Effective Space Quantization in Friedman-Robertson-Walker Models

Abstract

Quantum--mechanical operators corresponding to canonical momentum and position of a point--like particle, which follow from the quantum field theory in the general Riemannian space-time, satisfy generally to a deformation of the canonical commutation relations with $c^{-2}$ as the parameter of deformation. For operators of the quasi-Cartesian coordinates in the closed and open Friedman--Robertson--Walker models, the deformation reproduces the spatial part of the well--known Snyder formula for quantization of the Minkowsky space-time. The spatially-flat models are distinguished apart by that the deformation is reduced exactly to the standard canonical commutation relations, which correlates remarkably with the fact of the observed flatness of the Universe. Conditions are briefly discussed for which the deformation could have cosmological manifestations. Key words: quantum mechanics, cosmology, quantized space.