Ending and consequences of Terry Tao’s criticism



Summer days are gone and I am back to work. I thought that Terry Tao’s criticism to my work was finally settled and his intervention was a good one indeed. Of course, people just remember the criticism but not how the question evolved since then (it was 2009!). Terry’s point was that the mapping given here between the scalar field solutions and the Yang-Mills field in the classical limit cannot be exact as it is not granted that they represent an extreme for the Yang-Mills functional. In this way the conclusions given in the paper are not granted being based on this proof. The problem can be traced back to the gauge invariance of the Yang-Mills theory that is explicitly broken in this case.

Terry Tao, in a private communication, asked me to provide a paper, to be published on a refereed journal, that fixed the problem. In such a case the question would have been settled in a way or another. E.g., also a result disproving completely the mapping would have been good, disproving also my published paper.

This matter is rather curious as, if you fix the gauge to be Lorenz (Landau), the mapping is exact. But the possible gauge choices are infinite and so, there seems to be infinite cases where the mapping theorem appears to fail. The lucky case is that lattice computations are generally performed in Landau gauge and when you do quantum field theory a gauge must be chosen. So, is the mapping theorem really false or one can change it to fix it all?

In order to clarify this situation, I decided to solve the classical equations of the Yang-Mills theory perturbatively in the strong coupling limit. Please, note that today I am the only one in the World able to perform such a computation having completely invented the techniques to do perturbation theory when a perturbation is taken to go to infinity (sorry, no AdS/CFT here but I can surely support it). You will note that this is the opposite limit to standard perturbation theory when one is looking for a parameter that goes to zero. I succeeded in doing so and put a paper on arxiv (see here) that was finally published the same year, 2009.

The theorem changed in this way:

The mapping exists in the asymptotic limit of the coupling running to infinity (leading order), with the notable exception of the Lorenz (Landau) gauge where it is exact.

So, I sighed with relief. The reason was that the conclusions of my paper on propagators were correct. But these hold asymptotically in the limit of a strong coupling. This is just what one needs in the infrared limit where Yang-Mills theory becomes strongly coupled and this is the main reason to solve it on the lattice. I cited my work on Tao’s site, Dispersive Wiki. I am a contributor to this site. Terry Tao declared the question definitively settled with the mapping theorem holding asymptotically (see here).

In the end, we were both right. Tao’s criticism was deeply helpful while my conclusions on the propagators were correct. Indeed, my gluon propagator agrees perfectly well, in the infrared limit, with the data from the largest lattice used in computations so far  (see here)

Comparison with lattice dataAs generally happens in these cases, the only fact that remains is the original criticism by a great mathematician (and Terry is) that invalidated my work (see here for a question on Physics Stackexchange). As you can see by the tenths of papers I published since then, my work stands and stands very well. Maybe, it would be time to ask the author.

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical
Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Attilio Cucchieri, & Tereza Mendes (2007). What’s up with IR gluon and ghost propagators in Landau gauge? A puzzling answer from huge lattices PoS LAT2007:297,2007 arXiv: 0710.0412v1

Fooling with mathematicians



I am still working with stochastic processes and, as my readers know, I have proposed a new view of quantum mechanics assuming that at the square root of a Wiener process can be attached a meaning (see here and here). I was able to generate it through a numerical code. A square root of a number can always be taken, irrespective of any deep and beautiful mathematical analysis. The reason is that this is something really new and deserves a different approach much in the same way it happened to the Dirac’s delta that initially met with skepticism from the mathematical community (simply it did not make sense with the knowledge of the time). Here I give you some Matlab code if you want to try by yourselves:

nstep = 500000;
dt = 50;
B = normrnd(0,sqrt(dt/nstep),1,nstep);
dB = cumsum(B);
% Square root of the Brownian motion

Nothing can prevent you from taking the square root of  a number as is a Brownian displacement and so all this has a well sound meaning numerically. The point is just to understand how to give this a full mathematical meaning. The wrong approach in this case is just to throw all away claiming all this does not exist. This is exactly the behavior I met from Didier Piau. Of course, Didier is a good mathematician but simply refuses to accept the possibility that such concepts can have a meaning at all based on what has been so far coded in the area of stochastic processes. This notwithstanding that they can be easily computed on your personal computer at home.

