Los Alamos were there, from Bethe to Oppenheimer, and the young theoretical superstar Julian Schwinger.

It was at this small meeting, which began in suitably dramatic fashion, with the police escorting the famous war-hero “atomic” scientists through Long Island, that Lamb presented the results of his experiment. This was the highlight of the meeting, which Feynman later referred to as the most important conference he had ever attended.

As far as Feynman’s work was concerned, however, and probably for all theorists thinking about the problems of QED, the most important outcome of the conference was not a calculation that Feynman performed, but rather a calculation that his mentor, Hans Bethe, performed on the train trip back to Ithaca from New York City, where Bethe had stayed for a few days to visit his mother. Bethe was so excited by the result he had obtained that he phoned Feynman from Schenectady to tell him the result: In his typical fashion, when finally presented with an experimental number, Bethe found it irresistible to use whatever theoretical machinery was at his disposal, no matter how limited, to derive a quantitative prediction to be compared with the experimental result. To his immense surprise and satisfaction, even without a full understanding of how to deal with the strange infinities of QED, Bethe claimed to understand the magnitude and origin of the frequency shift that had already become known as the
Lamb shift
.

For Feynman, Schwinger, and the rest of the community, the gauntlet had been laid.

CHAPTER
9

Splitting an Atom

A very great deal more truth can become known than can be proven.

—R
ICHARD
F
EYNMAN
,
N
OBEL LECTURE
, 1965

W
hen Willis Lamb presented his result to begin the Shelter Island conference, the question immediately arose as to what could have caused the discrepancy between observations and Dirac’s QED theory. Oppenheimer, who dominated the meeting, suggested that perhaps the source of the frequency shift might be QED itself, if anyone could actually figure out how to tame the unphysically infinite higher-order corrections in the theory. Bethe’s effort to do just that built on ideas from Oppenheimer and the physicists H. A. Kramers and Victor Weisskopf, who later would take a leave from MIT to become the first director of the European Laboratory for Nuclear Research, called CERN, in Geneva.

Kramers emphasized that since the problem of infinite contributions in electromagnetism went all the way back to the classical self-energy of an electron, physicists should focus on observable quantities, which were of course finite, when expressing the results of calculations. For example, the electron mass term that appeared in the equations, and in turn received infinite self-energy corrections, should not be considered to represent the measured physical mass of the particle. Instead, call this the
bare mass
. If the bare mass term in the equation was infinite, perhaps the sum of this term and the infinite self-energy correction could be made to cancel, leaving a finite residue that could be equal to the experimentally measured mass.

Kramers proposed that all of the infinite quantities that one calculated in electrodynamics, at least for electrons moving nonrelativistically, could be expressed in terms of the infinite self-energy contribution to the electron rest mass. In this case as long as one removed this single infinite quantity by expressing all results in terms of the finite measured mass, then all calculations might yield finite answers. In doing so, one would change the magnitude, or the
normalization
, of the mass term appearing in the fundamental equations, and this process became known as
renormalization
.

Weisskopf and Schwinger explicitly considered the relativistic quantum theory of electrodynamics in an effort to implement this idea. In particular, they demonstrated that the infinity one calculated in the self-energy of the electron actually became somewhat less severe when one incorporated relativistic effects.

Motivated by these arguments, Bethe performed an approximate calculation of such a finite contribution. As Feynman later put it in his Nobel address, “Prof. Bethe . . . is a man who has this characteristic: If there’s a good experimental number you’ve got to figure it out from theory. So, he forced the quantum electrodynamics of the day to give him an answer to the separation of these two levels [in hydrogen].” Bethe’s reasoning was undoubtedly something like this: if the effects of electrons and holes and relativity seemed to tame infinities somewhat, then perhaps one could do a calculation with the nonrelativistic theory, which was much easier to handle, and then simply ignore the contribution of all virtual photons higher than an energy equal to approximately the measured rest mass of the electron. Only when total energies exceed this rest mass do relativistic effects kick in, and perhaps when they do, they ensure that the contribution from virtual particles of higher energy become irrelevant.

