Quantum Man: Richard Feynman's Life in Science Page 12

  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.

  IF 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.

  FEYNMAN’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 fie
ld 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:

  In order to calculate the quantum mechanical amplitude for such a process, we would have to consider all possible space-time paths corresponding to the exchange of a virtual photon between the two particles. Since we don’t observe the photon, the following process, in which the photon is emitted at B and absorbed at A, when B is earlier than A, also contributes to the final sum:

  Now, there is another way of thinking about the two separate diagrams. Remember that we are dealing with quantum mechanics, and therefore in the time between measurements, anything that is consistent with the Heisenberg uncertainty principle is allowed. Thus, for example, the virtual photon is not restricted to travel at exactly the speed of light for the entire time it travels between the two particles. But special relativity says that if it is traveling faster than light, then in some frame of reference it would appear to be going backward in time. If it is going backward in time, then it can be emitted at A and absorbed at B. In other words, the second diagram corresponds to a process identical to the first, except that in the latter case the virtual photon is traveling faster than light.

  In fact, while Feynman never explicitly described it at that time, as far as I know, this same effect explains why a relativistic theory of electrons—that is, Dirac’s theory—requires antiparticles. A photon, which doesn’t have any electric charge, traveling backward in time from A to B, just looks like a photon traveling forward in time from B to A. But a charged particle traveling backward in time looks like a particle of opposite charge traveling forward in time.

  Thus, the simple process where an electron (e–) travels between two points, pictured in this space-time diagram:

  must also be accompanied by this process:

  But the latter process could also be described with an intermediate positron (e+), as follows:

  In other words, it appears that a single electron begins its journey, and at another point an electron-positron pair is created from empty space, and a virtual positron travels forward in time, ultimately annihilating with the first electron, leaving only the single final electron at the end of the journey.

  Feynman later beautifully described this situation in his 1949 paper “The Theory of Positrons.” His famous analogy was a bombardier looking down at a road from his scope on an airplane (the recent world war had undoubtedly influenced Feynman’s choice of analogies here): “It is as though a bombardier watching a single road through the bomb-sight of a low flying plane suddenly sees three roads and it is only when two of them come together and disappear again that he realizes that he has simply passed over a long switchback in a single road.”

  Thus particle number in a relativistic quantum theory has to be indeterminate. Just when we think we have one particle, a particle-antiparticle pair can pop out of the vacuum, making it three. After the antiparticle annihilates one of the particles (either its partner or the original particle), once again there is only one, just as what the bombardier would see through his bomb-sight if he were counting roads. The key point again is not merely that this is possible but that it is required by relativity, so that with hindsight we see that Dirac had no other choice but to introduce antiparticles in his relativistic theory of electrons and light.

  That Feynman was the one to point out the possibility of treating positrons as time-reversed electrons in his diagrams is fascinating because it immediately implies that his earlier aversion to quantum field theory was misplaced. His diagrammatic space-time expansion for calculating physical effects in QED implicitly contained within it the physical content of a theory where particles could be created and destroyed and particle number during intermediate steps of a physical process was therefore indeterminate. Feynman had been forced by physics to reproduce the physical content of quantum field theory. (In fact, in an obscure 1941 paper that predated Feynman’s by eight years, the German physicist Ernst Stückelberg had independently been driven to consider space-time diagrams, and positrons as time-reversed electrons, although he was not sufficiently driven to carry through the program that Feynman ultimately carried out with these tools.)

  Now that we are familiar with the diagrams that would eventually become known as Feynman diagrams, we can depict the events that correspond to the otherwise infinite processes associated with the electron self-energy and vacuum polarization:

  Self-energy (an electron interacting with its own electromagnetic field)

  Vacuum polarization (splitting of a virtual photon into electron-positron pair)

  For Feynman there was a fundamental difference between these two diagrams, however. The first diagram he could imagine occurring naturally as an electron emitted a photon and later reabsorbed it. But the second diagram seemed unnatural because it would not result from the trajectory of a single electron moving and interacting backward and forward in space and time, and he felt that such trajectories were the only appropriate ones to incorporate in his calculations. As a result, he was wary of the need to include these new processes, and did not originally do so. This decision caused Feynman a number of problems as he tried to derive a framework in which all of
the infinities of Dirac’s theory could be obviated, and in which predictions for physical processes could be unambiguously derived.

  The first great success of Feynman’s methods involved calculating the self-energy of the electron. Most important, he found a way to alter the interaction of electrons and photons at very small scales and very high energies in a manner that was consistent with the requirements of relativity. Pictorially this results from considering the case where the loop in the self-energy diagram becomes very small, and then altering the interactions for all loops that are small and smaller. In this way a provisional result could be derived, which is finite. Moreover, this result could be shown to be independent of the form of the alteration of the interactions for small loops in the limit that the loops become smaller and smaller. Most important, as I stressed earlier, because the loops take into account an arbitrary time of emission and absorption and at the same time include objects going forward and backward in time, the form of his alteration did not spoil the relativistic behavior of the theory, which should not depend on any one observer’s definition of time.

  As Kramers and others had predicted, the key was making these altered-loop contributions finite, or regularizing them, as it was later called, in a way that was consistent with relativity. Then if one expressed the corrections to physical quantities, such as the energy of an electron in the field of a hydrogen atom, in terms of the physical mass and physical charge of the electron, the remainder, after canceling out the term that would otherwise become infinite without the alteration for small loops, was both finite and independent of the explicit form of the alteration one made. More important, it remained finite even if the size scale at which one alters the loop diagrams is decreased to zero, where the loop diagram would otherwise become infinite. Renormalization worked. The finite correction agreed reasonably well with the measured Lamb shift, and electrodynamics as a quantum theory was vindicated.