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

Abstract: 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.