A framework for nonrelativistic isotropic models based on generalized uncertainty principles

Abstract

The existence of a fundamental length scale in nature is a common prediction of distinct quantum gravity models. Discovery of such would profoundly change current knowledge of quantum phenomena and modifications to the Heisenberg uncertainty principle may be expected. Despite the attention given to this possibility in the past decades, there has been no common framework for a systematic investigation of so-called generalized uncertainty principles (GUP). In this work we provide such a framework in the context of nonrelativistic quantum mechanics. Our approach is based on very few assumptions: there is a fundamental length scale, space isotropy, invariance under parity and time reversal transformations, and symmetricity of the position and momentum operators. We show that simple dimensional analysis allows building a common framework for isotropic models based on GUP (we call those iGUP models). We discuss some popular GUP models in this context after elaborating on relevant theoretical aspects of the framework. At last, we translate current bounds on three often investigated GUP models into bounds on parameters of such common iGUP framework.