Relative locality and relative Co-locality as extensions of the Generalized Uncertainty Principle

TL;DR

This paper extends the Generalized Uncertainty Principle by incorporating spatial curvature effects through Relative Locality and Co-locality, revealing dual non-commutative structures in momentum and position spaces linked to quantum group symmetries.

Contribution

It introduces a novel framework combining Relative Locality and Co-locality with quantum groups, linking curvature effects to non-commutative geometry and the GUP.

Findings

Spatial curvature effects induce red-shift in Relative Locality.

Momentum space curvature causes blue-shift in Co-locality.

The formalism predicts a q-deformation parameter related to observer scales.

Abstract

I introduce the spatial curvature effects inside the formalism of Relative Locality as a non-commutative structure of the momentum space in agreement with the very well known concepts of Quantum Groups. This gives a natural red-shift effect in agreement with an extended version of the Generalized Uncertainty Principle (GUP) and in agreement with the standard notions of curvature inside General Relativity. I then introduce the concept of Relative Co-locality as a reinterpretation of the usual notions of spacetime curvature. From this redefinition, I obtain the momentum space curvature effects as a non-commutativity in position space. This introduce a natural blue-shift effect in agreement with the extended version of GUP. Both effects, Relative locality and Co-locality are dual each other inside the formalism of quantum groups $SU_q(n)$ symmetric Heisenberg algebras and their q-Bargmann Fock representations. When Relative locality and Co-locality are introduced, the q-deformation parameter takes the form $q\approx 1+\sqrt{\frac{\vert p\vert \vert x\vert}{r_\Lambda m_{pl}}}$ with the spatial curvature effects in Relative Locality appearing like $\Delta X\approx \frac{\vert x\vert}{m_{pl}}\Delta P$ and the momentum curvature effects in Relative Co-locality appearing like $\Delta P\approx \frac{\vert p\vert}{r_\Lambda}\Delta X$, where $r_\Lambda=\frac{1}{\sqrt{\Lambda}}$ is the scale defined by the Cosmological Constant $\Lambda$, $m_{pl}$ is the Planck mass and $\Delta X/\Delta P$ is a scale of position/momentum or time/energy associated with the event, p and x are the momentum and position of the observer relative to the event.