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.

Space and time transformations with a minimal length

“…Nonetheless, from a phenomenological point of views, the description of a minimal measurable length remains particularly relevant. One of the most common approaches for such phenomenological investigations consists of a modification of the Heisenberg algebra describing a modified uncertainty relation between position and momentum, generically called generalized uncertainty principle (GUP) [17][18][19][20][21][22][23][24][25][26][27]. Indeed, via the Robertson-Schrödinger relation, a modified commutation relation implies a modified lower bound for the product of position and momentum uncertainties which, in turn, may result in a minimal uncertainty in position.…”