Length Uncertainty in a Gravity's Rainbow Formalism

TL;DR

This paper investigates quantum length and time uncertainties within a doubly special relativity framework, revealing a fundamental limit in spatial resolution that contrasts with the vanishing time uncertainty in non-perturbative models.

Contribution

It extends previous work on time uncertainty to analyze length uncertainty, demonstrating a universal spatial resolution limit in a non-perturbative quantum gravity setting.

Findings

Existence of a minimum spatial uncertainty in generic cases.

Time uncertainty can vanish in non-perturbative models with unbounded energy.

Spatial resolution limit is consistent across different quantum evolution descriptions.

Abstract

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 non-perturbative 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 non-perturbative quantization.