Disclaimer: This post is somewhat technical.

Recently, I posted a paper on arXiv (see here) claiming that quantum mechanics is the square root of a Wiener process. In order to get my results I have to consider some exotic Itō integrals that Didier Piau showed not existent (see here and here). In my argument I have a critical definition and this is the process that I defined using the sum

so that I assumed the limit exists and is finite. This position appears untenable as Didier showed in the following way. In this case one has ()

and increments are independent so that

Now, if you want to compute the limit in you are in trouble. Just choose and you will get

that is

If you compute these sums you will get finally a term proportional to that blows  up in the limit of increasingly large . The integral simply does not exist from a mathematical standpoint.

Of course, a curse for a mathematician is a blessing for a theoretical physicist, mostly when an infinity appears. Indeed, let us consider the sum

People who have read Hardy’s book know for sure that this sum is just (see also discussion here). This series can be regularized and so the limit can be taken to be finite!

This average is just finite and this is what I would expect for this kind of process. With this idea of regularization, the generalized Itō integral exists and is meaningful. The same idea can be applied to the case with and my argument is just consistent as I show that for the absolute value process enters.

As a theoretical physicist I can say: Piau’s paradox is happily evaded!

Marco Frasca (2012). Quantum mechanics is the square root of a stochastic process arXiv arXiv: 1201.5091v1

There is a very good reason why I was silent in the past days. The reason is that I was involved in one of the most difficult article to write down since I do research (and are more than twenty years now!).  This paper arose during a very successful collaboration with two colleagues of mine: Alfonso Farina and Matteo Sedehi. Alfonso is a recognized worldwide authority in radar technology and last year has got a paper published here about the ubiquitous Tartaglia-Pascal triangle and its applications in several areas of mathematics and engineering. What was making Alfonso unsatisfied was the way the question of Tartaglia-Pascal triangle fits quantum mechanics. It appeared like this is somewhat an unsettled matter. Tartaglia-Pascal triangle gives, in the proper limit, the solution of the heat equation typical of Brownian motion, the most fundamental of all stochastic processes. But when one comes to the Schroedinger equation, notwithstanding the formal resemblance between these two equations, the presence of the imaginary term changes things dramatically. So, a wave packet of a free particle is seen to spread like the square of time rather than linearly. Then, Alfonso asked to me to try to clarify the situation and see what is the role of Tartaglia-Pascal triangle in quantum mechanics. This question is old almost as quantum mechanics itself. Several people tried to explain the probabilistic nature of quantum mechanics through some kind of Brownian motion of space and the most famous of these attempts is due to Edward Nelson. Nelson was able to show that there exists a stochastic process producing hydrodynamic equations from which the Schroedinger equation can be derived. This idea turns out to be a description of quantum mechanics similar to the way David Bohm devised it. So, this approach was exposed to criticisms that can be summed up in a paper by Peter Hänggi, Hermann Grabert and Peter Talkner (see here) denying any possible representation of quantum mechanics as a classical stochastic process.

So, it is clear that the situation appears rather difficult to clarify with such notable works. With Alfonso and Matteo, we have had several discussions and the conclusion was striking: Tartaglia-Pascal triangle appears in quantum mechanics rather with its square root! It appeared like quantum mechanics is not itself a classical stochastic process but the square root of it. This could explain why several excellent people could have escaped the link.

At this point, it became quite difficult to clarify the question of what a square root of a stochastic process as Brownian motion should be. There is nothing in literature and so I tried to ask to trained mathematicians to see if something in advanced research was known (see here). MathOverflow is a forum of discussion for advanced research managed by the community of mathematicians. It met a very great success and this is testified by the fact that practically all the most famous mathematicians give regular contributions to it. Posting my question resulted in a couple of favorable comments that informed me that this question was not known to have an answer. So, I spent a lot of time trying to clarify this idea using a lot of very good books that are available about stochastic processes. So, last few days I was able to get a finite answer: The square of Brownian motion is computable in a standard way with Itō integral reducing to a Brownian motion multiplied by a Bernoulli process. The striking fact is that the Bernoulli process is that of tossing a coin! The imaginary factor emerges naturally out of this mathematical procedure and now the diffusion equation is the Schroedinger equation. The identification of the Bernoulli process came out thanks to the help of Oleksandr Pavlyk after I asking this question at MathStackexchange. This forum is also for well-trained mathematicians but the kind of questions one can put there can also be at a student level. Oleksandr’s answer was instrumental for a complete understanding of what I was doing.

