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 that I defined using the sum

so that I assumed the limit exists and is finite. This position appears untenable as Didier showed in the following way. In this case one has ()

and increments are independent so that

Now, if you want to compute the limit in you are in trouble. Just choose and you will get

that is

If you compute these sums you will get finally a term proportional to that blows up in the limit of increasingly large . 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 (see also discussion here). This series can be regularized and so the limit can be taken to be finite!

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 exists and is meaningful. The same idea can be applied to the case with and my argument is just consistent as I show that for 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

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

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.

[…] 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 […]

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

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.

[…] 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 […]

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

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

[…] 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 […]

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

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

[…] 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 […]