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

Numerical evidence for the square root of a Wiener process


Brownian motion is a very kind mathematical object being very keen to numerical simulations. There are a plenty of them for any platform and software so that one is able to check very rapidly the proper working of a given hypothesis. For these aims, I have found very helpful the demonstration site by Wolfram and specifically this program by Andrzej Kozlowski. Andrzej gives the code to simulate Brownian motion and compute Itō integral to verify Itō lemma. This was a very good chance to check my theorems recently given here by some numerical work. So, I have written a simple code on Matlab that I give here (rename from .doc to .m to use with Matlab).

Here is a sample of output:

As you could note, the agreement is almost perfect. I have had to rescale with a multiplicative factor as the square root appears somewhat magnified after the square but the pattern is there. You can do checks by yourselves. So, all my equations are perfectly defined as is a possible square root of a Wiener process.

Of course, improvements, advices or criticisms are very welcome.

Update: I have simplified the code and added a fixed scale factor to make identical scale. The code is available at Simulation. Here is an example of output:

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

Evading Piau’s paradox


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


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)


and increments are independent so that i\ne k



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


that is


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


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

Dispersive Wiki


Since I was seventeen my great passion has been the solution of partial differential equations. I used an old book written by Italian mathematicians to face for the first time the technique of variable separation applied to the free Schrödinger equation. The article was written by Paolo Straneo, professor at University of Genova in the first part of the last century and Einstein’s friend, and from it I was exposed to quantum theories in a not too simpler way. At eighteen, some friends of mine, during my vacation in Camdridge, gave to me my first book of mathematics on PDEs: François Treves, Basic Linear Partial Differential Equations. You can find this book at low cost from Dover (see here).

Since then I have never given up with my passion with this fundamental part of mathematics and today I am a professional in this area of research.  As a professional in this area, important references come from the work of Terry Tao (see also his blog), the Fields medalist. Terry, together with Jim Colliander at University of Toronto, manage a Wiki, Dispersive Wiki, with the aim to collect all the knowledge about differential equations that are at the foundation of dispersive effects. Most of you have been exposed at their infancy with the wave equation. Well, this represents a very good starting point. On the other side, it would be helpful to add some contributions for Einstein or Yang-Mills equations. Indeed, Dispersive Wiki is open to all people that, like me, is addicted to PDEs and all matter around them.

I have had the chance to write some contributions to Dispersive Wiki. Currently, I am putting down some lines on Yang-Mills equations (I did it before but this was recognized as self-promotion… just look at the discussion there), Dirac-Klein-Gordon equations and other articles. I think it would be important to help Jim and Terry in their endeavor as PDEs are the bread and butter of our profession and to have on-line such a bookkeeping of results would be extremely  useful. Just take your time to give a look.

Physics of the Riemann Hypothesis


In this blog I discuss frequently about one of the Clay Institute’s Millenium Prize problems: Mass gap and existence of a quantum Yang-Mills theory. Sometime I also used the Perelman’s theorem containing Poincarè’s conjecture to discuss about some properties of quantum gravity and also Cramer-Rao statistical bound. Today on arxiv I have found a beautiful review paper by Daniel Schumayer and David Hutchinson about Riemann hypothesis, another Millenium problem, and physics (see here). This question remained unsolved for almost 150 years since now. The relevance of the understanding of this conjecture relies on the possibility to give a function decribing the distribution of prime numbers.

The formulation of Riemann hypothesis is embarassingly simple. Riemann function is defined in a very simple way as


This function has a set of trivial zeros at all even negative integers and a set of nontrivial zeros. Riemann hypothesis claims that

All nontrivial zeros of \zeta(s) have the form \rho=\frac{1}{2}+it, being t a real number.

This is the eighth problem of Hilbert that gave also the name we are using today to this question. Simple as may seem the question, it baffled mathematicians efforts since today. But, as happens to most mathematics, it can be found applied in Nature and it is tempting to think to reproduce in a lab what appears a complicated mathematical problem and read the answer directly from experiments. Indeed, such a road was definitely open in 1999 when Michael Berry (the one of the phase) and Jon Keating put forward an important conjecture relating quantum systems and Riemann hypothesis. You can find this cornerstone paper here. But since then the hunt was open to find other connections amenable to a treatment in physics. Schumayer and Hutchinson give an extensive review of them in their paper. This view opens up the possibility of a solution through physics of this fundamental question. Surely, we are assisting again at an interesting interwining between these fundamental disciplines of science.

