Annales Henri Poincaré

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

Causality for Nonlocal Phenomena

  • Michał EcksteinEmail author
  • Tomasz Miller
Open Access


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.


  1. 1.
    Abdo, A., et al.: Testing Einstein’s special relativity with Fermi’s short hard \(\gamma \)-ray burst GRB090510. Nature 462, 331 (2009)ADSCrossRefGoogle Scholar
  2. 2.
    Aichmann, H., Nimtz, G.: On the traversal time of barriers. Found. Phys. 44(6), 678–688 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Al-Hashimi, M., Wiese, U.-J.: Minimal position-velocity uncertainty wave packets in relativistic and non-relativistic quantum mechanics. Ann. Phys. 324(12), 2599–2621 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Amelino-Camelia, G., Ellis, J., Mavromatos, N., Nanopoulos, D., Sarkar, S.: Tests of quantum gravity from observations of \(\gamma \)-ray bursts. Nature 393(6687), 763–765 (1998)ADSCrossRefGoogle Scholar
  5. 5.
    Barat, N., Kimball, J.: Localization and causality for a free particle. Phys. Lett. A 308(2–3), 110–115 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beckman, D., Gottesman, D., Nielsen, M.A., Preskill, J.: Causal and localizable quantum operations. Phys. Rev. A 64, 052309 (2001)ADSCrossRefGoogle Scholar
  7. 7.
    Beem, J., Ehrlich, P., Easley, K.: Global Lorentzian Geometry. Volume 202 of Monographs and Textbooks in Pure and Applied Mathematics. CRC Press, Boca Raton (1996)Google Scholar
  8. 8.
    Bernal, A., Sánchez, M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257(1), 43–50 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berry, M.V.: Causal wave propagation for relativistic massive particles: physical asymptotics in action. Eur. J. Phys. 33(2), 279 (2012)CrossRefzbMATHGoogle Scholar
  10. 10.
    Bertrand, J., Puel, M.: The optimal mass transport problem for relativistic costs. Calc. Var. Partial Differ. Equ. 46(1), 353–374 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Besnard, F.: A noncommutative view on topology and order. J. Geom. Phys. 59(7), 861–875 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Beuthe, M.: Oscillations of neutrinos and mesons in quantum field theory. Phys. Rep. 375(2–3), 105–218 (2003)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Brenier, Y.: Optimal Transportation and Applications: Lectures given at the C.I.M.E. Summer School, held in Martina Franca, Italy, September 2–8, 2001. Chapter Extended Monge–Kantorovich Theory, pp. 91–121. Springer, Berlin (2003)Google Scholar
  14. 14.
    Brenier, Y., Frisch, U., Hénon, M., Loeper, G., Matarrese, S., Mohayaee, R., Sobolevskii, A.: Reconstruction of the early universe as a convex optimization problem. Mon. Not. R. Astron. Soc. 346(2), 501–524 (2003)ADSCrossRefGoogle Scholar
  15. 15.
    Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84(1), 1–54 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Buchholz, D., Yngvason, J.: There are no causality problems for Fermi’s two-atom system. Phys. Rev. Lett. 73, 613–616 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Buscemi, F., Compagno, G.: Non-locality and causal evolution in QFT. J. Phys. B At. Mol. Opt. Phys. 39(15), 695–709 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chruściel, P.T., Grant, J.D.E., Minguzzi, E.: On differentiability of volume time functions. Ann. Henri Poincaré 17(10), 2801–2824 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Doplicher, S., Fredenhagen, K., Roberts, J.E.: Spacetime quantization induced by classical gravity. Phys. Lett. B 331(1–2), 39–44 (1994)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Doplicher, S., Fredenhagen, K., Roberts, J.E.: The quantum structure of spacetime at the planck scale and quantum fields. Commun. Math. Phys. 172(1), 187–220 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Eckstein, M., Miller, T.: Causal evolution of wave packets. Phys. Rev. A 95, 032106 (2017). doi: 10.1103/PhysRevA.95.032106
  22. 22.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47(10), 777–780 (1935)ADSCrossRefzbMATHGoogle Scholar
  23. 23.
    Feinberg, G.: Possibility of faster-than-light particles. Phys. Rev. 159, 1089–1105 (1967)ADSCrossRefGoogle Scholar
  24. 24.
    Foldy, L.L., Wouthuysen, S.A.: On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Phys. Rev. 78(1), 29 (1950)ADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Franco, N., Eckstein, M.: An algebraic formulation of causality for noncommutative geometry. Class. Quantum Gravity 30(13), 135007 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Franco, N., Eckstein, M.: Exploring the causal structures of almost commutative geometries. Symmetry Integr. Geom. Methods Appl. 10, 010 (2014). (Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Franco, N., Eckstein, M.: Causality in noncommutative two-sheeted space-times. J. Geom. Phys. 96, 42–58 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Frisch, U., Matarrese, S., Mohayaee, R., Sobolevski, A.: A reconstruction of the initial conditions of the universe by optimal mass transportation. Nature 417(6886), 260–262 (2002)ADSCrossRefGoogle Scholar
  29. 29.
