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

1. Tony Smith says:

With respect to Bernoulli shift
how do you see connections between your work
and that of Christian Beck in his book
Spatio-Temporal Chaos and Vacuum Fluctuations
of Quantum Fields (World Scientific 2002)
summarized at
hep-th/0207081
?
Tony

• mfrasca says:

Dear Tony,

Thank you for pointing me out Beck’s work. I was not aware of it and possibly this should be cited as an avenue to link Brownian motion and quantum mechanics. But I should say that Beck’s goal is somewhat higher than mine. I just aim to reformulate quantum mechanics in a different mathematical framework and most of it I have had to rewrite by myself in some way.

Indeed, square root of a classical stochastic process could not mean at all that what one gets is a classical layer behind quantum behavior. This makes an important difference.

Marco

2. [...] 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 [...]

3. Rafael Frigori says:

Dear Marco,

it is a very interesting post!
So, do you mean that those mathematicians don’t consider regularization as a valid mathematical concept or do they just ignore Hardy’s book ?
Best

Rafael

• mfrasca says:

Hi Rafael,

fine to hear from you again. Piau’s argument is fine as, being him a mathematician, he is in need for rigorous definitions. Here the question is just how to redefine the Ito integral to extend its meaning to the cases I am interested on. A regularization technique as the one I propose, based on the concept of divergent series, works perfectly well as can be checked through some very simple computations.

But I went somewhat beyond this having a matlab simulation that shows how my equation for the square root of a Wiener process indeed work. You wrote me just few minutes before I posted it.

Best,

Marco

4. [...] 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 [...]