So here’s a slightly weird thing. So there’s been all this excitement over the web about the old old “shocker” that the sum of the positive integers is -1/12. You know, not even an integer, and not even positive. Apparently there have been articles in the New York Times and Slate and goodness knows where else… and I’ve been ignoring it all since I’m tired of what it often leads to: People wilfully using it as a device to manipulate people’s ignorance about subtleties with infinite processes to make the tired point that string theory is somehow wrong since it is based on “funny math”. I called Lawrence Krauss (who should have known better) out about it some years ago when he did that at an event I happened to attend. It’s a bit tedious, not the least because it is actually part of a wonderful field of mathematics that can get misrepresented, and of course because it has nothing to do with string theory.
So I ignored it all. Then some students in my class asked me about it. And I explained why it is interesting and so forth… Then I carried on ignoring it all.
Then a day or two ago a mathematician colleague emailed me to ask what physicists do with this and what think about it, and gave me a link to the YouTube video that started all the excitement. Turns out it is part of a series of fun short films about aspects of mathematics. It is called Numberphile in case you want to look it up. I think I saw one in the series a while back since my friend Ed Copeland is involved somehow, so I know its coming from a trustworthy and well meaning place… So, I looked. (Embed of the video at the end.)
Imagine my surprise. The young fellow who is doing a quick proof of how it works starts off by mentioning he first saw it back when he “started learning a bit of string theory”. He (Tony Padilla) was instantly familiar to me because it was I who actually taught him that result back in Durham in my string theory class 12 or so years ago! Small world indeed. Glad to see he has kept his enthusiasm and is getting the public excited about science.
Enjoy the video. He puts it all together in a fun way. The physics way of arriving at this result is usually a bit different (and by the way despite what you read elsewhere, at heart it has nothing to do with string theory, and more to do with making sense of quantum field theory in general). Ultimately, it all comes down to trying to represent in a tractable way a finite (physical and measurable) quantum effect associated to the lowest energy state of a quantum system, by describing it in terms of an infinite family of oscillators. It is a representation, and as such has to be treated carefully, and not be taken too literally as the real thing. There’s a longer story here, but I think I’ll skip it and simply say that the whole business of “summing up to infinity” is full of pitfalls if you think of it in terms of the sums we do normally. It is completely different, and often does not have the same meaning at all.