Quantum causality in $\kappa$-Minkowski and related constraints

TL;DR

This paper explores quantum causal structures in $$-Minkowski space-time using Lorentzian Spectral Triples, revealing conditions for causal propagation and quantum analogs of light-cone coordinates.

Contribution

It introduces a twisted Lorentzian Spectral Triple framework for $$-Minkowski space-time and identifies quantum constraints that enable causal propagation.

Findings

Existence of causal propagation under quantum constraints.

Identification of quantum analogs of light-cone coordinates.

Quantum speed-of-light constraint derived.

Abstract

We study quantum causal structures in $1+1$ $\kappa$-Minkowski space-time described by a Lorentzian Spectral Triple whose Dirac operator is built from a natural set of twisted derivations of the $\kappa$-Poincar\'e algebra. We show that the Lorentzian Spectral Triple must be twisted to accommodate the twisted nature of the derivations. We exhibit various interesting classes of causal functions, including an analog of the light-cone coordinates. We show in particular that the existence of a causal propagation between two pure states, the quantum analogs of points, can exist provided quantum constraints, linking the momentum and the space coordinate, are satisfied. One of these constraints is a quantum analog of the speed of light limit.