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Ordered Quantization and the Ehrenfest Time Scale

We propose a prescription to quantize classical monomials in terms of symmetric and ordered expansions of non-commuting operators of a bosonic theory. As a direct application of such quantization rules, we quantize a classically time evolved function $\mathcal{O}(q,p,t)$, and calculate its expectation value in coherent states. The result can be expressed in terms of the application of a classical operator which performs a {\em Gaussian smoothing} of the original function $\mathcal{O}$ evaluated at the center of the coherent state. This scheme produces a natural semi-classical expansion for the quantum expectation values at a short time scale. Moreover, since the classical Liouville evolution of a Gaussian probability density gives the {\bf same} form for the classical statistical mean value, we can calculate the first order correction in $\hbar$ entirely from the associated classical time evolved function. This allows us to write a general expression for the Ehrenfest time in terms of the departure of the centroid of the quantum distribution from the classical trajectory provided we start with an initially coherent state for each subsystem. In order to illustrate this approach, we have calculated analytically the Ehrenfest time of a model with $N$ coupled non-linear oscillators with non-linearity of even order.