n-dimensional non-commutative GUP quantization and application to the Bianchi I model

Abstract: We analyse a n-dimensional Generalized Uncertainty Principle (GUP) quantization framework, characterized by a non-commutative nature of the configurational variables. First, we identify a set of states which are maximally localized only along a single direction, at the expense of being less localized in all the other ones. Subsequently, in order to recover information about localization on the whole configuration space, we use the only state of the theory which exhibits maximal localization simultaneously in every direction to construct a satisfactory quasi-position representation, by virtue of a suitable translational operator. The resultant quantum framework is then applied to model the dynamics of the Bianchi I cosmology. The corresponding Wheeler-DeWitt equation is reduced to Schr\"odinger dynamics for the two anisotropy degrees of freedom, using a WKB representation for the volume-like variable of the Universe, in accordance with the Vilenkin scenario. The main result of our cosmological implementation of the constructed quantum theory demonstrates how the dynamics of a wave packet peaked at some point in the configuration space represented in the quasi-position variables, favours as the most probable configuration exactly the initial one for a relatively long time, if compared with the ordinary quantum theory. This preference arises from the distinct behavioral dynamics exhibited by wave packets in the two quantum theories.