Noncommutative Lightcones from Quantum SO(2,1) Conformal Groups

TL;DR

The paper addresses how to model causality and locality in noncommutative spacetimes by constructing five covariant noncommutative lightcones in 2+1 dimensions as quantum homogeneous spaces of the SO(2,1) conformal group. ItClassifies inequivalent Poisson-Lie structures on SO(2,1, including automorphism analyses, yielding five Poisson homogeneous lightcones, each covariant under a corresponding quantum group. The five models are quantized with a fuzziness scale ℓ, giving explicit commutators [ŷ, ẑ] and Hermitian representations, and their localization properties are tied to the geometry of the underlying Poisson spaces. The approach clarifies how noncommutativity can coexist with a sharp lightcone and offers a pathway to extend the construction to 3+1 dimensions and to lightlike worldlines, with potential implications for causality in quantum gravity phenomenology.

Abstract

Five new families of noncommutative lightcones in 2+1 dimensions are presented as the quantizations of the inequivalent Poisson homogeneous structures that emerge when the lightcone is constructed as a homogeneous space of the SO(2,1) conformal group. Each of these noncommutative lightcones maintains covariance under the action of the respective quantum deformation of the SO(2,1) conformal group. We discuss the role played by SO(2,1) automorphisms in the classification of inequivalent Poisson homogeneous lightcones, as well as the geometric aspects of this construction. The localization properties of the novel quantum lightcones are analyzed and shown to be deeply connected with the geometric features of the Poisson homogeneous spaces.

Figures (2)

• Figure 1: The future and past lightcones in the three-dimensional Minkowski ambient space. The coordinate lines of $y$ (in blue) are obtained through pure translations and lie on the intersection of the cone with $45^\circ$ planes. Those of $z$ (in red) correspond to dilations and lie on the intersections of the cone with vertical planes (parallel to the time axis $x^0$). The black dots represent the origin $(1,0,-1)$ of the future lightcone and the origin $(-1,0,1)$ of the past lightcone when constructed as the homogeneous spaces ${\cal L}_\pm$. Notice that the boundary of our coordinate patch is the light ray $x^0=x^2$ (black dashed line).
• Figure 2: The regions of the lightcones where the five Poisson brackets vanish are given by the black curves defined by the intersection of the lightcones with the planes shown in this figure.