    Frisch, U., Podvigina, O., Villone, B., Zheligovsky, V.: Optimal transport by omni-potential flow and cosmological reconstruction. J. Math. Phys. 53(3), 033703 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Frisch, U., Sobolevskii, A.: Application of optimal transport theory to reconstruction of the early universe. J. Math. Sci. 133(4), 1539–1542 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Garling, D.J.H.: Inequalities: A Journey into Linear Analysis. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  32. 32.
    Gell-Mann, M., Goldberger, M.L., Thirring, W.E.: Use of causality conditions in quantum theory. Phys. Rev. 95, 1612–1627 (1954)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. Theoretical and Mathematical Physics. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  34. 34.
    Hawking, S.W.: Chronology protection conjecture. Phys. Rev. D 46, 603–611 (1992)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Hegerfeldt, G.C.: Remark on causality and particle localization. Phys. Rev. D 10, 3320–3321 (1974)ADSCrossRefGoogle Scholar
  36. 36.
    Hegerfeldt, G.C.: Violation of causality in relativistic quantum theory? Phys. Rev. Lett. 54, 2395–2398 (1985)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Hegerfeldt, G.C.: Causality, particle localization and positivity of the energy. In: Bohm, A., Doebner, H.-D., Kielanowski, P. (eds.) Irreversibility and Causality Semigroups and Rigged Hilbert Spaces, Volume 504 of Lecture Notes in Physics, pp. 238–245. Springer, Berlin (1998)CrossRefGoogle Scholar
  38. 38.
    Hegerfeldt, G.C.: Particle localization and the notion of Einstein causality. In: Horzela, A., Kapuścik, E. (eds.) Extensions of Quantum Theory, pp. 9–16. Apeiron, Montreal (2001)Google Scholar
  39. 39.
    Hegerfeldt, G.C., Ruijsenaars, S.N.M.: Remarks on causality, localization, and spreading of wave packets. Phys. Rev. D 22, 377–384 (1980)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Miller, T.: Polish spaces of causal curves. J. Geom. Phys. 116, 295–315 (2017). doi: 10.1016/j.geomphys.2017.02.006 ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Miller, T.: On the causality and \(K\)-causality between measures. Preprint arXiv:1702.00702 [math-ph] (2017)
  42. 42.
    Minguzzi, E.: Time functions as utilities. Commun. Math. Phys. 298(3), 855–868 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Minguzzi, E., Sánchez, M.: The causal hierarchy of spacetimes. In: Alekseevsky, D.V., Baum, H. (eds.) Recent Developments in Pseudo-Riemannian Geometry. ESI Lectures in Mathematics and Physics, pp. 299–358. European Mathematical Society, Helsinki (2008)CrossRefGoogle Scholar
  44. 44.
    Moretti, V.: Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes. Rev. Math. Phys. 15(10), 1171–1217 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  46. 46.
    O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)zbMATHGoogle Scholar
  47. 47.
    Pawłowski, M., Paterek, T., Kaszlikowski, D., Scarani, V., Winter, A., Żukowski, M.: Information causality as a physical principle. Nature 461(7267), 1101–1104 (2009)ADSCrossRefGoogle Scholar
  48. 48.
    Penrose, R.: Techniques of Differential Topology in Relativity. Volume 7 of CBMS–NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1972)CrossRefGoogle Scholar
  49. 49.
    Peres, A., Terno, D.R.: Quantum information and relativity theory. Rev. Mod. Phys. 76, 93–123 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Ringström, H.: The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics. European Mathematical Society, Helsinki (2009)CrossRefzbMATHGoogle Scholar
  51. 51.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)zbMATHGoogle Scholar
  52. 52.
    Srivastava, S.M.: A Course on Borel Sets, Volume 180 of Graduate Texts in Mathematics. Springer, New York (2008)Google Scholar
  53. 53.
    Strange, P.: Relativistic Quantum Mechanics: with Applications in Condensed Matter and Atomic Physics. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  54. 54.
    Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and All That. Princeton Landmarks in Mathematics and Physics. Princeton University Press, Princeton (2000)zbMATHGoogle Scholar
  55. 55.
    Suhr, S.: Theory of optimal transport for Lorentzian cost functions. Preprint arXiv:1601.04532 [math-ph] (2016)
  56. 56.
    Thaller, B.: The Dirac Equation, Volume 31 of Theoretical and Mathematical Physics. Springer, Berlin (1992)Google Scholar
  57. 57.
    van Suijlekom, W.D.: Noncommutative Geometry and Particle Physics. Mathematical Physics Studies. Springer, New York (2015)zbMATHGoogle Scholar
  58. 58.
    Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics. American Mathematical Society, Providence (2003)zbMATHGoogle Scholar
  59. 59.
    Villani, C.: Optimal Transport: Old and New. Volume 338 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2008)Google Scholar
  60. 60.
    Wagner, R., Shields, B., Ware, M., Su, Q., Grobe, R.: Causality and relativistic localization in one-dimensional Hamiltonians. Phys. Rev. A 83, 062106 (2011)ADSCrossRefGoogle Scholar
  61. 61.
    Willard, S.: General Topology. Addison-Wesley, Reading (1970)zbMATHGoogle Scholar
  62. 62.
    Winful, H.G.: Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox. Phys. Rep. 436(1–2), 1–69 (2006)ADSCrossRefGoogle Scholar

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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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