When Bethe performed the calculation with this arbitrary cutoff in the energy of virtual particles, the predicted frequency shift between light emitted or absorbed by the two different orbital states in hydrogen was about 1,040 megacycles per second, which was in very good agreement with the observation of Lamb.

Feynman, as brilliant as he was, later remembered not fully appreciating Bethe’s result at the time. It was only later, at Cornell, when Bethe gave a lecture on the subject suggesting that if one had a fully relativistic way of handling the higher-order contributions in the theory one might be able to not only get a more accurate answer but also demonstrate the consistency of the ad hoc procedure he had employed, that Feynman understood both the significance of Bethe’s result and how all the work he had done up to that point could allow him to improve on Bethe’s estimate.

Feynman went up to Bethe after his lecture and told him, “I can do that for you. I’ll bring it in for you tomorrow.” His confidence was based on his years of labor reformulating quantum mechanics using his action principle and the sum over paths, which provided him with a relativistic starting point that he could use in his calculations. The formalism he had developed in fact allowed him to adjust the possible paths of particles in a way that would constrain the otherwise infinite terms in the quantum calculation by effectively limiting the maximum energy of the virtual particles that enter into the calculation, but did so in a way that was consistent also with relativity, just as Bethe had requested.

The only problem was that Feynman had never actually worked through a calculation of the self-energy of the electron in the quantum theory, so he went to Bethe’s office, where Bethe could explain to him how to do the calculation, and Feynman in turn could explain to Bethe how to use his formalism. In one of those serendipitous accidents that affect the future of physics, when Feynman went to visit Bethe and they worked their calculations out at the blackboard, they made a mistake. As a result, the answer they got at the time was not only not finite, but the infinities were actually worse than had appeared in the nonrelativistic calculation, making it harder to isolate what were the finite pieces.

Feynman went back to his room, certain that the correct calculation should be finite. Ultimately, in a typical Feynmanesque way, he decided he had to teach himself in excruciating detail how to do the self-energy calculation in the traditional complicated way using holes, negative-energy states, and so on. Once he knew in detail how to do the calculation the traditional way, he was confident he would be able to repeat it using his new path-integral for-malism, doing the modifications necessary to make the result finite but in a way where relativity remained manifestly obeyed.

When the dust had settled, the result was just what he had hoped for. Expressing everything in terms of the experimentally measured rest mass of the electron, Feynman was able to get finite results, including a highly accurate result for the Lamb shift.

As it turned out, others had also been able to do a relativistic calculation at around the same time, including Weisskopf and his student Anthony French, as well as Schwinger. Schwinger, moreover, was able to show that taming precisely the same infinities that resulted in a finite and calculable Lamb shift also allowed a calculation of another experimental deviation from the predictions of the uncorrected Dirac theory discovered by Rabi’s group at Columbia. This effect had to do with the measured magnetic moment of the electron.

Since the electron acts like it is spinning, and since it is charged, electromagnetism tells us that it should also behave like a tiny magnet. The strength of its magnetic field should therefore be related to the magnitude of the electron’s spin. But measurement of the strength revealed that it deviated from the simple lowest-order prediction by about 1 percent. This is a small amount, but nevertheless the accuracy of the measurement was such that the difference from the prediction was real and significant. One therefore needed to understand the theory to a higher order to know if it agreed with experimental data.

Schwinger showed that the same type of calculation, isolating the otherwise infinite pieces and modifying them in a well-defined way, and then expressing all of the calculated results in terms of quantities like the measured rest mass of the electron, produced a predicted shift in the magnetic moment of the electron that conformed with the experimental result.

Rabi wrote Bethe an elated note on hearing about Schwinger’s calculation, and Bethe replied, referring back to Rabi’s experiments, “It is certainly wonderful how these experiments of yours have given a completely new slant to a theory and the theory has blossomed out in a relatively short time. It is as exciting as in the early days of quantum mechanics.”