Daniel Schumayer, & David A. W. Hutchinson (2011). Physics of the Riemann Hypothesis arxiv arXiv: 1101.3116v1

Berry, M., & Keating, J. (1999). The Riemann Zeros and Eigenvalue Asymptotics SIAM Review, 41 (2) DOI: 10.1137/S0036144598347497

Current status of Yang-Mills mass gap question


I think that is time to make a point about the question of mass gap existence in the Yang-Mills theory. There are three lines of research in this area: Theoretical, numerical and experimental. I can suppose that the one that mostly interests my readers is the theoretical one. I would like to remember that, in order to get a Millenium Prize, one also needs to prove the existence of the theory. This makes the problem far from being trivial.

As for today, the question of existence of the mass gap both for scalar field theories and Yang-Mills theory should be considered settled. Currently there are two papers of mine, here and here both published in archival journals, proving the existence of the mass gap and give it in a closed analytical form. A proof has been also given by Alexander Dynin at Ohio State University here. Alexander does not give the mass gap in a closed form but gets a lower bound that permits him to conclude that Yang-Mills theory has a discrete spectrum with a mass gap. This is enough to declare this part of the problem solved. It is interesting to note that, differently from Poincaré conjecture, this solution does not require a mathematics that is too much complex. This can be understood from the fact that the corresponding classical equations of the theory already admit  massive solutions of free particle. The quantum theory can be built on these solutions and all this boils down to a trivial fixed point in the infrared for the quantum theory. Such a trivial fixed point, that explains also the lower bound Alexander is able to find, is a good news: We have a set of asymptotic states at diminishing momenta that can be used to do perturbation theory and do computations for physics! The reason why these relevant mathematical results did not get the proper exposition so far escape me and enters into the realm of things that I do not know. It is true that in this area there is a lot of caution and this can be understood as this problem received a lot of attention after Witten and Jaffe proposed it for a big money prize.

But, as I have already said, this problem has two questions to be answered and while computing the mass gap is quite easy, the other question is rather involved. To prove the existence of a quantum field theory is not a trivial matter and, for sure, we know that the Wiener integral exists and the Feynman integral does not (so far and only for mathematicians). What I prove in my papers is that the Euclidean theory exists for the scalar field theory (thanks to Glimm and Jaffe that already proved this) and that this theory matches the Yang-Mills theory in the limit of the gauge coupling going to infinity. It should be an asymptotic existence… Alexander by his side proves existence in a different way but here unfortunately I cannot say too much but I would appreciate that Alexander would write down some lines here about his work.

Other theoretical attempts are based on some educated guess as a starting point as could be the vacuum functional, the beta function or other parts of the theory that, for a full proof, should be derived instead. These attempts give a strong support to my work and that of Alexander. In these papers you will see a discrete spectrum and this is the one of a harmonic oscillator or simply the very existence of the mass gap itself. But, for physicists, the spectrum is the relevant conclusion as from it we can get the masses of physical states to be seen in accelerator facilities. This is the reason why I do not worry too much for mathematicians fussing about my papers.

Finally, I would like to spend a few words about numerical and experimental results. Experiments show clearly always bound states of quarks and gluons that are never seen as free. This is the better proof so far Nature gave us of the existence of the mass gap. Numerically, people computed both Green functions and the spectrum of the theory. I am convinced that these lines should merge. The spectrum on the lattice, both quenched and unquenched, displays the mass gap. Green functions, when one considers just the decoupling solution, are Yukawa-like, both on the lattice and from Dyson-Schwinger equations, and this again is a proof of existence of the mass gap.

I hope I have not forgotten anyone. Please, let me know. If you need explicit references here and there I will be pleased to post here. A lot of people is involved in this kind of research and I am happy to acknowledge the good work.

Finally, I would like to remember that one cannot be skeptical about mathematics as mathematics can only be either right or wrong. No other way.

Exact solutions go published!


My paper presenting exact solutions to classical scalar field theories, with a corresponding quantum formulation, has been accepted for publication in the Journal of Nonlinear Mathematical Physics.  The replacement on arxiv will appear tomorrow, the link is here. I would like to thank the Editor, Norbert Euler, and an anonymous referee that pointed out to me the existence of a zero mode in the quantum fluctuations.

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