The Infinity Puzzle: Quantum Field Theory and the Hunt for an Orderly Universe - Hardcover

Frank Close is a Professor of Theoretical Physics at Oxford University and Fellow and Tutor in Physics at Exeter College, Oxford. He is the winner of the Kelvin Medal for the public understanding of physics and the author of ten books. He lives in Abingdon, England.

Amsterdam, 1971

And now I introduce Mr. ’t Hooft, who has a theory that is at least as elegant as anything we have heard before.
—Tini Veltman, “amsterdam international conference on elementary particles,” 1971

Tini Veltman is a contrarian: a forthright man who has never shied away from controversy. His single-mindedness has brought him success where others either gave up or didn’t even dare to try. It is the characteristic that set him on course to a Nobel Prize for Physics. Part of the reason for his triumph was the fortune to have a student whose genius was in constructing a masterpiece by using tools that Veltman had forged. Veltman and his protege, Gerard ’t Hooft, are like chalk and cheese. Veltman is a big man, with a fulsome beard, often found with a cigar stuck in the corner of his mouth or waved between his fingers as he holds court. His near-perfect English resonates with Dutch vowels as he dismisses some rival’s work as “baloney” or “crap.” This blunt approach can mislead, obscuring a sensitive and thoughtful personality, with deeply held convictions about the way science should be conducted. His nickname,  “Tini”—an abbreviation of Martinus—is ironic given his stature, in all senses of the word.
 
’t Hooft, by contrast, slight in build, with thinning hair, dressed smartly in jacket and tie, and with a small mustache, could easily be mistaken for an English country doctor or an accountant. During discussions, I am often possessed by a sense that he already knows what he is being told and is politely waiting to hear something novel. When he speaks, there is no doubt that he is correct: His soft voice carries real force, aided by a dry sense of humor.
 
Forty years ago, their meeting would change the world of physics. However, today, Veltman—the teacher whose ideas enabled his star pupil to produce his magnum opus—and ’t Hooft have drifted apart. In Veltman’s own book about particle physics, ’t Hooft’s appearance is limited to a photograph and a few lines of text. He describes ’t Hooft’s breakthrough as “a splendid piece of work,” which, enigmatically, he was very happy with “at the time.” That is how it was in 1971, when Veltman “proudly introduced” his young maestro to the world.
 
The Infinity Puzzle

A half century or so ago, and more than two thousand years after the philosophers of ancient Greece had first conceived of atoms, these basic pieces of matter had been revealed to consist of smaller particles, of lightweight electrons remotely encircling a bulky central nucleus.
 
In the aftermath of Hiroshima, where the nuclear atom’s explosive power had been revealed, understanding the nature of the atomic nucleus and the mysterious forces that control it was what defined the new frontier. That the nucleus of an atom has a labyrinthine structure of its own was already apparent; the surprise was that the closer that scientists looked at it, the more complicated things appeared to be. And to cap it all, strange particles—similar to those found on Earth, yet behaving in other ways—were discovered to be pouring down from the heavens, as the result of cosmic rays from outer space smashing into the atmosphere above our heads. Exotic forms of matter, whose existence had not been dreamed of by scientists in their earthbound laboratories, were changing our whole perception of nature. Any theory of the universe had to explain them.
 
This was a time when the pursuit of breakthroughs had become the physics world’s equivalent of the Klondike gold rush. Some theoretical high-energy physicists staked their claims with half-baked theories, which they published in obscure journals. The logic seemed to be that if your idea turned out to be wrong, few would notice and the paper would be quietly forgotten. However, if it turned out that a discovery proved your idea to have been correct, you could then refer the world back to your paper and claim priority.
 
Throughout this febrile period, one problem stood out, resisting all attempts at a solution. This was what I call the “Infinity Puzzle.” Three great theories—Maxwell’s theory of electromagnetism of the nineteenth century, Einstein’s theory of special relativity of 1905, and Quantum Mechanics, developed in the 1920s—individually made profound predictions that turned out to be completely accurate: for example, the description of light as electromagnetic waves with a constant speed; the conversion of mass into energy via E=mc2, where c is the speed of light; and the explanation of the stability of atoms, with a quantitative description of their beautiful spectra. In the 1930s the union of these theories gave birth to a complete theory of electromagnetic force and how light interacts with atoms, known as Quantum Electrodynamics, or QED. Initially, it appeared beautifully seductive, but what at first had appeared to be a Cinderella soon threatened to become an Ugly Sister. When the equations of QED were applied beyond the simplest approximations, they seemingly kept predicting that the chance of some things occurring was “infinite percent.” Why is this a problem? The answer is that infinity is transcendent, beyond measure, signifying a failure of understanding rather than a real answer.
 
To put this into context, the probability of chance can range from zero (that I will never win the lottery, for instance, as I never buy a ticket) to an absolute certainty at 100 percent (death and taxes). “Infinity,” by contrast, is boundless and immeasurable; it has no quantifiable meaning. In the context of the questions that the scientists were posing, the answer was nonsense, analogous to your computer giving you an error message: “computer violation” or “overflow.” When this happens it is usually a hint that you have made some catastrophic error—such as instructing the machine to divide by zero. Or it may be a sign that there is a glitch in your computer, perhaps even that the machine itself has been assembled incorrectly.
 
 Without doubt “overflow”—or in our example, infinity—is telling you that something is wrong; the problem is: What to do about it?
 
Nor was this a nonsense confined to some arcane piece of atomic science, for this enigma touched upon our ability to understand the principles underlying some of the most basic and far-reaching phenomena. Plants grow as their atoms absorb energy from light, for instance; radio waves result when electric charges are disturbed by electric or magnetic forces; and much of modern electronic technology involves the interactions between electromagnetic radiation and electrons. Each of these— whole industries and indeed many forms of life itself—depends on a simple underlying mechanism: an electron absorbing or emitting a photon, which is the basic particle of light. Yet QED seemed unable to agree with even this most rudimentary of processes. If, as QED seemingly implied, the chance of a photon being absorbed by an atom was infinite, then photosynthesis and indeed many chemical reactions would happen instantaneously. Life would have burned itself out long ago, if indeed it had ever begun.
 
For physicists, infinity is a code word for disaster, the proof that you are trying to apply a theory beyond its realm of applicability. In the case of QED, if you can’t calculate something as basic as a photon being absorbed by an electron, you haven’t got a theory—it’s as fundamental as that.
 
One particular example of this catastrophe is the magnitude of an electron’s magnetism, which experiments could measure relative to some standard scale. By using the standard theory, that is, QED, physicists expected to be able to compute this number. All that is required is to solve the algebraic equation describing an electron absorbing a single photon.
 
This is standard fare in undergraduate physics, and I can well recall the joy I felt when, back in 1967, I first carried out the calculation myself. I thought that at last I had qualified as a theorist. Unfortunately, I then learned that this was just the first of a whole series of calculations that would be needed in order to arrive at the true answer; furthermore, my tutor had glossed over the fact that if I were somehow able to do this momentous task, and then to add up the total, the answer would turn out to be infinity. Unknown to me at that time, a few hundred miles away, in Holland, I had a contemporary named Gerard ’t Hooft, who was also being exposed to the mysteries of infinity and within five years would gain scientific immortality by solving them.