But this saga is not over yet. This time I was trying to compute the cubic root of a Wiener process and I posted this at Mathematics Stackexchange. I put this question with  the simple idea in mind to consider a stochastic process with a random mean and I did not realize that I was provoking a small crisis again. This time the question is the existence of the process {\rm sign}(dW). Didier Piau immediately wrote down that it does not exist. Again I give here the Matlab code that computes it very easily:

nstep = 500000;
dt = 50;
B = normrnd(0,sqrt(dt/nstep),1,nstep);
dB = cumsum(B);
% Sign and absolute value of a Wiener process
dS = sign(dB);
dA = dB./dS;

Didier Piau and a colleague of him just complain on the Matlab way the sign operation is performed. My view is that it is all legal as Matlab takes + or – depending on the sign of the displacement, a thing that can be made by hand and that does not imply anything exotic.  What it is exotic here it the strong opposition this evidence meets notwithstanding is easily understandable by everybody and, of course, easily computable on a tabletop computer. The expected distribution for the signs of Brownian displacements is a Bernoulli with p=1/2. Here is the histogram from the above code

Histogram sign(dW)This has mean 0 and variance 1 as it should for N=\pm 1 and p=\frac{1}{2} but this can be verified after some Montecarlo runs. This is in agreement with what I discussed here at Mathematics Stackexchange as a displacement in a Brownian motion is a physics increment or decrement of the moving particle and has a sign that can be managed statistically. My attempt to compare all this to the case of Dirac’s delta turns out into a complain of overstatement as delta was really useful and my approach is not (but when Dirac put forward his idea this was just airy-fairy for the time). Of course, a reformulation of quantum mechanics would be a rather formidable support to all this but this mathematician does not seem to realize it.

So, in the end, I am somewhat surprised by the behavior of the community against novelties. I can understand skepticism, it belongs to our profession, but for facing new concepts that can be easily checked numerically to exist I would prefer a more constructive behavior trying to understand rather than an immediate dismissal. It appears like history of science never taught anything leaving us with a boring repetition of stereotyped reactions to something that instead would be worthwhile further consideration. Meanwhile, I hope my readers will enjoy playing around with these new computations using some exotic mathematical operations on a stochastic process.

Marco Frasca (2012). Quantum mechanics is the square root of a stochastic process arXiv arXiv: 1201.5091v2

What is Science?



Reading this Lubos’ post about a very good site (this one) I entered into the comment area and I have found the following declaration by him:

Science is a meritocracy where answers are determined by objective criteria, and for most of the difficult questions, only one or a few people know the right answer and the scientific method exists to isolate this special right answer…

Of course, I subscribe this that is widely known to people doing research. I would just change the word “meritocracy” by “dictatorship of truth”. But there is an intermediate age where the truth takes time to become acclaimed and this is time for opinions and before to become aware of the people that firstly reached the goal, there is a struggle for the truth to be acquired. I would like to remember here the status of quantum field theory in the sixties when bootstrap and similar failures appeared as a paradigm and very few brave people were doing research in the right directions taking us to the triumph of today. In this kind of dynamics, at a first stage it is very difficult to be able to tell, also for very well trained people, where the right track is lying. In physics our luck resides in experiments. This makes things simpler when technology helps us to perform them otherwise time to decide for the best are increasingly longer. So, merit as claimed by Lubos is something that sets in at the very end of the process.

In my specific field of activity, QCD, we are in a better situation as a lot of laboratories around the World have facilities to perform important measurements to reach the goal. And this situation is even better as we can use powerful computers to solve the theory. My view as a physicist is that, without a sound comparison of the spectrum of the theory with experiments, nobody can claim to have properly solved the mass gap problem. All my present effort is going into this direction because there is nothing more exciting than having hit the right behavior of Nature (our mother not the bitch…). I take this chance to remember here the effort in this direction of Silvio Sorella, that with the help of other fine colleagues, is going to show how his approach indeed fulfill these expectations of glueball masses (see here). These authors give a correct idea about what is the  right approach to be followed for the problem of low-energy QCD.

