Evading Piau’s paradox



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


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)


and increments are independent so that i\ne k



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


that is


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


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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