Quantum Man: Richard Feynman's Life in Science

by Lawrence M. Krauss

  But remember that with virtual particles all bets are off, and energy and momentum need not be conserved as long as the virtual particle disappears in a time sufficiently short so that it cannot be measured directly. Thus, a virtual photon can spontaneously transform into an electron-positron pair, as long as the electron-positron pair then annihilates and transforms back into the single virtual photon on a short timescale.

  The process involving a photon momentarily splitting up into an electron-positron pair is called vacuum polarization. It gets this name because in a real medium such as any solid object made of atoms, which contains both positive and negative charges, if we turn on a large external electric field, we can “polarize” the medium by separating charges of different types—negative charges will be pushed in one direction by the field, while positive charges will be pulled in the other direction. Thus a neutral material will remain neutral, but the charges of different signs will spatially separate. This is what momentarily happens in empty space as a photon splits temporarily into a negatively charged electron and its positively charged antiparticle, a positron. Thus, empty space gets briefly polarized.

  Whatever we call it, an electron, which previously had to be thought of as having a cloud of virtual photons around it, now had to be thought of as being surrounded by a cloud of virtual photons plus electron-positron pairs. In some sense, this picture is just another way of thinking about Dirac’s interpretation of positrons as “holes” in an infinite sea of electrons in the vacuum. Either way, once we include relativity, and the existence of positrons, the theory of a single electron turns into a theory of an infinite number of electrons and positrons.

  Moreover, just as emission and absorption of virtual photons by a single electron produced an infinite electron self-energy in calculations, the production of virtual particle-antiparticle pairs produced a new infinite correction in QED calculations. Recall once again that the electric force between particles can be thought of as being due to the interchange of virtual photons between those particles. If the photons could now split up into electron-positron pairs, this process would change the strength of the interaction between particles and would thus shift the calculated energy of interaction between an electron and a proton in an atom like hydrogen. The problem was, the calculated shift was infinite.

  The frustrating fact was that the Dirac theory produced very accurate predictions for the energy levels of electrons in atoms as long as only the exchange of single photons was considered and the annoying higher-order effects that produced infinities were not. Morever Dirac’s prediction of positrons had been vindicated by experimental data. Were it not for these facts, many physicists, as Dirac implied, would have preferred to simply dispense with QED altogether.

  IT TURNED OUT that what was needed to resolve all of these difficulties was neither a wholesale disposal of quantum mechanics, nor dispensing with all these virtual particles, but rather developing a deeper understanding of how to implement the basic principles of quantum theory in the context of relativity. It would take a long circuitous route, and the guidance brought by key experiments, before this fact, hidden in a mire of crushingly complex calculations, would become clear, both to Feynman and to the rest of the world.

  The process of discovery began slowly and confusedly, as it usually does. After his Reviews of Modern Physics paper was completed, Feynman turned his attention once again to Dirac’s theory. He had decided physics was fun again, and in spite of his otherwise unsettled personal situation, his focus never strayed far from the problem that had obsessed him since he had been an undergraduate—the infinite self-energy of an electron. It was a puzzle he hadn’t yet solved, and it was contrary to his nature to let it go.

  He began with a warm-up problem. Since the spin of the electron makes sense only quantum mechanically, Feynman began by trying to understand whether he could account for spin directly within his sum-over-paths formalism. One complication of Dirac’s theory is that a single equation had four separate pieces: one to describe spin-up electrons, one to describe spin-down electrons, one to describe spin-up positrons, and one to describe spin-down positrons. Since the normal conception of spin requires three dimensions (a two-dimensional plane to spin in and a perpendicular axis to spin around), Feynman reasoned that he could make the problem simpler if he first tried to consider a world with just one spatial dimension and one time dimension, where the different sorts of allowed paths were also trivial. Paths would just involve travel back and forth in the one space dimension, namely, a line.

  He was able to derive the simplified version of the Dirac equation appropriate for such a two-dimensional world if every time an electron “turned around” from rightward motion to leftward motion, the probability amplitude for that path got multiplied by a “phase factor,” which in this case was a “complex number,” an exotic number that involved the square root of –1. Complex numbers can appear in probability amplitudes, and actual probabilities depend on the square of these numbers, so that only real numbers appear in the final result.

  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.

  THE 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.

  —RICHARD FEYNMAN,

  NOBEL LECTURE, 1965

  When 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.