Finally, I would like to emphasize the relevance of sites like the one pointed out by Lubos. This site has also been posted by Sean Carroll (see here) in his blog. I have pointers to my blog there and in the more successful Mathoverflow. Unfortunately, I have no much time to spend on contributing to these sites but these are very good places to know about science and the right one. So, this is also my invitation for my readers to contribute to them actively.

D. Dudal, M. S. Guimaraes, & S. P. Sorella (2010). Glueball masses from an infrared moment problem and nonperturbative
Landau gauge arxiv arXiv: 1010.3638v3

Quest for truth


In these days, in Italian theaters and after a two year struggle for distribution, it is possible to see Agora. This is the story of Hypatia,

Hypatia by Raphael

the first female being a scientist and a philosopher. She taught in Alexandria toward the end of the fourth century and the beginning of the fifth. Her struggle against the forces pushing toward Middle Age ended with a failure and she was brutally killed, all her works were burnt and today we know some of her ideas because some of her disciples wrote about that. There is a deep lesson to be learned here. Doing science is a quest for truth but this quest happens inside our world, with forces pushing to avert us from the right track, to stop us again and again or to use us or our ideas for their not so clear aims. So, our community must protect and help gifted people and have their ideas, when we are sure these are correct, spread as soon as possible for the common good and mankind at large. We must avoid to ask ourselves, after sixteen centuries: How many times must Hypatia die again?

Wikipedia and abuses


Till now, I have avoided to feed this flooding about abuses on Wikipedia. But I think that a few words are needed in order to clarify my position and to let my own point of view widely known.

The problem started when Peter Woit took a look at the Yang-Mills entry of Wikipedia. He has found a section apparently self-promoting my work. What was about this section? The title said “Integrable solutions of classical Yang-Mills equations and QFT “. I think that there is a lot to say about this matter as classical solutions of Y-M exist and is well acquired matter. But in this section a class of solutions were put, cited in the Smilga’s book, that I have generalized and introduced in my papers.  Should they be there? I think yes as they belong to the class of solutions stated in the title. They appear to be too recent for inclusion but this is plain mathematics. Mathematics is a two-way switch: It is either right or wrong and so, if these solutions are right, they should be there as a bookkeeping for the readers.

The worst question is anyhow self-promotion. On this ground Peter Woit did worst: He is self-promoting his book (see here , thank you Lubos). Promoting a book means to earn money for the author while promoting a scientific idea may have the right side that, being the idea good, a good service has been done to the community.

The worst aspect of the story has been the intervention of Woit through his blog. This is a perfect war machine that when activated may leave a lot of casualties. People should be smart at their defense as otherwise the risk is to be counted in that number. A flood of people moved toward Wikipedia with any means trying to remove the questioned section and attacking me and whoever has written it. A lot of comments in Woit’s blog was posted attacking me. I was forced to introduce moderation for comments in my blog. Of course this appears like a kind of lynching without any understanding of scientific merit. Curators of Wikipedia decided that majority was right and Woit have had his win: The section was finally removed.

What next? This situation is quite interesting by my side. The reason is that the physical matter is Yang-Mills theory that is one of the biggest open problems both in phyics and mathematics. There is a lot of very good people working on that in this moment and my view is that a complete understanding is at hand. Ask yourself this question: What would be Woit’s position if I am right? This is not like string theory that we do not know when a confirmation will be at hand. Here we have computers, accelerators and a lot of smart people crunching this problem. In a very short time an eventual Woit’s error will be exposed. And by irony, Wikipedia’s entry will be updated with my ideas. Much better than now.

The question to be asked is: Should Wikipedia support new material? Since the editors of scientific entries in Wikipedia are scientists themselves one cannot ask them impartiality. Science is a dynamic endeavor and Wikipedia a dynamic source of information. They should be merged to meet each other in the right way.

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