Finally, I decided to verify with the community of mathematicians if all this was nonsense or not and I posted again on MathStackexchange a derivation of the square root of a stochastic process (see here).  But, with my great surprise, I discovered that some concepts I used for the Itō calculus were not understandable at all. I gave them for granted but these were not defined in literature! So, after some discussions, I added important clarifications there and in my paper making clear what I was doing from a mathematical standpoint. Now, you can find all this in my article. Itō calculus must be extended to include all the ideas I was exploiting.

The link between quantum mechanics and stochastic processes is a fundamental one. The reason is that, if one get such a link, an understanding of the fundamental behavior of space-time is obtained. This appears a fluctuating entity but in an unexpected way. This entails a new reformulation of quantum mechanics with the language of stochastic processes. Given this link, any future theory of quantum gravity should recover it.

I take this chance to give publicly a great thank to all these people that helped me to reach this important understanding and that I have cited here. Also mathematicians that appeared anonymously were extremely useful to improve my work. Thank you very much, folks!

Update: After an interesting discussion here with Didier Piau and George Lowther, we reached the conclusion that the definitions I give in my paper to extend the definition of the Ito integral are not mathematically consistent. Rather, when one performs the corresponding Riemann sums one gets diverging results for the interesting values of the exponent and the absolute value. Presently, I cannot see any way to get a sensible definition for this and so this paper should be considered mathematically not consistent. Of course, the idea of quantum mechanics as the square root of a stochastic process is there to stay and to be eventually verified, possibly with different approaches and better mathematics.

Marco Frasca (2012). Quantum mechanics is the square root of a stochastic process arXiv arXiv: 1201.5091v1

Farina, A., Giompapa, S., Graziano, A., Liburdi, A., Ravanelli, M., & Zirilli, F. (2011). Tartaglia-Pascal’s triangle: a historical perspective with applications Signal, Image and Video Processing DOI: 10.1007/s11760-011-0228-6

Grabert, H., Hänggi, P., & Talkner, P. (1979). Is quantum mechanics equivalent to a classical stochastic process? Physical Review A, 19 (6), 2440-2445 DOI: 10.1103/PhysRevA.19.2440

People working in science are well aware that severe criteria are generally used to scrutinize their work, work that must appear on reputable journals where a review by peers decides the goodness or the rejection. This is generally the start of a procedure that can last several years and that should end up with the output of some experiments, at least for experimental science like physics. So, when some people, with none or very few publications get instantaneous fame by media hype the matter is suspicious since the start and some caution is in order. As an example, I would like to remember what happened to my work when someone took the braveness to put it in Wikipedia and the discussion that followed (see here). The final result was its removal after Peter Woit and all his gang claimed my head. This work just stand  up through passing time and Terry Tao agreed on the correctness of the main theorem supporting all this and that was the foundation of the entry into the Yang-Mills article in Wikipedia (see here) after I provided a correct proof (see here). Currently, I keep on working on this and I keep on giving talks in international conferences about.

Lisi’s case is completely different and belongs to those with immediate hype with no substance at all. No serious file of publications just someone that, for some reason very difficult to understand, after a preprint appeared on arXiv became an immediate star. After all that fuss, serious people in the scientific community found serious drawbacks in that preprint that never saw the light in a reputable journal. Rather, Distler and Garibaldi showed that it was simply flawed in its claims as its author (see here). This paper appeared in a very prestigious mathematical physics journal.

In the world of mathematicians, after such a proof of wrongness, one should go off with his tail between his legs. This happened in the case of Deolalikar and the Np vs P Millenium problem and this is the way a sane community just works. But this did not happen in physics as we are coping with this matter even after it was proven wrong and was never seen on any refereed journal. There is an ongoing discussion at Wikipedia and an edit war at the Lisi’s article (see here and here). An interesting criticism is that Lisi’s page is wider than Nobel winners while he does not appear to have similar merits. By my side, I would just add that there are a lot of very good people with tons of publications and citations that would be worth a Wikipedia article and Lisi obtained a large one just thanks to a lot of media fuss. There is very few to say because this is Wikipedia but this is also not the right way to convey scientific information.

In the end, we are just tired of nothingness in science getting all this room. The right information should be conveyed and wrong theories should be simply forgotten everywhere independently on the fact that somebody used someone else.

