Readers of The Huffington Post have heard much about the great discovery of the Higgs boson, announced to the world with much fanfare on July 4 by the international laboratory CERN. And everyone knows that this important last particle in the reigning theory in particle physics, the Standard Model, performs the mysterious and crucial task of "giving mass" to everything in the universe.
But few outside a small community of specialized mathematicians and theoretical physicists know that, in fact, the discovery of the Higgs boson is even more significant than has been popularly explained. This grand experimental achievement in the largest, most powerful machine ever built, the Large Hadron Collider, marks a far wider scientific, philosophical and intellectual triumph -- and one that spans human history from the dawn of civilization. It has to do with the idea of symmetry: Amazingly, the Higgs boson was predicted to exist not for any physical reasons, but on strictly mathematical grounds based on arcane symmetries usually studied in "pure" mathematics. And its story involves a major quest that began with the Babylonians and Egyptians and continued to the ancient Greeks, the Arabs, medieval Europe, and on through the 19th century to our own time.
Symmetry is something that is so prevalent in nature that it underlies even things we may not suspect. An infant only a few weeks old will already recognize a stylized smiley face because of its intrinsic two-sided symmetry. And experiments with monkeys have shown that a monkey will often fill in the other side of an asymmetrical drawing to complete a symmetrical design. Studies have shown that people we perceive as beautiful tend to have more perfectly symmetrical features, and this seems to extend to the animal world, where more symmetrical individuals breed more frequently. But symmetry is even deeper than meets the eye and in fact rules the structure of the universe. Its story is long and involved, and originated in the mathematical study of equations and their solutions.
Both the ancient Babylonians and the Egyptians of the second millennium B.C. could derive and solve linear equations, as well as some quadratic equations, but went no further. The ancient Greeks, too, could solve similar equations. Then a more general solution method for quadratic equations came with the Arab-Persian mathematician Al-Khowarizmi (c. 780-850) who worked in the Caliph's "House of Wisdom" in Baghdad, and from whose name we get the word algorithm, and from the name of his book, Al Gabr wa al Muqabbala, we get the word algebra. In the 16th century, the Italian mathematicians Cardano, del Ferro, Fior, and Tartaglia developed methods for solving the cubic and quartic equations. But for 300 years afterwards no one knew a general method for solving a quintic -- or fifth-order -- equation, no matter how hard they tried! Then in 1829-1830, a brilliant young French mathematician by the name of Evariste Galois applied himself to this age-old mystery. Galois was only a teenager at the time and smarter than all his teachers: No one could understand his highly advanced theories. He became depressed by lack of recognition, left the university, became involved in revolution against the King, was arrested, became entangled with "an infamous coquette," as he referred to her in a desperate letter to a friend--and died in a bloody duel, perhaps for her "honor," outside Paris in 1831 at the age of 20!
Before he died, Galois wrote down in letters an entire theory that explained why the fifth-order equation cannot be solved. Galois' amazing answer had to do with symmetry: in this case, the symmetry of the set of solutions to an equation. The fifth-order equation simply has the wrong kind of symmetry. In developing his incredibly insightful theory, Galois invented the mathematical concept of a group. Groups are today the mathematician's way of explaining and handling symmetry.
Group theory has applications to a huge variety of problems in both pure and applied mathematics. For example, the great French anthropologist Claude Lévi-Strauss could not solve the hard problem of explaining the complicated marriage laws of the Murngin aboriginal tribe in Australia until the mathematician André Weil attacked this problem using abstract group theory. It turns out that a particular kind of groups, called continuous, or Lie, groups (named after the Norwegian mathematician Sophus Lie), are extremely useful in theoretical physics.
By the middle of the 20th century, theoretical physicists came to the conclusion that the deepest mysteries in nature could only be analyzed and solved using deep symmetries defined in abstract mathematical settings. The culmination of this approach took place in 1967, when the American physicist Steven Weinberg was able to show theoretically that mass was obtained by a certain kind of particle (called a W or a Z boson, which acts inside a nucleus of matter to produce radioactive decay) when a primeval symmetry of the very early universe was broken through something he termed "the Higgs mechanism," after the Scottish physicist Peter Higgs. What Peter Higgs, and independently of him Robert Brout, Francois Englert, Tom Kibble, Carl Hagen and Gerald Guralnik, all did in the same year, 1964, was to remove a technical hurdle called the Goldstone-Weinberg-Salam theorem, which stood in the way of producing mass by breaking a mathematical symmetry. Higgs and the others proved in three independent papers that the particular symmetry believed to rule the early universe (a continuous Lie group symmetry called a Yang-Mills symmetry) was immune to the Goldstone-Weinberg-Salam theorem.
In doing so they provided a mechanism by which the symmetry of the early universe -- a tiny fraction of a second after the Big Bang -- can be broken. The actual way this was done was explained by Weinberg three years later. This is what gave us mass during the very early life of the universe -- and it all goes back to the pure mathematics of continuous symmetries and the idea of a group developed by Galois and originally proposed in order to understand why some equations can be solved and others not.
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