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Annales Henri Poincaré

, Volume 18, Issue 9, pp 3049–3096 | Cite as

Causality for Nonlocal Phenomena

  • Michał EcksteinEmail author
  • Tomasz Miller
Open Access
Article

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 Lorentzian geometry, 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 outline the possible applications of the developed formalism in both classical and quantum physics.

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Physics, Astronomy and Applied Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Copernicus Center for Interdisciplinary StudiesKrakówPoland
  3. 3.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

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