Gravitational effects on the Heisenberg Uncertainty Principle: a geometric approach

TL;DR

This paper introduces a geometric approach to generalize the Heisenberg Uncertainty Principle within relativistic contexts, applying it to curved spacetimes and linking it to quantum gravity concepts like discrete spacetime.

Contribution

It proposes a semiclassical geometric formalism extending the HUP to general relativity scenarios, connecting it with deformations and the Generalized Uncertainty Principle.

Findings

Uncertainty relations are mapped to known deformations of HUP.

Derived a perturbed metric indicating discrete spacetime at Planck scale.

Connected results with recent approaches in quantum gravity literature.

Abstract

The Heisenberg Uncertainty Principle (HUP) limits the accuracy in the simultaneous measurements of the position and momentum variables of any quantum system. This is known to be true in the context of non-relativistic quantum mechanics. Based on a semiclassical geometric approach, here we propose an effective generalization of this principle, which is well-suited to be extended to general relativity scenarios as well. We apply our formalism to Schwarzschild and de Sitter space-time, showing that the ensuing uncertainty relations can be mapped into well-known deformations of the HUP. We also infer the form of the perturbed metric that mimics the emergence of a discrete spacetime structure at Planck scale, consistently with the predictions of the Generalized Uncertainty Principle. Finally, we discuss our results in connection with other approaches recently appeared in the literature.