Evading Piau’s paradox

ResearchBlogging.org

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

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6 Responses to Evading Piau’s paradox

  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 […]

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