Assigning a value to infinity This is what happens when you mess with infinity.“You might think that if you simply started adding the natural numbers, one plus two plus three and so on all the way to infinity, you would get a pretty big number. At least I always did.“So it came as a shock to a lot of people when, in a recent video, a pair of physicists purported to prove that this infinite series actually adds up to … minus 1/12. To date some 1.5 million people have viewed this calculation, which plays a key role in modern physics and quantum theory; the answer, as absurd as it sounds, has been verified to many decimal places in lab experiments. After watching the video myself, I checked to make sure I still had my wallet and my watch. Even the makers of the video – Brady Haran, a journalist, and Ed Copeland and Antonio Padilla, physicists at the University of Nottingham in England – admit there is a certain amount of “hocus-pocus,” or what some mathematicians have called dirty tricks, in their presentation. Which has led to some online grumbling. But there is broad agreement that a more rigorous approach to the problem gives the same result, as shown by a formula in Joseph Polchinski’s two-volume textbook “String Theory.”“So what’s going on with infinity? “This calculation is one of the best-kept secrets in math,” said Edward Frenkel, a mathematics professor at the University of California, Berkeley, and author of “Love and Math: The Heart of Hidden Reality” (Basic Books, 2013). No one on the outside knows about it.”“The great 18th-century mathematician Leonhard Euler, who was born in Switzerland but did most of his work in Berlin and St. Petersburg, Russia, was the first one down this road. Euler wanted to know if you could find an answer to endless sums of numbers like 1 plus 1/2 plus 1/3 plus 1/4 on up to infinity, or the squares of those fractions. “These are all different versions of what has become known as the Riemann zeta function, after Bernhard Riemann, who came along about a century after Euler. The zeta function is one of the more mysterious and celebrated subjects in mathematics, important in the theory of prime numbers, among other things. It was one of the plot threads, for example, in Thomas Pynchon’s 2006 novel, “Against the Day.” “In 1749, Euler used a bag of mathematical tricks to solve the problem of adding the natural numbers from 1 to infinity, a so-called divergent series because the terms keep growing without limit as you go along. Clearly, if you stop adding anywhere along the way – at a quintillion (1 with 18 zeros after it), say, or a googolplex (10 to the 100th power zeros) – the sum will be enormous. The problem with infinity is that you can’t stop. You never get there. It’s more of a journey than a destination. As Padilla says to Haran at the end of their video, “You have to face infinity, Brady.” The method in the video is essentially the same as Euler’s. It involves nothing more complicated than addition and subtraction (although the things being added and subtracted are more infinite series) and a small piece of algebra that my sixth-grade daughter would breeze through. You are not alone in wondering how this can make sense. Norwegian mathematician Niels Henrik Abel, whose notion of an Abel sum plays a role here, once wrote, “The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.” In modern terms, Frenkel explained, the gist of the calculations can be interpreted as saying that the infinite sum has three parts: One blows up when you go to infinity, one goes to zero, and minus 1/12. The infinite term, he said, is just thrown away.“And it works. A hundred years later, Riemann used a more advanced and rigorous method, involving imaginary as well as real numbers, to calculate the zeta function and got the same answer: minus 1/12. “So Euler guessed it right,” Frenkel said. Those of us who are not mathematicians probably wouldn’t care so much about infinity except that it crops up again and again in calculations of things, like the energy of the electron, that we know are finite, or in string theory, which physicists would like to hope is finite. In this case, our current understanding of the very solidity of reality depends on coming up with a consistent way to assign values to infinite sums. In the process known as regularisation, which is a part of many calculations in quantum theory, physicists do something similar to what Euler did, arriving at a real number that corresponds to the quantity they want to know and an infinite term, which they throw away. The process works so well that theoretical predictions in quantum electrodynamics, the fancy version of the familiar force of electromagnetism, agree with experiments to a precision of one part in a trillion. -New York Times Service. February 18, 2014 at 04:18AM
Posted on: Tue, 18 Feb 2014 03:25:19 +0000
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