Causality for nonlocal phenomena

TL;DR

This paper introduces a rigorous causal relation for probability measures on spacetimes using optimal transport theory, linking causality, topology, and measure theory, and explores its properties and implications in quantum causality.

Contribution

It proposes a new notion of causality for probability measures on spacetimes and introduces the Lorentz-Wasserstein distance, connecting causality with optimal transport.

Findings

Characterisations of the causal relation are equivalent under robust causal structures.

The Lorentz-Wasserstein distance has well-defined basic properties.

Connections to quantum causality and Hegerfeldt's theorem are established.

Abstract

Drawing from the theory of optimal transport we propose a rigorous notion of a causal relation for Borel probability measures on a given spacetime. To prepare the ground, we explore the borderland between causality, topology and measure theory. We provide various characterisations of the proposed causal relation, which turn out to be equivalent if the underlying spacetime has a sufficiently robust causal structure. We also present the notion of the 'Lorentz-Wasserstein distance' and study its basic properties. Finally, we discuss how various results on causality in quantum theory, aggregated around Hegerfeldt's theorem, fit into our framework.