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

Jacques Distler, & Skip Garibaldi (2009). There is no “Theory of Everything” inside E8 Commun.Math.Phys.298:419-436,2010 arXiv: 0905.2658v3

We have left 2011 with a lot of exciting results from experiments. Neutrinos appear to move a bit faster than expected and Higgs provided some glimpses at CERN. Of course, this kind of Higgs appears somewhat boring at first being in the range of what Standard Model expected. But it is really too early to say something for sure. We expect definite answer for the next summer with a lot more data analyzed by people at CERN.

With the new year, I would like to point out to my readers a couple of nice papers that are really worthwhile reading. About CUDA and lattice QCD, my Portuguese friends, Pedro Bicudo and Nuno Cardoso,  made a relevant step beyond and made available their code for working for a generic SU(N) gauge group (see here, their code is here). As I have some time I will try their code. The work of these people is excellent and making their code worldwide available is really helpful for all our community.

Finally, Axel Maas put forward a revision of his very good review paper (see here). Axel gave important contributions to the current understanding of Yang-Mills theory and his paper yields a lucid description of these ideas that rely on a large effort on lattice computations and functional methods. Often, I complain about the fact that the community at large seems to not consider these lines of research reliable yet to work with. This is not true as the results they were able to get give since now sound results to work with and the most important of these are that Yang-Mill theory has indeed a mass gap and that this theory appears to display a running coupling reaching zero lowering momenta, a completely unexpected result that goes against common wisdom but this is just what lattice put out.

So, let me wish to you a great 2012 and I hope to share with you the excitement physics research is promising.

Nuno Cardoso, & Pedro Bicudo (2011). Generating SU(Nc) pure gauge lattice QCD configurations on GPUs with
CUDA and OpenMP arXiv arXiv: 1112.4533v1

Axel Maas (2011). Describing gauge bosons at zero and finite temperature arXiv arXiv: 1106.3942v2

The conference raised a lot of enthusiasm (see here) to some caution (see here) or skepticism (see here).

Fabiola Gianotti, Rolf Heuer and Guido Tonelli

What makes these hints striking is the fact that both experiments see the excess in the same region where the particle was expected and with the proper rates. It should also be said that, with these data and energy, people at CERN have done an excellent work with the analysis of them. But, of course, it is still possible that we are coping with a fluctuation and the particle is hiding elsewhere or is something else. For sure, next year the puzzle will be completed and also this part of the Standard Model will be part of our textbooks in the right way. What we have here is a completely new situation holding the premises for a clear understanding of one of the greatest question of mankind ever. So, when a child will ask to you: “Mom, what are we made of?” this question will have an answer, an answer arising from the work of a lot of smart people running one of the greatest technological achievement of our history: LHC.

While CERN is calming down rumors (see here), research activity on Yang-Mills theories keeps on going on.  A few days ago, a paper by Axel Weber appeared on arxiv  (see here). As my readers know, having discussed this at length, in these last years there has been a hot debate between the proponents of the so called “scaling solution” and the “decoupling solution” for the propagators and the running coupling of a pure Yang-Mills theory in the infrared limit. Scaling solution describes a scenario with the gluon propagator reaching zero with lowering momenta, a ghost propagator enhanced with respect to the tree level one and the running coupling reaching a finite non zero value in the same limit. Decoupling solution instead is given by a gluon propagator reaching a finite non-zero value at lower momenta, a ghost propagator behaving like the one of a free particle (tree level) and the running coupling going to zero in this limit.. It is quite easy to recognize in the decoupling solution all the chrisms of a trivial infrared fixed point for a pure Yang-Mills theory against common wisdom that pervaded the community for a lot of years. So, for some years, having lattice computations unable to tell which solution was the right one, scaling solution seemed the only one to be physically viable and almost accepted by a large part of the community.

Things started to change after the Lattice Conference in Regensburg on 2007 when some groups where able to display lattice computations on very huge volumes. The striking result was that lattice computations confirmed the decoupling solution against common wisdom. What was really shocking here is that the gluon becomes massive at the expenses of the BRST sysmmetry that seems now to acquire an even more relevant role in the understanding of Yang-Mills theory.

The idea of Axel Weber is to perform an -expansion for the Yang-Mills Lagrangian with a massive term to fix the scale. The striking result he gets is that both the scaling and the decoupling solutions are there but the former is unstable with respect to the renormalization group flow in dimensions greater than 2. So, this computation confirms again the scenario that I and other authors were able to devise.

