Length uncertainty in a gravity’s rainbow formalism

It is commonly accepted that the combination of quantum mechanics and general relativity gives rise to the emergence of a minimum uncertainty both in space and time. The arguments that support this conclusion are mainly based on perturbative approaches to the quantization, in which the gravitational interactions of the matter content are described as corrections to a classical background. In a recent paper, we analyzed the existence of a minimum time uncertainty in the framework of doubly special relativity. In this framework, the standard definition of the energy-momentum of particles is modified appealing to possible quantum gravitational effects, which are not necessarily perturbative. Demanding that this modification be completed into a canonical transformation determines the implementation of doubly special relativity in position space and leads to spacetime coordinates that depend on the energy-momentum of the particle. In the present work, we extend our analysis to the quantum length uncertainty. We show that, in generic cases, there actually exists a limit in the spatial resolution, both when the quantum evolution is described in terms of the auxiliary time corresponding to the Minkowski background or in terms of the physical time. These two kinds of evolutions can be understood as corresponding to perturbative and nonperturbative descriptions, respectively. This result contrasts with that found for the time uncertainty, which can be made to vanish in all models with unbounded physical energy if one adheres to a nonperturbative quantization.