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

## A briefer history of Stephen Hawking

10/05/2011

I am always happy to point out to my readers worthwhile readings from the web and mostly from significant sites. One of my preferred ones is New York Times. This time there is an interview by Claudia Dreifus to the great physicist Stephen Hawking. Hawking is well-known for his fundamental contributions to cosmology and our current understanding of black hole physics positing the foundations to any future theory of quantum gravity. Hawking is also known for his enduring struggle against the motor neuron disease that afflicts him since the times of his youth. Notwithstanding such a hurdle he was able to find his way becoming one of the greatest living theoretical physicists. Hawking has been Lucasian Professor at Cambridge University and left the chair due to the age succeeded by Michael Green, a well-known string theorist.

Claudia in this interview gives relevance to Hawking’s disease and tries to give a picture on how Stephen was able to reach such high goals despite of this. It is also interesting to point out a couple of questions about LHC and the recent finding at Fermilab of a claim for a new particle discover (see here). All this makes the interview a worthwhile reading.

## Physicists and finance

10/03/2009

Just to point out an interesting article in the New York Times (see here) about physicists working for financial markets. I have known some years ago a physicist that took this decision rather than keeping on living as a postdoc with very few bucks to maintain his family. I have never seen him again but I think he did not regret his choice.

The article is interesting as points out as a physicist working for certainties can become a scientist on uncertainties. Looking at their salaries you should interchange above adjectives.

## The whisper of the universe

01/12/2008

Current experimental situation about dark matter is becoming increasingly exciting as results are coming out. I would like to emphasize the work of Piergiorgio Picozza that has been my professor teaching me the foundations of nuclear physics. He is currently the leader of PAMELA, an experiment that produced a lot of rumors and striking results as well. The New York Times has, as usual, published a beautiful article on the current situation about dark matter (see here).  It is worthwhile reading and makes quite clear that the situation is becoming stunning day after day but avoids anyway to cite the recent results of CDF Collaboration. Indeed, the link between dark matter and CDF finding is yet in its infancy. We are all awaiting for other confirmations as always happens to experiments. I am confident that these will not miss for too long. Whoever is interested about the relevant discussions about I just point you to the Dorigo’s blog being him one of the authors of the CDF’s paper. This provoked a lot of rumors in the blogosphere that sometime degenerated in hot discussions. We just stay tuned to learn more about.

## Dumbness does not pay

30/09/2008

Today I have read on New York Times (see here) that the lawsuit to halt LHC has been dismissed by a federal judge in Honolulu. A wasting of time and resource against missing of common sense and serching for limelight. Very bad. A story to be forgotten as fastly as possible.