Quantum gravity

Quantum gravity appears today as the Holy Grail of physics. This is so far detached from any possible experimental result but with a lot of attentions from truly remarkable people anyway. In some sense, if a physicist would like to know in her lifetime if her speculations are worth a Nobel prize, better to work elsewhere. Anyhow, we are curious people and we would like to know how does the machinery of space-time work this because to have an engineering of space-time would make do to our civilization a significant leap beyond.

A fine recount of the current theoretical proposals has been rapidly presented by Ethan Siegel in his blog. It is interesting to notice that the two most prominent proposals, string theory and loop quantum gravity, share the same difficulty: They are not able to recover the low-energy limit. For string theory this is a severe drawback as here people ask for a fully unified theory of all the interactions. Loop quantum gravity is more limited in scope and so, one can think to fix the problem in a near future. But of all the proposals Siegel is considering, he is missing the most promising one: Non-commutative geometry. This mathematical idea is due to Alain Connes and earned him a Fields medal. So far, this is the only mathematical framework from which one can rederive the full Standard Model with all its particle content properly coupled to the Einstein’s general relativity. This formulation works with a classical gravitational field and so, one can possibly ask where quantized gravity could come out. Indeed, quite recently, Connes, Chamseddine and Mukhanov (see here and here), were able to show that, in the context of non-commutative geometry, a Riemannian manifold results quantized in unitary volumes of two kind of spheres. The reason why there are two kind of unitary volumes is due to the need to have a charge conjugation operator and this implies that these volumes yield the units in the spectrum. This provides the foundations for a future quantum gravity that is fully consistent from the start: The reason is that non-commutative geometry generates renormalizable theories!

The reason for my interest in non-commutative geometry arises exactly from this. Two years ago, I, Alfonso Farina and Matteo Sedehi obtained a publication about the possibility that a complex stochastic process is at the foundations of quantum mechanics (see here and here). We described such a process like the square root of a Brownian motion and so, a Bernoulli process appeared producing the factor 1 or i depending on the sign of the steps of the Brownian motion. This seemed to generate some deep understanding about space-time. Indeed, the work by Connes, Chamseddine and Mukhanov has that understanding and what appeared like a square root process of a Brownian motion today is just the motion of a particle on a non-commutative manifold. Here one has simply a combination of a Clifford algebra, that of Dirac’s matrices, a Wiener process and the Bernoulli process representing the scattering between these randomly distributed quantized volumes. Quantum mechanics is so fundamental that its derivation from a geometrical structure with added some mathematics from stochastic processes makes a case for non-commutative geometry as a serious proposal for quantum gravity.

I hope to give an account of this deep connection in a near future. This appears a rather exciting new avenue to pursue.

Ali H. Chamseddine, Alain Connes, & Viatcheslav Mukhanov (2014). Quanta of Geometry: Noncommutative Aspects Phys. Rev. Lett. 114 (2015) 9, 091302 arXiv: 1409.2471v4

Ali H. Chamseddine, Alain Connes, & Viatcheslav Mukhanov (2014). Geometry and the Quantum: Basics JHEP 12 (2014) 098 arXiv: 1411.0977v1

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, 8 (1), 27-37 DOI: 10.1007/s11760-013-0473-y

Advertisement

Evidence of the square root of Brownian motion

A mathematical proof of existence of a stochastic process involving fractional exponents seemed out of question after some mathematicians claimed this cannot exist. This observation is strongly linked to the current definition and may undergo revision if nature does not agree with it. Stochastic processes are very easy to simulate on a computer. Very few lines of code can decide if something works or not. I and Alfonso Farina, together with Matteo Sedehi,  have introduced the idea that the square root of a Wiener process yields the Schroedinger equation (see here or download a preprint here). This implies that one has to attach a meaning to the equation

In a paper appeared today on arxiv (see here) we finally have provided this proof: We were right. The idea is to solve such an equation by numerical methods. These methods are themselves a proof of existence. We used the Euler-Maruyama method, the simplest one and we compared the results as shown in the following figure

a) Original Brownian motion. b) Same but squaring the formula for the square root. c) Formula of the square root taken as a stochastic equation. d) Same from the stochastic equation in this post.

There is now way to distinguish each other and the original Brownian motion is completely recovered by taking the square of the square root process computed in three different ways. Each one of these completely supports the conclusions we have drawn in our published paper. You can find the code to recover this figure in our arxiv paper. It is obtained by a Monte Carlo simulation with 10000 independent paths. You can play with it changing the parameters as you like.

This paper has an important consequence: Our current mathematical understanding of stochastic processes should be properly extended to account for our results. As a by-product, we have shown how, using Pauli matrices, this idea can be generalized to include spin introducing a new class of stochastic processes in a Clifford algebra.

In conclusion, we would like to remember that, it does not matter what your mathematical definition could be, a stochastic process is always a well-defined entity on a numerical ground. Tests can be easily performed as we proved here.

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, 8 (1), 27-37 DOI: 10.1007/s11760-013-0473-y

Marco Frasca, & Alfonso Farina (2014). Numerical proof of existence of fractional Wiener processes arXiv arXiv: 1403.1075v1

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

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;
t=0:dt/nstep:dt;
B = normrnd(0,sqrt(dt/nstep),1,nstep);
dB = cumsum(B);
% Square root of the Brownian motion
dB05=(dB).^(1/2);

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 . 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;
t=0:dt/nstep:dt;
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

This has mean 0 and variance 1 as it should for and 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

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