The Infinity Puzzle

Quantum Field Theory and the Hunt for an Orderly Universe


By Frank Close

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Speculation is rife that by 2012 the elusive Higgs boson will be found at the Large Hadron Collider. If found, the Higgs boson would help explain why everything has mass. But there’s more at stake — what we’re really testing is our capacity to make the universe reasonable.

Our best understanding of physics is predicated on something known as quantum field theory. Unfortunately, in its raw form, it doesn’t’t make sense — its outputs are physically impossible infinite percentages when they should be something simpler, like the number 1. The kind of physics that the Higgs boson represents seeks to “renormalize” field theory, forcing equations to provide answers that match what we see in the real world.

The Infinity Puzzle is the story of a wild idea on the road to acceptance. Only Close can tell it.


The Void
The New Cosmic Onion:
Quarks and the Nature of the Universe
Particle Physics: A Very Short Introduction
Lucifer's Legacy: The Meaning of Asymmetry

Whose appearance out of The Void
Is an Infinite Puzzle

Old men forget. . . . But he'll remember with advantages, what feats he did that day.

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 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 protégé, 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.1 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."2 That is how it was in 1971, when Veltman "proudly introduced" his young maestro to the world.


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.3
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.4 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. 5 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.
The reason that there was so much more to do lies with the fact that, according to QED, the electron in question is not alone in the void: A vacuum is not empty but seethes with transient particles of matter and antimatter, which bubble in and out of existence. Although these will-o'-the-wisps are invisible to our normal senses, they disturb the photon and electron in the moment of their union and contribute to the number that the experiment measures.
QED contains the means of calculating the effect of each of these disturbances, one by one. There is an infinity of them, the contributions of all but a few being so trifling that they can be ignored—so long as you are prepared to accept some limit to the precision of what you are computing. The trick is to start with the most important (which is what my student calculation had done, naively thinking it to be the lot), then add in the next, and then to continue by including the effects of smaller and smaller contributions, the sum total approaching the "true" answer ever more accurately.
This can be difficult to do, but there is nothing necessarily wrong here, as an infinite sum can have a finite answer (such as 1 + ½ + ¼ + ⅛ + . . . = 2). After the first two terms you are already within 25 percent of the answer; add in the next couple, and your inaccuracy is less than 10 percent. It is merely a pragmatic question of how precise an answer you need as to how many terms, and how much work, you have to do.
Or so physicists thought in their early explorations of the implications of quantum mechanics and QED. However, by contrast to the previous sum, which gave the desired answer of 2, what they found instead was a series that was more like 1 + ½ + ⅓ + ¼ + . . . . At first sight this looks good too—after just three terms the sum is already within 10 percent of 2. But add in the next one, ¼, and you will find that the running total has overshot: 2.08. Add in further terms and it continues to get worse: infinitely worse. The sum 1 + ½ + ⅓ + ¼ + . . . = infinity.
In their search for precision, physicists had utterly lost accuracy. Attempt to calculate an electron's electrical properties, such as the size of its charge or magnetism, and your answer would turn out to be infinity; if you wanted to know what would happen when a photon hit an electron, and listed the odds of this or that possibility, each one would turn out to have the chance "infinite percent."
While QED describes how light interacts with matter, it alone cannot confront the stability of matter itself. There are two other forces acting in and around the atomic nucleus, known as the weak and strong nuclear forces, their names alluding to their strengths relative to that of the electromagnetic force when acting on atoms here on Earth. The strong force is the binding force that holds atomic nuclei together; the weak force, by contrast, destabilizes nuclei, causing a form of radioactivity that plays an essential role in the way that the sun produces its energy (see Figure at right). The theories of these forces also ran into problems.
The theory of the weak force gave a series of diminishing terms, similar to QED, which led to infinity also. The strong force was an even greater enigma, as in its case the infinite sum explodes; instead of a gentle approach to infinity, like 1 + ½ + ⅓ + ¼ . . . , there was an unnerving sum like 1 + 4 + 9 + 16 + . . . , where each successive term is bigger than all that went before it. This was so daunting a result that physicists decided some other way of explaining the strong force was needed.
For the particular case of QED, a way of abstracting useful numbers from the morass was found in 1948, as we shall see in Chapter 2. The basic trick, which works but has never made everyone, including those who created it, totally satisfied, is as follows.6
