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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Abdo, A., et al.: Testing Einstein’s special relativity with Fermi’s short hard \(\gamma \)-ray burst GRB090510. Nature 462, 331 (2009)
Aichmann, H., Nimtz, G.: On the traversal time of barriers. Found. Phys. 44(6), 678–688 (2014)
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)
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)
Barat, N., Kimball, J.: Localization and causality for a free particle. Phys. Lett. A 308(2–3), 110–115 (2003)
Beckman, D., Gottesman, D., Nielsen, M.A., Preskill, J.: Causal and localizable quantum operations. Phys. Rev. A 64, 052309 (2001)
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)
Bernal, A., Sánchez, M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257(1), 43–50 (2005)
Berry, M.V.: Causal wave propagation for relativistic massive particles: physical asymptotics in action. Eur. J. Phys. 33(2), 279 (2012)
Bertrand, J., Puel, M.: The optimal mass transport problem for relativistic costs. Calc. Var. Partial Differ. Equ. 46(1), 353–374 (2013)
Besnard, F.: A noncommutative view on topology and order. J. Geom. Phys. 59(7), 861–875 (2009)
Beuthe, M.: Oscillations of neutrinos and mesons in quantum field theory. Phys. Rep. 375(2–3), 105–218 (2003)
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)
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)
Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84(1), 1–54 (1982)
Buchholz, D., Yngvason, J.: There are no causality problems for Fermi’s two-atom system. Phys. Rev. Lett. 73, 613–616 (1994)
Buscemi, F., Compagno, G.: Non-locality and causal evolution in QFT. J. Phys. B At. Mol. Opt. Phys. 39(15), 695–709 (2006)
Chruściel, P.T., Grant, J.D.E., Minguzzi, E.: On differentiability of volume time functions. Ann. Henri Poincaré 17(10), 2801–2824 (2016)
Doplicher, S., Fredenhagen, K., Roberts, J.E.: Spacetime quantization induced by classical gravity. Phys. Lett. B 331(1–2), 39–44 (1994)
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)
Eckstein, M., Miller, T.: Causal evolution of wave packets. Phys. Rev. A 95, 032106 (2017). doi:10.1103/PhysRevA.95.032106
Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47(10), 777–780 (1935)
Feinberg, G.: Possibility of faster-than-light particles. Phys. Rev. 159, 1089–1105 (1967)
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)
Franco, N., Eckstein, M.: An algebraic formulation of causality for noncommutative geometry. Class. Quantum Gravity 30(13), 135007 (2013)
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)
Franco, N., Eckstein, M.: Causality in noncommutative two-sheeted space-times. J. Geom. Phys. 96, 42–58 (2015)
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)
Frisch, U., Podvigina, O., Villone, B., Zheligovsky, V.: Optimal transport by omni-potential flow and cosmological reconstruction. J. Math. Phys. 53(3), 033703 (2012)
Frisch, U., Sobolevskii, A.: Application of optimal transport theory to reconstruction of the early universe. J. Math. Sci. 133(4), 1539–1542 (2006)
Garling, D.J.H.: Inequalities: A Journey into Linear Analysis. Cambridge University Press, Cambridge (2007)
Gell-Mann, M., Goldberger, M.L., Thirring, W.E.: Use of causality conditions in quantum theory. Phys. Rev. 95, 1612–1627 (1954)
Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. Theoretical and Mathematical Physics. Springer, Berlin (1996)
Hawking, S.W.: Chronology protection conjecture. Phys. Rev. D 46, 603–611 (1992)
Hegerfeldt, G.C.: Remark on causality and particle localization. Phys. Rev. D 10, 3320–3321 (1974)
Hegerfeldt, G.C.: Violation of causality in relativistic quantum theory? Phys. Rev. Lett. 54, 2395–2398 (1985)
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)
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)
Hegerfeldt, G.C., Ruijsenaars, S.N.M.: Remarks on causality, localization, and spreading of wave packets. Phys. Rev. D 22, 377–384 (1980)
Miller, T.: Polish spaces of causal curves. J. Geom. Phys. 116, 295–315 (2017). doi:10.1016/j.geomphys.2017.02.006
Miller, T.: On the causality and \(K\)-causality between measures. Preprint arXiv:1702.00702 [math-ph] (2017)
Minguzzi, E.: Time functions as utilities. Commun. Math. Phys. 298(3), 855–868 (2010)
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)
Moretti, V.: Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes. Rev. Math. Phys. 15(10), 1171–1217 (2003)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)
Pawłowski, M., Paterek, T., Kaszlikowski, D., Scarani, V., Winter, A., Żukowski, M.: Information causality as a physical principle. Nature 461(7267), 1101–1104 (2009)
Penrose, R.: Techniques of Differential Topology in Relativity. Volume 7 of CBMS–NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1972)
Peres, A., Terno, D.R.: Quantum information and relativity theory. Rev. Mod. Phys. 76, 93–123 (2004)
Ringström, H.: The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics. European Mathematical Society, Helsinki (2009)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)
Srivastava, S.M.: A Course on Borel Sets, Volume 180 of Graduate Texts in Mathematics. Springer, New York (2008)
Strange, P.: Relativistic Quantum Mechanics: with Applications in Condensed Matter and Atomic Physics. Cambridge University Press, Cambridge (1998)
Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and All That. Princeton Landmarks in Mathematics and Physics. Princeton University Press, Princeton (2000)
Suhr, S.: Theory of optimal transport for Lorentzian cost functions. Preprint arXiv:1601.04532 [math-ph] (2016)
Thaller, B.: The Dirac Equation, Volume 31 of Theoretical and Mathematical Physics. Springer, Berlin (1992)
van Suijlekom, W.D.: Noncommutative Geometry and Particle Physics. Mathematical Physics Studies. Springer, New York (2015)
Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics. American Mathematical Society, Providence (2003)
Villani, C.: Optimal Transport: Old and New. Volume 338 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2008)
Wagner, R., Shields, B., Ware, M., Su, Q., Grobe, R.: Causality and relativistic localization in one-dimensional Hamiltonians. Phys. Rev. A 83, 062106 (2011)
Willard, S.: General Topology. Addison-Wesley, Reading (1970)
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)
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Communicated by James A. Isenberg.
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Eckstein, M., Miller, T. Causality for Nonlocal Phenomena. Ann. Henri Poincaré 18, 3049–3096 (2017). https://doi.org/10.1007/s00023-017-0566-1
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DOI: https://doi.org/10.1007/s00023-017-0566-1