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 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
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
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