There are many properties of atoms and their constituent particles that you may compute in QED, each of which gives the answer infinity, but the key discovery was that whatever you calculated, the way that infinity emerged from the mathematics was the same from one process to the next. For example, when physicists calculated one quantity, they found a horrible infinite thing multiplied by, say, the number 1. Then they calculated some other quantity and found the very same "horrible infinite thing," but this time multiplied by, let's suppose, 2. So this second quantity was predicted to be twice the size of the first. If an experiment had already measured the true (finite!) value for the first quantity, QED could then confidently predict the magnitude of the second as being twice as great, and experiment confirmed this to be true. So the horrible "infinity" could be subsumed, hidden from view as if it didn't exist, leaving an apparently pristine theory on display. As I said, no one was entirely happy, yet it worked.
Gravity is attractive and controls the large-scale motion of galaxies, planets, and falling apples. Electric and magnetic forces hold electrons in the outer reaches of atoms. They can be attractive or repulsive and tend to counterbalance in bulk matter, leaving gravity dominant at large distances. The electrical repulsion between protons in an atomic nucleus would prevent the existence of stable nuclei but for the existence of the strong force, which gives a powerful attraction between protons or neutrons when they touch. The weak force can change one form of particle into another. This can cause transmutation of the elements, such as turning hydrogen into helium in the heart of the sun. (Credit: Reproduced with permission of Oxford University Press)
This is how. The values of an electron's electric charge and its mass have been measured. The miracle is that these two known quantities are sufficient to provide benchmarks for anything else that we may wish to compute in QED. We cannot use QED to calculate the electron's charge or mass from theory—for were we to do so, we would get infinity—but we can use QED to calculate everything else relative to these experimentally determined quantities. The marvel is that instead of infinity, all the answers now turn out to be finite, and, even better, the values are correct. Today, some quantities have been calculated this way that agree with experiment to an accuracy of one part in a trillion, which is an order of magnitude much like the diameter of a hair when compared to the width of the Atlantic Ocean.
Although inelegant, the recipe for extracting sensible answers for QED worked. However, the explosive numbers that erupted in the case of the strong force convinced people that some other route was needed there (we shall see in Chapters 12 and 13 how that breakthrough came about). For three decades, both the weak and the strong nuclear forces appeared to be beyond quantitative description. However, in the case of the weak force there was a tantalizing hope that something akin to the miracle of QED might be replicated. Unfortunately, here, too, everyone who tried soon hit a seemingly impenetrable barrier.
The Infinity Puzzle for the weak force resisted the physics world's greatest minds for a quarter century. Some tried to solve the problem but failed; most ignored it and hoped that it would go away. The nature of this impasse, how it was defeated, and the arguments over priority for Nobel Prizes that it has spawned are the themes of this book.
The saga is a paradigm of how science happens in the real world, as opposed to the steady heroic progression portrayed in some winners' accounts. Instead of a direct line linking theoretical idea and experimental discovery, there are numerous wrong turns, partial answers, and mislaid arguments. The picture of science as a sequence of great discoveries and Nobel Prizes, which is presented in some narrative histories, and which forms many people's idea of the field, is really an attempt to make easy narrative sense of the whole saga, with hindsight. In practice, scientific research is a series of twists and turns; scientists experience the same emotions, pressures, and temptations as any other group of people and respond in as many ways.
You may experience the euphoria of making a great discovery, only to find out that someone else has beaten you to it. Or you may have been first, but not been ready, or brave enough, to go out on a limb and publish—perhaps wanting more time in order to be certain, or even not realizing at the time the significance of what you had achieved. As we shall see, even at the top level people often don't know if their idea is world changing or a mere fancy until later events determine which. This is like Paul McCartney, years later, admitting that at the time of writing his songs he didn't know which would sell millions and which would fail.
For composers of music, or literature, there is no limit to the number of possible creations—it is infinite. If you don't make your composition public, it is unlikely that anyone else will create the very same symphony. For theoretical physics, on the other hand, nature already has the solution, and it is we who are trying to reveal it for ourselves. So there is a sense of uniqueness, a right or wrong, which experiment or further advances in theoretical understanding may ultimately reveal. Discover what it is, publish first, and the credit will be yours. However, if you do not, and someone else independently, later, publishes what you might have done, how do you react when the world takes notice? History records the winners' names in the pantheons of science; the names of "Nobel Prize runners-up" are as memorable as those of the losing semifinalists in Grand Slams or World Series.
Such are the realities of science, where scientists' emotional responses to these pressures may be far from the dispassionate ideal of popular belief. Our story, spanning more than a half century, has examples of all of these, and more.


