Quantum causality in kappa Minkoswki

TL;DR

This work addresses how to define causality in noncommutative spacetimes by developing a twisted Lorentzian Spectral Triple for 1+1D κ-Minkowski and formulating a preserved algebraic causal order via causal functions and pure states. The authors construct a twisted Dirac operator using $X_0=(1-\mathcal{E})$, $X_1=P_1$ with $\mathcal{E}=e^{-P_0/\kappa}$, realize κ-Minkowski through a star-product realization on the affine group, and establish a Krein-style Lorentzian structure with a twist implemented by $\mathcal{E}$. A key result is a quantum constraint, $\delta\langle P \rangle \geq |\delta\langle X \rangle|$, arising from the causal cone when comparing pure states, which serves as a quantum analog of the classical speed-of-light bound and allows for potential average superluminal fluctuations at the Planck scale. Collectively, the work provides a concrete framework for quantum causal structure in quantum gravity models, linking algebraic causality in noncommutative geometry to observable-like constraints on κ-Minkowski dynamics with possible phenomenological implications.

Abstract

Recent results on causality in noncommutative space-time are reviewed. We study, in particular, quantum causal structures in 1+1 dimensional kappa Minkowski space-time. This later is described by a twisted Lorentzian Spectral Triple build with a twisted set of derivatives. Investigation of causality provides a quantum constraint, which is a quantum analog of the speed light limits

Figures (1)

• Figure 1: Two events $x$ and $y$ are causally related by a curve with only future time-directed tangent vectors. One can describe a light cone structure for every event.