27 | January | 2012 | The Gauge Connection

Evading Piau’s paradox

27/01/2012

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.