Causality for nonlocal phenomena

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.

42 pages, 3 figures

References in corpus (6)

• The causal hierarchy of spacetimes
• Causal evolution of wave packets
• Causality in noncommutative two-sheeted space-times
• Polish spaces of causal curves
• Theory of optimal transport for Lorentzian cost functions
• Non locality and causal evolution in QFT

Cited by in corpus (19)

• An optimal transport formulation of the Einstein equations of general relativity
• Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications
• Quantum Monge-Kantorovich problem and transport distance between density matrices
• A review of Lorentzian synthetic theory of timelike Ricci curvature bounds
• Causal evolution of wave packets
• Geometry of weighted Lorentz-Finsler manifolds I: Singularity theorems
• The noncommutative geometry of Zitterbewegung
• Gravity from thermodynamics: optimal transport and negative effective dimensions
• Polish spaces of causal curves
• Convergence of Combinatorial Gravity
• Generally covariant $N$-particle dynamics
• Operational causality in spacetime
• Towards a Probabilistic Foundation of Relativistic Quantum Theory: The One-Body Born Rule in Curved Spacetime
• Optimal transport on null hypersurfaces and the null energy condition
• A No-go theorem for device-independent security in relativistic causal theories
• Relativistic Causality vs. No-Signaling as the limiting paradigm for correlations in physical theories
• Time functions and $K$-causality between measures
• Physical models from noncommutative causality
• Causal evolution of probability measures and continuity equation