Today, we have reached a deep understanding of the infrared physics of a Yang-Mills field theory. Scientific community is urged to take a look to the work of these people that could accelerate progress in a large body of physics research.

Axel Weber (2011). Epsilon expansion for infrared Yang-Mills theory in Landau gauge arXiv arXiv: 1112.1157v1

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

Alexander (Sasha) Migdal was a former professor at Princeton University. But since 1996, he is acting as a CEO of a small company. You can read his story from that link. Instead, Marco Bochicchio was a former colleague student of mine at University of Rome “La Sapienza”. He was a couple of years ahead of me. Now, he is a researcher at Istituto Nazionale di Fisica Nucleare, the same of OPERA and a lot of other striking contributions to physics. With Marco we shared a course on statistical mechanics held by Francesco Guerra at the department of mathematics of our university. Today, Marco posted a paper of him on arXiv (see here). I am following these works by Marco with a lot of interest because they contain results that I am convinced are correct, in the sense that are describing the right scenario for Yang-Mills theory. Marco, in this latter work, is referring to preceding publications from Sasha Migdal about the same matter that go back till ’70s! You can find a recollection of these ideas in a recent paper by Sasha (see here). So, what are these authors saying? Using somewhat different approaches than mine (that you can find well depicted here), they all agree that a Yang-Mills theory has a propagator going like

being some numbers and is given by the zeros of some Bessel functions. This last result seems quite different from mine that I get explicitly but this is not so because, in the asymptotic regime, and zeros for the cosine go like and then, my spectrum is easily recovered in the right limit. The right limit is properly identified by Sasha Migdal from Padè approximants for the propagator that start from the deep Euclidean region , being an arbitrary energy scale entering into the spectrum. So, the agreement between the scenario proposed by these authors and mine is practically perfect, notwithstanding different mathematical approaches are used.

The beauty of these conclusions is that such a scenario for a Yang-Mills theory is completely unexpected but it is what is needed to grant confinement. So, the conclusion about the questions of mass gap and confinement is approaching. As usual, we hope that the community will face these matters as soon as possible making them an important part of our fundamental knowledge.

Marco Bochicchio (2011). Glueballs propagators in large-N YM arXiv arXiv: 1111.6073v1

Alexander Migdal (2011). Meromorphization of Large N QFT arXiv arXiv: 1109.1623v2

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3

The great news for me, in this week, has been the acceptance of my paper of QCD at finite temperature in Physical Review C (see here). This chance materialized after the excellent work of the referee that helped me to improve the paper in a significant way. For a good paper, such a way to review is a fundamental one and should be a rule. I have discussed this paper previously in my blog (see here and here) and I presented its content in a conference in Paris this year (see here). The contribution to proceedings is this.

I think that the main conclusions of this paper that should be emphasized is that the low-energy limit of QCD is a non-local Nambu-Jona-Lasinio model, a critical point exists at finite temperature with zero mass and zero chemical potential and that the instanton liquid picture of the vacuum of QCD is a very good one. These are fundamental questions in QCD that were waiting for an answer for so long.

I would like to thank all the people that, with their efforts and interest about my work, helped me to get these results published, in the end, in such an important journal.

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature arXiv arXiv: 1105.5274v4

Marco Frasca (2011). Low-energy limit of QCD at finite temperature arXiv arXiv: 1110.0096v1

A couple of fundamental great news, well one is just a rumor, is hitting scientific community today.

Higgs search

At Paris Conference, Gigi Rolandi addressed his talk on combination for LHC and Tevatron. This picture has been waited for a long time since the excellent work of Phil Gibbs at his blog (see here for an account of this). So far, this combination accounted just for a luminosity and what is obtained is that no excess greater than is observed on all the range starting from 114 GeV to near 600 GeV. I give here, as done by other bloggers, the picture

Now, the most promising region seems to be at high mass but we are always around . The great news here, but it is an uncontrolled rumor, is given at Jester’s blog: Also with no excess greater than is seen in the low mass region! Standard model Higgs seems to be ruled out and the physics here is somewhat different. My view is that if it is proven true that such a scalar particle exists and has a high mass, something unacceptable so far for the standard model, also supersymmetry will be proven true (see here).

