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Magic, Mystery, and Matrix
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Short Description: stands for magic, mystery, or matrix, according to. taste. The magic and mystery are clear enough,. while “matrix” refers to a new noncommutativity, ...
Content Inside: witten.qxp 8/12/98 8:18 AM Page 1124 Magic, Mystery, and Matrix Edward Witten In the twentieth century, the quest for deeper a much more complex history than general rela- understanding of the laws of nature has tivity, and in some sense most of its influence on largely revolved around the development of mathematics belongs to the twenty-first century. two great theories: namely, general relativity The quantum theory of particles--which is more and quantum mechanics. commonly called nonrelativistic quantum me- General relativity is, of course, Einstein's theory chanics--was put in its modern form by 1925 and according to which gravitation results from the cur- has greatly influenced the development of func- vature of space and time; the mathematical frame- tional analysis, and other areas. work is that of Riemannian geometry. While pre- But the deeper part of quantum theory is the viously spacetime was understood as a fixed arena, quantum theory of fields, which arises when one given ab initio, in which physics unfolds, in gen- tries to combine quantum mechanics with special eral relativity spacetime evolves dynamically, ac- relativity (the precursor of general relativity, in cording to the Einstein equations. Part of the prob- which the speed of light is the same in every in- lem of physics, according to this theory, is to ertial frame but spacetime is still flat and given ab determine, given the initial conditions as input, how initio). This much more difficult theory, developed spacetime will develop in the future. from the late 1920s to the present, encompasses The influence of general relativity in twentieth- most of what we know of the laws of physics, ex- century mathematics has been clear enough. Learn- cept gravity. In its seventy years there have been ing that Riemannian geometry is so central in many milestones, ranging from the theory of "an- physics gave a big boost to its growth as a math- timatter", which emerged around 1930, to a more ematical subject; it developed into one of the most precise description of atoms, which quantum field fruitful branches of mathematics, with applica- theory provided by 1950, to the "standard model tions in many other areas. of particle physics" (governing the strong, weak, While in physics general relativity is used to and electromagnetic interactions), which emerged understand the behavior of astronomical bodies and the universe as a whole, quantum mechanics by the early 1970s, to new predictions in our own is used primarily to understand atoms, molecules, time that one hopes to test in present and future and subatomic particles. Quantum theory has had accelerators. Quantum field theory is a very rich subject for mathematics as well as physics. But its develop- Edward Witten is professor of physics at the Institute for ment in the last seventy years has been mainly by Advanced Study. His e-mail address is witten@ias.edu. physicists, and it is still largely out of reach as a rigorous mathematical theory despite important ef- This article is based on the Josiah Willard Gibbs Lecture given at the Joint Meetings in Baltimore in January 1998. forts in constructive field theory. So most of its im- pact on mathematics has not yet been felt. Yet in The author's work is supported in part by NSF Grant PHY- 9513835. many active areas of mathematics, problems are 1124 NOTICES OF THE AMS VOLUME 45, NUMBER 9
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