Quantum-spacetime effects on nonrelativistic Schr\"odinger evolution

Abstract

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. Both approaches imply equivalent results for single-particle quantum mechanics. Based on a map between our approach and the generalized uncertainty principle (GUP), we recognise in the latter the MDR which is least amenable to low-energy experiments. Consequently, importing data from 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, from which we use the ensuing deviation from the equivalence principle to bound some MDRs to up to one order of magnitude below the Planck scale, while constraining those customarily associated with the bicrossproduct basis of the $\kappa$-Poincar\'e algebra to energy scales beyond $10^{15}$GeV.