QED had at last begun to emerge from a long murky initiation. In the years since that time, the predictions of the theory have agreed with the results of experiments to an accuracy that is unparalleled anywhere else in all of science. There is simply no better scientific theory in nature from this point of view.

I
F FEYNMAN HAD
been only one of a crowd that had correctly shown how to calculate the Lamb shift we probably would not be memorializing his contributions today. But the real value of his efforts at calculating the Lamb shift, and his understanding of how to tame the infinities involved in the calculation, was that he began to calculate more and more things. And in the process, he used his formidable mathematical skills, along with the intuition he had developed in the process of reformulating quantum mechanics, to gradually develop a whole new way of picturing the phenomena involved in QED. And he did so in a way that produced a remarkable new way of calculating with the theory, based on diagrammatic space-time pictures, which were themselves founded on a sum-over-paths approach.

F
EYNMAN’S APPROACH TO
resolving the problems of QED was both highly original and highly scattershot. He often simply guessed what the likely formulas should be and then compared his guesses in different contexts with available known results. Furthermore, while his space-time approach allowed him to write down mathematics in a manner that was in accord with relativity, the actual calculations he performed were not derived directly from any systematic mathematical framework in which relativity and quantum mechanics were unified—even if after the fact everything worked out correctly.

A systematic framework for combining relativity and quantum mechanics had in fact existed since at least the 1930s. It was called quantum field theory, and it was intrinsically a theory of infinitely many particles, which is why Feynman probably steered away from it. In classical electromagnetism, the electromagnetic field is a quantity that is described at every point in space and time. When treating the field quantum mechanically, one finds that it can be thought of in terms of elementary particles—in this case, photons. In quantum field theory, the field can be thought of as a quantum object, with a certain probability of creating (or destroying) a photon at each point in space. This allows the existence of a possibly infinite number of virtual photons to be temporarily produced by fluctuations in the electromagnetic field. It was precisely this complication that motivated Feynman to originally reformulate electrodynamics in a way in which these photons disappeared completely and in which there were direct interactions of charged particles. These direct interactions he then handled quantum mechanically using his sum-over-paths approach.

In the midst of his tinkering with how to incorporate Dirac’s relativistic theory of electrons into his calculational framework for QED, Feynman stumbled upon a beautiful mathematical trick that simplified tremendously the calculations and did away with the need to think of particles and “holes” as separate entities. But at the same time, this trick makes manifest the fact that the moment relativity is incorporated into quantum mechanics, one can no longer live in a world where the number of possible particles is finite. Relativity and quantum mechanics simply
require
a theory that can handle a possibly infinite number of virtual particles existing at any instant.

The trick Feynman used hearkened back to the old idea that John Wheeler proposed to him one day when he argued that all electrons in the world could be thought of as arising from a single electron, as long as that electron was allowed to go backward as well as forward in time. An electron going backward in time would appear just like a positron going forward in time. In this way, a single electron going forward and backward in time (and masquerading as a positron when it was doing the latter) could reproduce itself a huge number of times at any instant. Naturally, Feynman pointed out the logical flaw in this picture when he argued that if it were true, there would be as many positrons as electrons around at any instant and there aren’t.

Nevertheless, the idea that a positron could be thought of as an electron traveling backward in time was an idea— Feynman suddenly realized all those years later—that he could exploit in a different context. When trying to do relativistic calculations, where both electrons and holes normally had to be incorporated, he recognized that he could get the same results by including just electrons in his space-time picture, but allowing processes where the electrons went both forward and backward in time. (The idea that my high school physics teacher mangled a bit when he tried to get me more interested in physics that summer afternoon long ago.)

To understand how Feynman’s unified treatment of positrons and electrons arose, it is easiest to begin to think in terms of the diagrams that Feynman eventually began to draw for himself to depict the space-time processes that arose in his sum-over-paths view of quantum mechanics.

Consider a diagram describing the space-time process of two electrons exchanging a virtual photon, emitted at
A
and absorbed at
B
: