The Schr$\ddot{o}$dinger-Poisson equations as the large-N limit of the
  Newtonian N-body system: applications to the large scale dark matter dynamics
  In this paper it is argued how the dynamics of the classical Newtonian N-body
system can be described in terms of the Schr$\ddot{o}$dinger-Poisson equations
in the large $N$ limit. This result is based on the stochastic quantization
introduced by Nelson, and on the Calogero conjecture. According to the Calogero
conjecture, the emerging effective Planck constant is computed in terms of the
parameters of the N-body system as $\hbar \sim M^{5/3} G^{1/2}
(N/<\rho>)^{1/6}$, where is $G$ the gravitational constant, $N$ and $M$ are the
number and the mass of the bodies, and $<\rho>$ is their average density. The
relevance of this result in the context of large scale structure formation is
discussed. In particular, this finding gives a further argument in support of
the validity of the Schr$\ddot{o}$dinger method as numerical double of the
N-body simulations of dark matter dynamics at large cosmological scales.
