Metrics and causality on Moyal planes

TL;DR

This paper explores the metric and causal structures of Moyal planes within noncommutative geometry, demonstrating the existence of causality relations similar to Minkowski space in certain states.

Contribution

It introduces a framework for defining causality on Moyal planes, linking quantum metric spaces with Lorentzian noncommutative geometry.

Findings

Metrics from Connes distance make Moyal planes quantum metric spaces

Causal relations are established between specific pure states on Moyal plane

Causal structure resembles that of Minkowski space in certain states

Abstract

Metrics structures stemming from the Connes distance promote Moyal planes to the status of quantum metric spaces. We discuss this aspect in the light of recent developments, emphasizing the role of Moyal planes as representative examples of a recently introduced notion of quantum (noncommutative) locally compact space. We move then to the framework of Lorentzian noncommutative geometry and we examine the possibility of defining a notion of causality on Moyal plane, which is somewhat controversial in the area of mathematical physics. We show the actual existence of causal relations between the elements of a particular class of pure (coherent) states on Moyal plane with related causal structure similar to the one of the usual Minkowski space, up to the notion of locality.