Quantum dynamics beyond lorentz invariance

Abstract

Many approaches to quantum gravity predict the existence of a minimal length associated with the Planck scale, leading to modifications in the relations between energy, momentum, and position, which may in turn affect Lorentz symmetry. In this thesis, we investigate these effects in both systematic and stochastic scenarios, with particular emphasis on their low-energy manifestations. In the first part, we analyze systematic effects derived from modified dispersion relations (MDRs) motivated by quantum gravity. We show that the nonrelativistic limit of general MDRs leads to a deformed nonrelativistic Schrödinger equation containing fractional and nonlocal terms in the Hamiltonian. These terms modify quantum kinematics and allow the establishment of experimental bounds on MDR parameters using low-energy systems. We further demonstrate that certain MDRs with fractional powers give rise to a dynamics described by fractional quantum mechanics at low energies, associated with an effectively fractal geometry in the infrared. We also study stochastic effects associated with Planck-scale fluctuations in MDRs, modeling a stochastic violation of Lorentz symmetry. Performing the stochastic average leads to a Lindblad equation that describes a mechanism of gravitational decoherence in the nonrelativistic regime. We apply this formalism to gravitational Schrödinger cat states (gravcats), revealing the competition between gravitationally induced entanglement and decoherence driven by Planckian fluctuations, and allowing us to estimate the scales at which such effects become relevant. In the second part, we analyze an additional scenario based on noncommutative spacetimes, showing that the deformation of the energy composition rule characteristic of the κ-Minkowski spacetime induces quantum correlations between two systems, even in the nonrelativistic limit. These results demonstrate that low-energy quantum systems provide a novel avenue to test Lorentz symmetry, either through systematic deformations or stochastic violations. The infrared regime has the potential to amplify Planck-suppressed corrections, thereby providing new opportunities to test effective quantum gravity models in tabletop experiments.