Quantum mechanics and stochastic processes: Revised paper posted

31/01/2012

After having fixed the definition of the extended Itō integral, I have posted a revised version of my paper on arXiv (see here). The idea has been described here. A full account of this story is given here. The interesting aspect from a physical standpoint is the space that is fluctuating both for a Wiener process and a Bernoulli process, the latter representing simply the tossing of a coin. We can sum up everything in the very simple formula

$dX(t)=[dW(t)+\beta dt]^\frac{1}{2}.$

The constant $\beta$ to be properly fixed to recover Schrödinger equation.

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

27/01/2012

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 $|dW(t)|$ that I defined using the sum

$S_n=\sum_{i=1}^n|W(t_i)-W(t_{i-1})|$

so that I assumed the limit $\lim_{n\rightarrow\infty}\langle S_n^2\rangle$ exists and is finite. This position appears untenable as Didier showed in the following way. In this case one has ($s,\ t>0$)

$\langle|W(t+s)-W(t)|\rangle=\sqrt{2s/\pi}$

and increments are independent so that $i\ne k$

$\langle|W(t_i)-W(t_{i-1})||W(t_k)-W(t_{k-1})|\rangle=$

$\langle|W(t_i)-W(t_{i-1})|\rangle\langle|W(t_k)-W(t_{k-1})|\rangle=\frac{2}{\pi}\sqrt{t_i-t_{i-1}}\sqrt{t_k-t_{k-1}}.$

Now, if you want to compute the limit in $L^2$ you are in trouble. Just choose $t_i=i/n$ and you will get

$\langle\left(\sum_{i=1}^n|W(t_i)-W(t_{i-1})|\right)^2\rangle$

that is

$\frac{2}{\pi}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n.$

If you compute these sums you will get finally a term proportional to $n$ that blows  up in the limit of increasingly large $n$. 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

$\sum_{i=1}^\infty=1+1+1+1+\ldots$

People who have read Hardy’s book know for sure that this sum is just $-1/2$ (see also discussion here). This series can be regularized and so the limit can be taken to be finite!

$\langle S_n^2\rangle\rightarrow\ {\rm finite\ value}.$

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 $\int_{t_0}^tG(t')|dW(t')|$ exists and is meaningful. The same idea can be applied to the case $\int_{t_0}^tG(t')(dW(t'))^\alpha$ with $0<\alpha<1$ and my argument is just consistent as I show that for $(dW(t))^\frac{1}{2}$ 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

Quantum mechanics and the square root of Brownian motion

25/01/2012

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 $0<\alpha<1$ 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.

Further update:  I have posted a revised version of the paper with a proper definition of this generalized class of Ito integrals (see here).

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

Nothingness in science: Lisi’s case

08/01/2012

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

A new year full of promises

03/01/2012

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