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

I also had a question answered at math stack exchange, one having to do with the bases of a finite Hilbert space: http://math.stackexchange.com/questions/28413 . The paper associated with it is now under review at Jour. Math. Phys.

I have a question on the equation (10), which you call the “Schroedinger equation” for a free particle. But doesn’t the term proportional to the first spatial derivative of psi affect its temporal evolution? This way it looks like some kind of damped oscillator with complex damping constant: with a straightforward calculation you get solutions which indeed oscillate like the plain-wave solutions of the free particle equation, but for great times the solutions tend to the trivial zero solution. Did I interpretate in a wrong way? I’m very interested in understanding this article, because I was thinking about it on my own, and with your permission I would like to talk about it during a seminar within a course of Statistical Physics of Complex Systems I’m attending.

Dear Giulio,

Thank you a lot for your interest on my work. Please, note that there is a problem in the definition of the stochastic integral as it is given there. Sums like do not appear to converge for $0<\alpha<1$ and so the integral in the Riemann sense does not exist. So, I am in need for a proper definition of this integral to make all the argument consistent. I am open to whatever good proposal.

About the form of the Schroedinger equation you are right, there a term proportional to multiplied by the derivative of the wave function. This should be corrected using a potential that can remove it. It is interesting to note that, in 3 dimension, I would expect a gauge coupled equation to come out.

Regards,

Marco

The presence of the square root suggests that pure density matrices are a better description of pure states than state vectors, and of course mixed density matrices can represent states that state vectors alone cannot.

I think that the primary utility of state vectors and spinors is that their mathematics is linear. That is, they’re advantageous for our representation of reality rather than (necessarily) being what most naturally represents reality. So I see them as a mathematical tool; the fundamental physics is in density matrix form. (Or propagators.)

One way of seeing that QM is inherently nonlinear is to note that tripling a wave function doesn’t create a model of a quantum state in any way different from the original one. This is in distinction to a truly linear theory like classical E&M where tripling the charges, currents, voltages, etc., creates a new state that represents a different physical state than the original.

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Hi,

Big fan of this work. No words to tell how much.

Would love to cooperate in research with you.

one thing I didn’t figure out yet:

in (8), the Kolmogorov (btw, how did you choose bwd/fwd?) is about the pdf. but you mention psi there. so, is that psi the wave function or its norm squared?

Hi Ohad,

Thank you for your interest about my work. In this case or pdf are the same thing but here happens to be complex and the interpretation should be somewhat changed.

Marco

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