I am currently working on the understanding of the stochastic nature of the Schroedinger equation. This has a notable history dating back to Nelson's works and relative criticisms. But one can take a different path and, starting from a random walk process with probability


\begin{equation}
 P(k;N) = \binom{N}{k}\left(\frac{1}{2}\right)^N,
\end{equation}


one can assume the existence of a "square root" of this process with a complex amplitude


\begin{equation}
 A(k;N) = \binom{N}{k}^\frac{1}{2}\left(\frac{1}{2}\right)^{N/2}e^{i\phi(k,N)}
\end{equation}


such that $P(k;N)=|A(k;N)|^2$ and $\phi(k,N)$ are some phases exactly determined. These are quantum amplitudes. Is this anything making sense? Does any literature exist about?



Square root of a stochastic process
**Note:** This mostly debunks an answer posted by the OP and now deleted. As a consequence of this unfortunate deletion, the argument below might be a little difficult to follow.




---


As explained on the [MSE page](https://math.stackexchange.com/questions/101917/square-root-of-a-wiener-process) you are referring to, very basic arguments show that the limits of the sums $S\_n$ you write in your "answer" cannot exist. 


Consider for example the second version of $S\_n$ (the one based on absolute values) in the simplest possible situation, that is, on the interval $[0,1]$ and when $G\equiv1$. Note that the Gaussian random variables $W(t\_i)-W(t\_{i-1})$ are independent with variances $t\_i-t\_{i-1}$. Hence, for every $i$,
$$
E((W(t\_i)-W(t\_{i-1}))^2)=t\_i-t\_{i-1},
$$
and, for every $i\ne k$,
$$
E(|W(t\_i)-W(t\_{i-1})|\cdot|W(t\_k)-W(t\_{k-1})|)=c\cdot\sqrt{t\_i-t\_{i-1}}\cdot\sqrt{t\_k-t\_{k-1}},
$$
where it happens that $c=2/\pi$ but this is irrelevant. Again in the simplest case, that is, when $t\_i=i/n$ for every $i$, this shows that
$$
E(S\_n^2)=1+c\cdot(n-1).
$$
Hence, **in contradiction with what you assert**, the sequence $(S\_n)\_{n\geqslant1}$ diverges in $L^2$.


To sum up, integrals such as 
$$
\int G\cdot|\mathrm dW|\qquad\text{or}\quad\int G\cdot(\mathrm dW)^{\alpha},
$$
whatever their precise definitions would be, cannot exist due to the (ir)regularity properties of the Brownian paths (except, of course, the second one when $\alpha=1$).


