2017 **Abstract:** We compare two different approaches for quantization of the Bianchi I model: a reduced phase space quantization, in which the isotropic Misner variables is taken as time, and the Vilenkin proposal, in which a semiclassical approximation is performed for the same variable. We outline the technical and interpretative issues of these two methods and we demonstrate that they provide equivalent results only if the dynamics is essentially dictated by the isotropic matter contribution.

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“…This approach, which can be applied to any set of variables (see, e.g., [9]), is particularly appropriate to reconstruct the limit of quantum field theory on a classical curved background from quantum gravity. For a discussion on the necessary conditions to adopt the BO approximation and applications, see [10].…”

confidence: 99%

“…This approach, which can be applied to any set of variables (see, e.g., [9]), is particularly appropriate to reconstruct the limit of quantum field theory on a classical curved background from quantum gravity. For a discussion on the necessary conditions to adopt the BO approximation and applications, see [10].…”

confidence: 99%

“…On the contrary, we identified in the spatial curvature the ingredient responsible for the anisotropy quantum suppression. By other words, when the Universe can be characterized by small quantum anisotropies, in the sense discussed in [11] and in [12], the scalar potential takes the form of a harmonic oscillator, which frequency increases with time as the Universe volume expands. This potential term is then responsible for the damping of the anisotropy, providing a valuable paradigm for the Universe isotropization.…”

confidence: 99%

“…Moreover, according to the Vilenkin idea of a small quantum system (see also [12]), we consider the quasiisotropic regime |β + | 1 so that the potential term gets a quadratic form…”

confidence: 99%

“…By other words, we are inferring that the variable β − is enough small to explore the uncertainty principle with ∆β − ≤ 2 √ β + and ∆p − ≥ 2 √ /β + . The quantum subsystem shows to possess the "smallness" requirement postulated in [25] and precised in [26]. Under the hypotheses above, the Universe state functional can be written as follows…”

confidence: 99%