OPERA

OPERA Collaboration confirmed their measurements on the speed of neutrinos. This is a major breakthrough in physics and a new version of their preprint is appeared on arXiv today (see here).  This will soon be published on JHEP. So, no more discussions whatsoever but the last word is left to other independent measurements. This is really a breaking news for physics and my personal view is that this should represent a first example of a measurement that could have some impact in the area of quantum gravity. For a fine account, as usual, you can read here.

These are the promises for exciting time ahead. Stay tuned!

Update: Dennis Overbye commented on OPERA new results on New York Times (see here). A few comments from reputable scientists are worth reading.

Marco Frasca (2010). Mass generation and supersymmetry arXiv arXiv: 1007.5275v2

The OPERA Collaboraton: T. Adam, N. Agafonova, A. Aleksandrov, O. Altinok, P. Alvarez Sanchez, A. Anokhina, S. Aoki, A. Ariga, T. Ariga, D. Autiero, A. Badertscher, A. Ben Dhahbi, A. Bertolin, C. Bozza, T. Brugière, R. Brugnera, F. Brunet, G. Brunetti, S. Buontempo, B. Carlus, F. Cavanna, A. Cazes, L. Chaussard, M. Chernyavsky, V. Chiarella, A. Chukanov, G. Colosimo, M. Crespi, N. D’Ambrosio, G. De Lellis, M. De Serio, Y. Déclais, P. del Amo Sanchez, F. Di Capua, A. Di Crescenzo, D. Di Ferdinando, N. Di Marco, S. Dmitrievsky, M. Dracos, D. Duchesneau, S. Dusini, J. Ebert, I. Efthymiopoulos, O. Egorov, A. Ereditato, L. S. Esposito, J. Favier, T. Ferber, R. A. Fini, T. Fukuda, A. Garfagnini, G. Giacomelli, M. Giorgini, M. Giovannozzi, C. Girerd, J. Goldberg, C. Göllnitz, D. Golubkov, L. Goncharov, Y. Gornushkin, G. Grella, F. Grianti, E. Gschwendtner, C. Guerin, A. M. Guler, C. Gustavino, C. Hagner, K. Hamada, T. Hara, M. Hierholzer, A. Hollnagel, M. Ieva, H. Ishida, K. Ishiguro, K. Jakovcic, C. Jollet, M. Jones, F. Juget, M. Kamiscioglu, J. Kawada, S. H. Kim, M. Kimura, E. Kiritsis, N. Kitagawa, B. Klicek, J. Knuesel, K. Kodama, M. Komatsu, U. Kose, I. Kreslo, C. Lazzaro, J. Lenkeit, A. Ljubicic, A. Longhin, A. Malgin, G. Mandrioli, J. Marteau, T. Matsuo, N. Mauri, A. Mazzoni, E. Medinaceli, F. Meisel, A. Meregaglia, P. Migliozzi, S. Mikado, D. Missiaen, K. Morishima, U. Moser, M. T. Muciaccia, N. Naganawa, T. Naka, M. Nakamura, T. Nakano, Y. Nakatsuka, V. Nikitina, F. Nitti, S. Ogawa, N. Okateva, A. Olchevsky, O. Palamara, A. Paoloni, B. D. Park, I. G. Park, A. Pastore, L. Patrizii, E. Pennacchio, H. Pessard, C. Pistillo, N. Polukhina, M. Pozzato, K. Pretzl, F. Pupilli, R. Rescigno, F. Riguzzi, T. Roganova, H. Rokujo, G. Rosa, I. Rostovtseva, A. Rubbia, A. Russo, O. Sato, Y. Sato, J. Schuler, L. Scotto Lavina, J. Serrano, A. Sheshukov, H. Shibuya, G. Shoziyoev, S. Simone, M. Sioli, C. Sirignano, G. Sirri, J. S. Song, M. Spinetti, L. Stanco, N. Starkov, S. Stellacci, M. Stipcevic, T. Strauss, S. Takahashi, M. Tenti, F. Terranova, I. Tezuka, V. Tioukov, P. Tolun, N. T. Tran, S. Tufanli, P. Vilain, M. Vladimirov, L. Votano, J. -L. Vuilleumier, G. Wilquet, B. Wonsak, J. Wurtz, C. S. Yoon, J. Yoshida, Y. Zaitsev, S. Zemskova, & A. Zghiche (2011). Measurement of the neutrino velocity with the OPERA detector in the CNGS beam arXiv arXiv: 1109.4897v2