Time Evolution in Quantum Mechanics with a Minimal Time Scale

Abstract

The existence of a minimum measurable length scale was suggested by various theories of quantum gravity, string theory and black hole physics. Motivated by this, we examine a quantum theory exhibiting a minimum measurable time scale. We use the Page-Wootters formalism to describe time evolution of a quantum system with the modified commutation relations between the time and frequency operator. Such modification leads to a minimal uncertainty in the measurement of time. This causes breaking of the time-translation symmetry and results in a modified version of the Schr\"odinger equation. A minimal time scale also allows us to introduce a discrete Schr\"odinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We demonstrate the developed theory on a couple simple quantum systems.