Tartaglia-Pascal triangle and quantum mechanics

The paper I wrote with Alfonso Farina and Matteo Sedehi about the link between the Tartaglia-Pascal triangle and quantum mechanics is now online (see here). This paper contains as a statement my theorem that provides a connection between the square root of a Wiener process and the Schrödinger equation that arose a lot of interest and much criticisms by some mathematicians (see here). So, it is worthwhile to tell how all this come about.

On fall 2011, Alfonso Farina called me as he had an open problem after he and his colleagues got published a paper on Signal, Image and Video Processing, a journal from Springer, where it was shown how the Tartaglia-Pascal triangle is deeply connected with diffusion and the Fourier equation. The connection comes out from the binomial coefficients, the elements of the Tartaglia-Pascal triangle, that in some limit give a Gaussian and this Gaussian, in the continuum, is the solution of the Fourier equation of heat diffusion. This entails a deep connection with stochastic processes. Stochastic processes, for most people working in the area of radar and sensors, are essential to understand how these device measure through filtering theory. But, in the historic perspective Farina & al. put their paper, they were not able to get a proper connection for the Schrödinger equation, notwithstanding they recognized there is a deep formal analogy with the Fourier equation. This was the open question: How to connect Tartaglia-Pascal triangle and Schrödinger equation?

People working in quantum physics are aware of the difficulties researchers have met to link stochastic processes a la Wiener and quantum mechanics. Indeed, skepticism is the main feeling of all of us about this matter. So, the question Alfonso put forward to me was not that easy. But Alfonso & al. paper contains also a possible answer: Just start from discrete and then go back to continuum. So, the analog of the heat equation is the Schrödinger equation for a free particle and its kernel and, indeed, the evolution of a Gaussian wave-packet can be managed on the discrete and gives back the binomial coefficient. What you get in this way are the square root of binomial coefficients. So, the link with the Tartaglia-Pascal triangle is rather subtle in quantum mechanics and enters through a square root, reminiscent of the Dirac’s work and his greatest achievement, Dirac equation. This answered Alfonso’s question and in a way that was somewhat unexpected.

Then, I thought that this connection could be deeper than what we had found. I tried to modify Itō calculus to consider fractional powers of a Wiener process. I posted my paper on arxiv and performed both experimental and numerical computations. All this confirms my theorem that the square root of a Wiener process has as a diffusion equation the Schrödinger equation. You can easily take the square root of a natural noise (I did it) or compute this on your preferred math software. It is just interesting that mathematicians never decided to cope with this and still claim that all this evidence does not exist, basing their claims on a theory that can be easily amended.

We have just thrown a seed in the earth. This is our main work. And we feel sure that very good fruits will come out. Thank you very much Alfonso and Matteo!

Farina, A., Frasca, M., & Sedehi, M. (2013). Solving Schrödinger equation via Tartaglia/Pascal triangle: a possible link between stochastic processing and quantum mechanics Signal, Image and Video Processing DOI: 10.1007/s11760-013-0473-y

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

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, 7 (1), 173-188 DOI: 10.1007/s11760-011-0228-6

A first paper on square root of a Brownian motion and quantum mechanics gets published!

Following my series of posts on the link between the square root of a stochastic process and quantum mechanics (see here, here, here, here, here), that I proved to exist both theoretically and experimentally, I am pleased to let you know that the first paper of my collaboration with Alfonso Farina and Matteo Sedehi was finally accepted in Signal, Image and Video Processing. This paper contains the proof of what I named the “Farina-Frasca-Sedehi proposition” in my paper that claims that for a well localized free particle there exists a map between the wave function and the square root of binomial coefficients. This finally links the Pascal-Tartaglia triangle, given through binomial coefficients, to quantum mechanics and closes a question originally open by Farina and collaborators on the same journal (see here). My theorem about the square root of a stochastic process also appears in this article but without a proof.

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

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

Quantum mechanics and the square root of Brownian motion

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.

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