Two Arrows of Time in Nonlocal Particle Dynamics

  • Roderich TumulkaEmail author


Considering what the world would be like if backwards causation were possible is usually mind-bending. Here I discuss something that is easier to study: a toy model that incorporates a very restricted sort of backwards causation. It defines particle world lines by means of a kind of differential delay equation with negative delay. The model presumably prohibits signaling to the past and superluminal signaling, but allows nonlocality while being fully covariant. And that is what constitutes the model’s value: it is an explicit example of the possibility of Lorentz-invariant nonlocality. That is surprising in so far as many authors thought that nonlocality, in particular nonlocal laws for particle world lines, must conflict with relativity. The development of this model was inspired by the search for a fully covariant version of Bohmian mechanics.


Bohmian mechanics Relativity Quantum nonlocality Backwards causation Differential delay equations 



I wish to thank Sheldon Goldstein for his comments on a draft of this paper.


  1. 1.
    Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987) Google Scholar
  2. 2.
    Dirac, P.A.M., Fock, V.A., Podolsky, B.: On quantum electrodynamics. Phys. Z. Sowjetunion 2, 468 (1932). Reprinted in Schwinger, J. (ed.): Quantum Electrodynamics. Dover Publishing, New York (1958) Google Scholar
  3. 3.
    Dürr, D., Goldstein, S., Münch-Berndl, K., Zanghì, N.: Hypersurface Bohm–Dirac models. Phys. Rev. A 60, 2729 (1999). quant-ph/9801070 CrossRefGoogle Scholar
  4. 4.
    Goldstein, S.: Bohmian mechanics. In: Zalta, E.N. (ed.) Stanford Encyclopedia of Philosophy (2001). Published online by Stanford University. Google Scholar
  5. 5.
    Goldstein, S., Tumulka, R.: Opposite arrows of time can reconcile relativity and nonlocality. Class. Quantum Gravity 20, 557–564 (2003). quant-ph/0105040 CrossRefGoogle Scholar
  6. 6.
    Tumulka, R.: A relativistic version of the Ghirardi–Rimini–Weber model. J. Stat. Phys. 125, 821–840 (2006). quant-ph/0406094 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA

Personalised recommendations