Among those who thought that they knew how to solve the Infinity Puzzle for the weak force were Abdus Salam and Tsung-Dao (T.-D.) Lee. The person who actually did solve it, however, was Gerard 't Hooft, Veltman's brilliant student. It was 1971 when 't Hooft convinced his teacher that he had indeed succeeded. Veltman decided to launch his apprentice, already a master craftsman, in dramatic fashion.
In June that year a major international physics conference was scheduled to take place in Amsterdam. Veltman, a senior physicist at the University of Utrecht, had been asked to organize a series of presentations in theoretical physics. He invited Lee and Salam to present their ideas on how to solve the Infinity Puzzle.
Chinese American theoretical physicist T.-D. Lee had already won a Nobel Prize, shared with his colleague Chen-Ning Yang, for showing that the world behind the mirror is essentially different from the real world. Whatever it is that is responsible for the radioactive decays of atoms seems in the real world to be controlled by a mysterious subatomic left-handed screw. Viewed in a mirror this would appear to be right-handed. Had Alice known of radioactivity, she would have been able to tell whether she was in the world behind the "Looking Glass" or in the real one. The discovery in 1956 that nature is left-handed was a huge cultural shock and guaranteed Lee and Yang scientific immortality. By 1971, Lee had decided that the Infinity Puzzle was the one to crack, and he thought he knew how to solve it. However, thanks to 't Hooft, Veltman knew better.
In 1971 Abdus Salam had not yet won a Nobel Prize but was ambitious to do so. Later, Veltman would not be slow to remind people of this, hinting at the lengths Salam would go to lobby the committees. Salam was a visionary, head in the clouds, ideas flowing as if he were in a perpetual brainstorm, ready to publish anything and hope for the best. His style didn't gel with Veltman, who held Salam's oeuvre in less high esteem than some. Salam thought that he knew how to solve the Infinity Puzzle for the weak force, and from some ambiguous remarks in talks may even have convinced himself that he had the answer. However, he never quite convinced others, certainly not Veltman.
That summer, Salam believed that the key to finding the solution was to incorporate gravity into the mix. Here too, thanks to 't Hooft, Veltman knew better. He invited Salam to open the proceedings.
The venue was a small room off the main hall in the Amsterdam Congress Centre. More than 2,000 scientists attended the conference, but of them only a few dozen were present at what at first appeared to be a sideshow to the main proceedings.
Salam spoke first, saying that he was convinced that gravitation was the key. Veltman let Salam talk about his "baloney" before calling on T.-D. Lee, who then described his own attempts to solve the puzzle by inventing hitherto unknown particles with weird properties.7 Lee completed his presentation, answered questions, and returned to his seat in the auditorium. The moment had at last arrived: "And now I introduce Mr. 't Hooft," Veltman announced, "who has a theory that is at least as elegant as anything we have heard before."
't Hooft's talk lasted just ten minutes, and to those in the audience, unaware of the significance of what they were witnessing, the occasion appeared to be simply a means for Veltman to push a promising student to wider attention. The "before" in Veltman's introduction was assumed by members of the audience to mean "in this session,"8 and as few regarded Salam's or Lee's ideas with much enthusiasm, this introduction did not seem unreasonable, nor did it heighten expectations. However, what Veltman meant by "before" was "in the past thirty years," for 't Hooft had found the philosopher's stone.
Most did not understand his talk, let alone realize that they were present at a singular moment in the history of science. Salam certainly did not. In the written version of his own talk, which he revised after the conference, he added a note "welcoming G 't Hooft's theory," also advertising that "the same theory" had been proposed in 1964 by himself and a colleague, J. C. Ward, and then included this afterthought: "Gravity . . . is likely [to be needed] to give the right numerical values."9 However, as the passage of time has shown, incorporating gravity would not be necessary. Salam's postscript shows how even an expert failed to appreciate the full import of what he had just heard.
't Hooft, by contrast, did not write up his talk. He was still completing his thesis and wanted all the arguments to be presented there, carefully, where they could be spelled out like a legal document for experts to examine the logic of the proof until convinced that it was watertight. As one colleague present recalled years afterward, some in the audience had caught a flavor of what had happened, and the delegates were asking one another, "Veltman's student—'t Hooft—is he really claiming to have solved the Infinity Puzzle?"10 Discussions in the corridors afterward convinced them that indeed he was.
When the news began to spread, the reactions of two Nobel laureates were typical. Steven Weinberg remarked, "I had never heard of him so my first reaction was: this can't be right."11 Sheldon Glashow retorted, "Either the guy's a total idiot [to be making such an outrageous claim] or he's the biggest genius to hit physics in years."12
"Genius" was correct. 't Hooft and Veltman would share the Nobel Prize for Physics in 1999 for this achievement. Given their initial reactions, there is irony too that Glashow, Weinberg, and Salam (but not Ward) would themselves share a Nobel Prize in 1979 for their own work, which't Hooft's breakthrough was about to bring to center stage, for 't Hooft's entrance was a pivotal moment in the development of understanding during the second half of the twentieth century.
In simplified accounts, Veltman's role was much like that of John the Baptist, preparing the way with the tools, the blueprints, and the machinery to fit everything together; 't Hooft was the true Messiah, the genius that physics had awaited for years who built the theory, and the structure, that would lead to a golden age. Forty years later, their legacy includes the largest and most ambitious experiments in physics that have ever been attempted: the simulation at the Large Hadron Collider (LHC) at CERN in Geneva of the first moments in the universe after the Big Bang.
For more than two thousand years, until 't Hooft, a central aim of philosophy and science had been to identify the fundamental pieces of matter, the "atoms," and, latterly, the elementary particles. Following that breakthrough, the focus has changed: Our conceit today is that we may be able to reveal how matter itself was created and how our universe of shape and form came to be.
The first half of this narrative describes how 't Hooft, and others, made the crucial breakthroughs that culminated in the triumph of 1971. The remarkable developments that have come to pass since that seminal moment will be the theme of the later chapters. There I shall trace the path from a sideshow of a talk in Amsterdam to a multibillion-dollar worldwide scientific collaboration that hopes to answer such questions at the Large Hadron Collider.



