Quantum-spacetime effects on nonrelativistic Schr\"odinger evolution

The last three decades have witnessed the surge of quantum gravity phenomenology in the ultraviolet regime as exemplified by the Planck-scale accuracy of time-delay measurements from highly energetic astrophysical events. Yet, recent advances in precision measurements and control over quantum phenomena may usher in a new era of low-energy quantum gravity phenomenology. In this study, we investigate relativistic modified dispersion relations (MDRs) in curved spacetime and derive the corresponding nonrelativistic Schr\"odinger equation using two complementary approaches. First, we take the nonrelativistic limit, and canonically quantise the result. Second, we apply a WKB-like expansion to an MDR-inspired deformed relativistic wave equation. Within the area of applicability of single-particle quantum mechanics, both approaches imply equivalent results. Surprisingly, we recognise in the generalized uncertainty principle (GUP), the prevailing approach in nonrelativistic quantum gravity phenomenology, the MDR which is least amenable to low-energy experiments. Consequently, importing data from the mentioned time-delay measurements, we constrain the linear GUP up to the Planck scale and improve on current bounds to the quadratic one by 17 orders of magnitude. MDRs with larger implications in the infrared, however, can be tightly constrained in the nonrelativistic regime. We use the ensuing deviation from the equivalence principle to bound some MDRs, for example the one customarily associated with the bicrossproduct basis of the $\kappa$-Poincar\'e algebra, to up to four orders of magnitude below the Planck scale.Comment: 34 pages, one figur

Quantum Configuration and Phase Spaces: Finsler and Hamilton Geometries

In this paper, we review two approaches that can describe, in a geometrical way, the kinematics of particles that are affected by Planck-scale departures, named Finsler and Hamilton geometries. By relying on maps that connect the spaces of velocities and momenta, we discuss the properties of configuration and phase spaces induced by these two distinct geometries. In particular, we exemplify this approach by considering the so-called $q$-de Sitter-inspired modified dispersion relation as a laboratory for this study. We finalize with some points that we consider as positive and negative ones of each approach for the description of quantum configuration and phases spaces.Comment: 22 pages. Matches published version. Invited contribution for Physics. Special Issue "New Advances in Quantum Geometry