Reinterpreting deformed Heisenberg algebras

TL;DR

This paper explores a covariant framework linking deformed Heisenberg algebras with curved momentum and position spaces, revealing new noncommutative geometries and their implications for quantum gravity models.

Contribution

It introduces a Born reciprocal, covariant approach to describe deformed uncertainty principles via quantum dynamics on curved cotangent manifolds, uncovering new classes of noncommutative geometries.

Findings

Curvature tensors relate to noncommutativity in position and momentum spaces.

The approach constrains models to a novel subclass of noncommutative geometries.

Application to (anti)-de Sitter spacetime illustrates the formalism.

Abstract

Minimal and maximal uncertainties of position measurements are widely considered possible hallmarks of low-energy quantum as well as classical gravity. While General Relativity describes interactions in terms of spatial curvature, its quantum analogue may also extend to the realm of curved momentum space as suggested, e. g. in the context of Relative Locality in Deformed Special Relativity. Drawing on earlier work, we show in an entirely Born reciprocal, i. e. position and momentum space covariant, way that the quadratic Generalized Extended Uncertainty principle can alternatively be described in terms of quantum dynamics on a general curved cotangent manifold. In the case of the Extended Uncertainty Principle the curvature tensor in position space is proportional to the noncommutativity of the momenta, while an analogous relation applies to the curvature tensor in momentum space and the noncommutativity of the coordinates for the Generalized Uncertainty Principle. In the process of deriving this map, the covariance of the approach constrains the admissible models to an interesting subclass of noncommutative geometries which has not been studied before. Furthermore, we reverse the approach to derive general anisotropically deformed uncertainty relations from general background geometries. As an example, this formalism is applied to (anti)-de Sitter spacetime.