Fear of Physics (5 page)

After getting over the shock of such a possibility, you may think it sounds suspiciously close to trying to count the number of angels on the head of a pin. But there is one important difference. The electron-positron pair does not vanish without a trace. Much like the famous Cheshire cat, it leaves behind a calling card. The presence of the pair can subtly alter the properties you attribute to the electron when you assume it is the only particle traveling during the whole affair.

By the 1930s it was recognized that such phenomena, including the very existence of antiparticles such as the positron, must occur as a consequence of the merging of quantum mechanics and relativity. The problem of how to incorporate such new possibilities into the calculation of physical quantities remained unsolved, however. The problem was that if you consider smaller and smaller scales, this kind of thing can keep recurring. For example, if you consider the motion of the original electron over an even shorter time, during which its energy is uncertain by even a larger
amount, it becomes momentarily possible for not one electron-positron pair but two to hang around, and so on as one considers ever shorter intervals. In trying to keep track of all these objects, any estimate yielded an infinite result for physical quantities such as the measured electric charge of an electron. This, of course, is most unsatisfactory.

It was against this background that, in April 1947, a meeting was convened at an inn on Shelter Island, an isolated community off the eastern tip of Long Island, New York. Attending was an active group of theoretical and experimental physicists working on fundamental problems on the structure of matter. These included grand old men as well as Young Turks, most of whom had spent the war years working on the development of the atomic bomb. For many of these individuals, returning to pure research after years of directed war effort was not easy. Partly for this reason, the Shelter Island conference was convened to help identify the most important problems facing physics.

Things began auspiciously. The bus containing most of the participants was met by police troopers on motorcycles as they entered Nassau County on western Long Island, and, to their surprise, they were escorted with wailing sirens across two counties to their destination. Later they found out that the police escort had been provided as thanks by individuals who had served in the Pacific during the war and had felt that their lives had been saved by these scientists who had developed the atomic bomb.

The preconference excitement was matched by sensational developments on the opening day of the meeting. Willis Lamb, an experimental atomic physicist, using microwave technology developed in association with war work on radar at Columbia University, presented an important result. Quantum mechanics,
as one of its principal early successes, had allowed the calculation of the characteristic energies of electrons orbiting around the outside of atoms. However, Lamb’s result implied that the energy levels of electrons in atoms were shifted slightly from those calculated in the quantum theory, which had been developed up to that time without explicitly incorporating relativity. This finding came to be known as the “Lamb shift.” It was followed up by a report by the eminent experimental physicist I.I. Rabi on work of his, and also of P. Kusch, showing similar deviations in other observables in hydrogen and other atoms compared to the predictions of quantum mechanics. All three of these U.S. experimentalists—Lamb, Rabi, and Kusch—would later earn the Nobel Prize for their work.

The challenge was out. How could one explain such a shift, and how could one perform calculations that could somehow accommodate the momentary existence of a possibly infinite number of “virtual” electron-positron pairs, as they became called? The thought that the merging of relativity and quantum mechanics that caused the problem would also lead to an explanation was at that time merely a suspicion. The law of relativity complicated the calculations so much that up to that time no one had found a consistent way to perform them. The young rising stars of theoretical physics, Richard Feynman and Julian Schwinger, were both at the meeting. Each was independently developing, as was the Japanese physicist Sin-itiro Tomonaga, a calculational scheme for dealing with “quantum field theory,” as the union of quantum mechanics and relativity became known. They hoped, and their hopes later proved correct, that these schemes would allow one safely to isolate, if not remove entirely, the effects of the virtual electron-positron pairs that appeared to plague the theory, while
at the same time giving results that were consistent with relativity. By the time they were finished, they had established a new way of picturing elementary processes and demonstrated that the theory of electromagnetism could be consistently combined with quantum mechanics and relativity to form the most successful theoretical framework in all of physics—an accomplishment for which these three men deservedly shared the Nobel Prize almost 20 years later. But at the time of the meeting, no such scheme existed. How could one handle the interactions of electrons in atoms with the myriad of “virtual” electron-positron pairs that might spontaneously burp out of the vacuum in response to the fields and forces created by the electrons themselves?

Also attending the meeting was Hans Bethe, already an eminent theorist and one of the leaders in the atomic bomb project. Bethe would also go on to win the Nobel Prize for work demonstrating that nuclear reactions are indeed the power source of stars such as the sun. At the conference he was inspired by what he heard from both the experimentalists and the theorists to return to Cornell University to try to calculate the effect observed by Lamb. Five days after the meeting ended, he had produced a paper with a calculated result, which he claimed was in excellent agreement with the observed value for the Lamb shift. Bethe had always been known for his ability to perform complex calculations in longhand at the board or on paper flawlessly. Yet his remarkable calculation of the Lamb shift was not in any sense a completely self-consistent estimate based on sound fundamental principles of quantum mechanics and relativity. Instead, Bethe was interested in finding out whether current ideas were on the right track. Since the complete set of tools for dealing with the quantum theory including relativity were not yet at hand, he used tools that were.

