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Multivariate Central Limit Theorem in Quantum Dynamics

We consider the time evolution of $N$ bosons in the mean field regime for factorized initial data. In the limit of large $N$, the many body evolution can be approximated by the non-linear Hartree equation. In this paper we are interested in the fluctuations around the Hartree dynamics. We choose $k$ self-adjoint one-particle operators $O_1, \dots, O_k$ on $L^2 (\R^3)$, and we average their action over the $N$-particles. We show that, for every fixed $t \in \R$, expectations of products of functions of the averaged observables approach, as $N \to \infty$, expectations with respect to a complex Gaussian measure, whose covariance matrix can be expressed in terms of a Bogoliubov transformation describing the dynamics of quantum fluctuations around the mean field Hartree evolution. If the operators $O_1, \dots, O_k$ commute, the Gaussian measure is real and positive, and we recover a "classical" multivariate central limit theorem. All our results give explicit bounds on the rate of the convergence (we obtain therefore Berry-Ess{é}en type central limit theorems).