09/02/2014

Dennis Overbye is one of the best science writer around. Recently, he wrote a beautiful piece on the odd behavior of non-converging series like $1+2+3+4+\ldots$ and so on to infinity (see here). This article contains a wonderful video, this one

where it shown why $1+2+3+4+\ldots=-1/12$ and this happens only when this series is taken going to infinity. You can also see a 21 minutes video on the same argument from these authors

This is really odd as we are summing up all positive terms and in the end one gets a negative result. This was a question that already bothered Euler and is generally fixed with the Riemann zeta function. Now, if you talk with a mathematician, you will be warned that such a series is not converging and indeed intermediate results become even more larger as the sum is performed. So, this series should be generally discarded when you meet it in your computations in physics or engineering. We know that things do not stay this way as nature already patched it. The reason is exactly this: Infinity does not exist in nature and whenever one is met nature already fixed it, whatever a mathematician could say. Of course, smarter mathematicians are well aware of this as you can read from Terry Tao’s blog. Indeed, Terry Tao is one of the smartest living mathematicians. One of his latest successes is to have found a problem in the presumed Otelbaev’s proof of the existence of solutions to Navier-Stokes equations, a well-known millennium problem (see the accepted answer and comments here).

This idea is well-known to physicists and when an infinity is met we have invented a series of techniques to remove it in the way nature has chosen. This can be seen from the striking agreement between computed and measured quantities in some quantum field theories, not last the Standard Model. E.g. the gyromagnetic ratio of the electron agrees to one part on a trillion with the measured quantity (see here). This perfection in the computations was never seen before in physics and belongs to the great revolution that was completed by Feynman, Schwinger, Tomonaga and Dyson that we have inherited in the Standard Model, the latest and greatest revolution seen so far in particle physics. We just hope that LHC will uncover the next one at the restart of operations. It is possible again that nature will have found further ways to patch infinities and one of these could be $1+2+3+4+\ldots=-1/12$.

So, we recall one of the greatest principles of physics: Nature patches infinities and use techniques to do it that are generally disgusting mathematicians. I think that diverging series should be taught at undergraduate level courses. Maybe, using the standard textbook by Hardy (see here). These are not just pathologies in an otherwise wonderful world but rather these are the ways nature has chosen to behave!

The reason for me to write about this matter is linked to a beautiful work I did with my colleagues Alfonso Farina and Matteo Sedehi on the way the Tartaglia-Pascal triangle generalizes in quantum mechanics. We arrived at the conclusion that quantum mechanics arises as the square root of a Brownian motion. We have got a paper published on this matter (see here or you can see the Latest Draft). Of course, the idea to extract the square root of a Wiener process is something that was disgusting mathematicians, mostly Didier Piau, that was claiming that an infinity goes around. Of course, if I have a sequence of random numbers, these are finite, I can arbitrarily take their square root. Indeed, this is what one sees working with Matlab that easily recovers our formula for this process. So, what does it happen to the infinity found by Piau? Nothing, but nature already patched it.

So, we learned a beautiful lesson from nature: The only way to know her choices is to ask her.

A. Farina,, M. Frasca,, & M. Sedehi (2014). Solving Schrödinger equation via Tartaglia/Pascal triangle: a possible link between stochastic processing and quantum mechanics Signal, Image and Video Processing, 8 (1), 27-37 DOI: 10.1007/s11760-013-0473-y

## Lubos and divergent series

12/02/2009

I have read this nice post and I have found it really interesting. The reason is the kind of approach of Lubos Motl, being physicist-like, on such somewhat old mathematical matter. The question of divergent series and their summation is as old as at least Euler and there is a wonderful book written by a great British mathematician, G. H. Hardy, that treats this problem here. Hardy is well-known for several discoveries in mathematics and one of this is Ramanujan. He had a long time collaboration with John Littlewood.

Hardy’s book is really shocking for people that do not know divergent series. In mathematics several well coded resummation techniques exist for these series. With a proper choice of one of these techniques a meaning can be attached to them. A typical example can be

$1-1+1-1+\ldots=\frac{1}{2}$

and this is true exactly in the same way is true that the sum of all  integers is -1/12. Of course, this means that discoveries by string theorists are surely others and most important than this one that is just good and known mathematics.

I agree with Lubos that these techniques are not routinely taught to physics students and come out as a surprise also to most mathematics students. I am convinced that Hardy’s book can be used for a very good series of lectures, for a short time, to make people acquainted with this deep matter that can have unexpected uses.

I think that mathematicians have something to teach us that is really profound: Do not throw anything out of the window. It could turn back in an unexpected way.