Evading Piau’s paradox

27/01/2012

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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