  • "Close chronicles the search for the elusive Higgs Boson particle (the 'God Particle').... Throughout, the author chronicles the winners and losers in the annual Nobel sweepstakes, giving them recognition for their achievements and providing a lively thread for readers."—Kirkus Reviews
  • "The Infinity Puzzle is eminently readable. It has no equations--only a few Feynman diagrams--and a glossary in the back so you don't get your bosons confused with your hadrons.... All of the luminaries of twentieth century particle physics are here, along with many unsung heroes whose contributions Dr. Close explains and puts into context so they can be better appreciated by a public hitherto ignorant of their work.... [T]he entire book is a very manageable introduction to quantum physics for those who are interested in, but possibly intimidated by, understanding the inner workings of the fabric of our Universe."—Ars Technica
  • "Building the standard model, the flagship theory of modern particle physics, was no mean task. It took decades of painstaking work to bring the forces and elementary particles that make up the universe together in a single framework (which still doesn't include gravity). Close, a theoretical physicist, chronicles this history from an insider's perspective.... the story doesn't unfold as a simple, clearly developing line of thought. Instead, the reader witnesses scientific progress in all its real-world messiness. It's a comedy of errors at times, full of dead ends, missed opportunities and ideas that lie dormant for years, unproven or unnoticed."—Science News
  • "In his new book, Oxford physicist Frank Close reviews decades' worth of brain-teasing theories and looks ahead to puzzles yet to be solved.... Close's tale illustrates that the course of true science doesn't always run smooth. It may well turn out that the long-sought Higgs boson is a will-o'-the-wisp, and physicists will have to go back to square one. But even that won't render The Infinity Puzzle out of date."—Alan Boyle,'s Cosmic Log
  • "As someone who can deftly explain abstruse quantum field theory to a lay reader, Frank Close is a rarity among physicists. Rarer still, he knows how to weave a compelling tale--that of the 'infinity problem,' which has bedeviled the field of quantum electrodynamics and subsequent attempts to unify the forces of nature. The result is a great scientific whodunit, replete with a large, engaging cast of characters, behind-the-scenes maneuvering, and unexpected twists and turns. Here is proof that Close belongs among the very first rank of scientist-authors. I strongly recommend The Infinity Puzzle."—Steve Nadis, coauthor of The Shape of Inner Space
  • "It is a pleasure to read a book on recent advances in our understanding of the structure of matter by an author who not only understands the subject but also takes care to investigate conflicting accounts of how these advances came about."—Peter Higgs, emeritus professor of physics, The University of Edinburgh
  • "Superb.... The Infinity Puzzle presents in light and fetching prose a (and you should pardon the pun) close-up of a wonderful set of episodes in contemporary science centered around one of the single most beautiful edifices of modern theoretical physics, quantum field theory, and leading up to the hottest example of big science to be found on the globe today."—MAA Reviews
  • "Mr Close, an accomplished particle physicist in his own right, enjoyed unprecedented access to all the principal players, many of whom he either knows well or, like Mr Higgs, has spoken to at length. He also appears to have left no relevant academic paper, no conference proceedings, memoir or other publicly available source unturned. This painstaking attention to historical detail yields many gems.... Mr Close's magisterial work is sure to become the definitive account of the story. It offers no unambiguous advice to the Nobel committee. But the judges would be wise to give it a thorough read anyway."—Economist
  • "[A] thoroughly researched and well-crafted narrative.... [Close] focuses on the triumphs and failures of the physicists behind the equations, providing a realistic view of how theoretical physics really progresses--the all-too-human endeavour fraught with personal ambitions, rivalries, alliances, errors and plain historical accident.... It's refreshing to read a popular physics book that doesn't revisit the same well-trodden ground of so many before it."—New Scientist

On Sale
Nov 29, 2011
Page Count
464 pages
Basic Books

Frank Close

About the Author

Frank Close, OBE, FRS is a particle physicist and an emeritus professor of physics at the University of Oxford. He is the author of over a dozen books, including Infinity Puzzle and Half-Life. He lives in Oxford, England. 

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