He reasoned that if one could not handle the relativistic motion of the electron consistently, one could perform a “hybrid” calculation that incorporated explicitly the new physical
phenomena
made possible by relativity—such as the virtual electron-positron pairs—while still using equations for the motion of electrons based on the standard quantum mechanics of the 1920s and 1930s, in which the mathematical complexity of relativity was not explicitly incorporated. However, he found that the effects of virtual electron-positron pairs were then still unmanageable. How did he deal with that? Based on a suggestion he heard at the meeting, he did the calculation twice, once for the motion of an electron inside a hydrogen atom and once for a free electron, without the atom along with it. While the result in each case was mathematically intractable (due to the presence of the virtual particle pairs), he subtracted the two results. In this way he hoped that the difference between them—representing the shift in the energy for an electron located in an atom compared to that of a free electron not in an atom, precisely the effect observed by Lamb—would be tractable. Unfortunately, it wasn’t. He next reasoned that this final intractable answer must be unphysical, so the only reasonable thing to do was to simplify it by some means, using one’s physical intuition. He suggested that while relativity allowed exotic new processes due to the presence of virtual electron-positron pairs to affect the electron’s state inside an atom, the effects of relativity could not be large for those processes involving many virtual electron-positron pairs whose total energy was much greater than energy corresponding to the rest mass of the electron itself.

I remind you that quantum mechanics allows such processes, in which many energetic virtual particles are present, to take place only over very small times because it is only over such
small time intervals that the uncertainty in the measured total energy of the system becomes large. Bethe argued that if the theory including relativity was to be sensible, one should be able to ignore the effects of such exotic processes acting over very small time intervals. Thus he proposed to ignore them. His final calculation for the Lamb shift, in which only those processes involving virtual pairs whose total energy was less than or equal to the rest mass energy of the electron were considered, was mathematically tractable. Moreover, it agreed completely with the observations. At the time, there was no real justification for his approach, except that it allowed him to perform the calculation and it did what he assumed a sensible theory incorporating relativity should do.

The later work of Feynman, Schwinger, and Tomonaga would resolve the inconsistencies of Bethe’s approach. Their results showed how in the complete theory, involving both quantum mechanics and relativity explicitly at every stage, the effects of energetic virtual particle-antiparticle pairs on measurable quantities in atoms would be vanishingly small. In this way, the final effects of incorporating virtual particles in the the theory would be manageable. The calculated result is today in such good agreement with the measured Lamb shift that this is one of the best agreements between theory and observation in physics! But Bethe’s early hybrid approximation had confirmed what everyone already knew about him. He was, and is, a “physicist’s physicist.” He cleverly figured out how to use the available tools to get results. In the spirit of spherical cows, his boldness in ignoring extraneous details associated with processes involving virtual particles in quantum mechanics helped carry us to the threshold of modern research. It has become a central part of the way physicists approach
the physics of elementary particles, a subject I shall return to in the final chapter of this book.

 

 

We have wandered here from cows to solar neutrinos, from exploding stars to an exploding universe, and finally to Shelter Island. The common thread tying these things together is the thread that binds physicists of all kinds. On the surface, the world is a complicated place. Underneath, certain simple rules seem to operate. It is one of the goals of physics to uncover these rules. The only hope we have of doing so is to be willing to cut to the chase—to view cows as spheres, to put complicated machines inside black boxes, or to throw away an infinite number of virtual particles, if need be. If we try and understand everything at once, we often end up understanding nothing. We can either wait and hope for inspiration, or we can act to solve the problems we
can
solve, and in so doing gain new insights on the physics we are really after.

TWO

Language, a human invention, is a mirror for the soul. It is through language that a good novel, play, or poem teaches us about our own humanity. Mathematics, on the other hand, is the language of nature and so provides a mirror for the physical world. It is precise, clean, diverse, and rock-solid. While these very qualities make it ideal for describing the workings of nature, they are the same qualities that appear to make it ill suited to the foibles of the human drama. So arises the central dilemma of the “two cultures.”

Like it or not, numbers are a central part of physics. Everything we do, including the way we think about the physical world, is affected by the way we think about numbers. Thankfully, the way we think about them is completely dependent upon how these quantities arise in the physical world. Thus, physicists think about
numbers very differently than do mathematicians. Physicists use numbers to extend their physical intuition, not to bypass it. Mathematicians deal with idealized structures, and they really don’t care where, or whether, they might actually arise in nature. For them a pure number has its own reality. To a physicist, a pure number usually has no independent meaning at all.

Numbers in physics carry a lot of baggage because of their association with the measurement of physical quantities. And baggage, as anyone who travels knows, has a good side as well as a bad side. It may be difficult to pick up and tiresome to carry, but it secures our valuables and makes life a lot easier when we get to our destination. It may confine, but it also liberates. So, too, numbers and the mathematical relations among them confine us by fixing how we picture the world. But the baggage that numbers carry in physics is also an essential part of simplifying this picture. It liberates us by illuminating exactly what we can ignore and what we cannot.

Such a notion, of course, is in direct contradiction with the prevailing view that numbers and mathematical relations only complicate things and should be avoided at all costs, even in popular science books. Stephen Hawking even suggested, in
A Brief History of Time,
that each equation in a popular book cuts its sales by half. Given the choice of a quantitative explanation or a verbal one, most people would probably choose the latter. I think much of the cause for the common aversion to mathematics is sociological. Mathematical illiteracy is worn as a badge of honor—someone who can’t balance his or her checkbook, for example, seems more human for this fault. But the deeper root, I think, is that people are somehow taught early on not to think about what numbers represent in the same way they think about what words represent. I was flabbergasted several years ago when teaching a
physics course for nonscientists at Yale—a school known for literacy, if not numeracy—to discover that 35 percent of the students, many of them graduating seniors in history or American studies, did not know the population of the United States to within a factor of 10! Many thought the population was between 1 and 10 million—less than the population of New York City, located not even 100 miles away.