Advertisement

Born’s rule for arbitrary Cauchy surfaces

  • Matthias LienertEmail author
  • Roderich Tumulka
Article
  • 11 Downloads

Abstract

Suppose that particle detectors are placed along a Cauchy surface \(\Sigma \) in Minkowski space-time, and consider a quantum theory with fixed or variable number of particles (i.e., using Fock space or a subspace thereof). It is straightforward to guess what Born’s rule should look like for this setting: The probability distribution of the detected configuration on \(\Sigma \) has density \(|\psi _\Sigma |^2\), where \(\psi _\Sigma \) is a suitable wave function on \(\Sigma \), and the operation \(|\cdot |^2\) is suitably interpreted. We call this statement the “curved Born rule.” Since, in any one Lorentz frame, the appropriate measurement postulates referring to constant-t hyperplanes should determine the probabilities of the outcomes of any conceivable experiment, they should also imply the curved Born rule. This is what we are concerned with here: deriving Born’s rule for \(\Sigma \) from Born’s rule in one Lorentz frame (along with a collapse rule). We describe two ways of defining an idealized detection process and prove for one of them that the probability distribution coincides with \(|\psi _\Sigma |^2\). For this result, we need two hypotheses on the time evolution: that there is no interaction faster than light and that there is no propagation faster than light. The wave function \(\psi _\Sigma \) can be obtained from the Tomonaga–Schwinger equation, or from a multi-time wave function by inserting configurations on \(\Sigma \). Thus, our result establishes, in particular, how multi-time wave functions are related to detection probabilities.

Keywords

Detection probability Particle detector Tomonaga–Schwinger equation Interaction locality Multi-time wave function Spacelike hypersurface 

Mathematics Subject Classification

81P05 81P15 81P16 81T99 

Notes

Acknowledgements

We thank Detlev Buchholz, Carla Cederbaum, Eddy Keming Chen, Sheldon Goldstein, Sören Petrat, Nicola Pinamonti, Reiner Schätzle, and Stefan Teufel for helpful discussions. Open image in new window This project has received funding from the European Union’s Framework for Research and Innovation Horizon 2020 (2014–2020) under the Marie Skłodowska-Curie Grant Agreement No. 705295.

References

  1. 1.
    Bloch, F.: Die physikalische Bedeutung mehrerer Zeiten in der Quantenelektrodynamik. Phys. Z. Sowjetunion 5, 301–305 (1934)zbMATHGoogle Scholar
  2. 2.
    Cauchy surface. In: Wikipedia, the Free Encyclopedia. http://en.wikipedia.org/wiki/Cauchy_surface Accessed 2 Mar 2018
  3. 3.
    Crater, H.W., Van Alstine, P.: Two-body Dirac equations. Ann. Phys. 148, 57–94 (1983)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Deckert, D.-A., Merkl, F.: External field QED on Cauchy surfaces for varying electromagnetic fields. Commun. Math. Phys. 345, 973–1017 (2016). arXiv:1505.06039 ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Dimock, J.: Dirac quantum fields on a manifold. Trans. AMS 269, 133–147 (1982)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dirac, P.A.M.: Relativistic quantum mechanics. Proc. R. Soc. Lond. A 136, 453–464 (1932)ADSCrossRefGoogle Scholar
  7. 7.
    Dirac, P.A.M., Fock, V.A., Podolsky, B.: On Quantum Electrodynamics. Phys. Z. Sowjetunion 2(6), 468–479 (1932). Reprinted in J. Schwinger: Selected Papers on Quantum Electrodynamics, New York: Dover (1958)Google Scholar
  8. 8.
    Dixmier, J.: Les algèbres d’opérateurs dans l’espace hilbertien. Gauthier-Villars, Paris (1957)zbMATHGoogle Scholar
  9. 9.
    Droz-Vincent, Ph: Second quantization of directly interacting particles. In: Llosa, J. (ed.) Relativistic Action at a Distance: Classical and Quantum Aspects, pp. 81–101. Springer, Berlin (1982)Google Scholar
  10. 10.
    Droz-Vincent, Ph: Relativistic quantum mechanics with non conserved number of particles. J. Geom. Phys. 2(1), 101–119 (1985)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dürr, D., Goldstein, S., Münch-Berndl, K., Zanghì, N.: Hypersurface Bohm–Dirac models. Phys. Rev. A 60, 2729–2736 (1999). arXiv:quant-ph/9801070 ADSCrossRefGoogle Scholar
  12. 12.
    Dürr, D., Goldstein, S., Tumulka, R., Zanghì, N.: Quantum Hamiltonians and stochastic jumps. Commun. Math. Phys. 254, 129–166 (2005). arXiv:quant-ph/0303056 ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Dürr, D., Goldstein, S., Tumulka, R., Zanghì, N.: Bell-type quantum field theories. J. Phys. A: Math. Gen. 38, R1–R43 (2005). arXiv:quant-ph/0407116 ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Dürr, D., Pickl, P.: Flux-across-surfaces theorem for a dirac particle. J. Math. Phys. 44, 423–456 (2003). arXiv:math-ph/0207010 ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Dürr, D., Teufel, S.: On the exit statistics theorem of many particle quantum scattering. In: Blanchard, P., Dell’Antonio, G. (eds.) Multiscale Methods in Quantum Mechanics. Birkhäuser, Basel (2004)zbMATHGoogle Scholar
  16. 16.
    Dürr, D., Teufel, S.: Bohmian Mechanics. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  17. 17.
    Goldstein, S., Taylor, J., Tumulka, R., Zanghì, N.: Fermionic wave functions on unordered configurations (2014). Preprint arXiv:1403.3705
  18. 18.
    Halvorson, H., Clifton, R.: No place for particles in relativistic quantum theories? Philos. Sci. 69, 1–28 (2002). arXiv:quant-ph/0103041 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hegerfeldt, G.: Instantaneous spreading and Einstein causality in quantum theory. Ann. Phys. 7, 716–725 (1998)CrossRefGoogle Scholar
  20. 20.
    Leinaas, J., Myrheim, J.: On the theory of identical particles. Il Nuovo Cimento 37 B, 1–23 (1977)ADSGoogle Scholar
  21. 21.
    Lienert, M.: A relativistically interacting exactly solvable multi-time model for two mass-less Dirac particles in 1 + 1 dimensions. J. Math. Phys. 56, 042301 (2015). arXiv:1411.2833 ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Lienert, M.: On the question of current conservation for the Two-Body Dirac equations of constraint theory. J. Phys. A: Math. Theor. 48, 325302 (2015). arXiv:1501.07027 MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lienert, M.: Lorentz invariant quantum dynamics in the multi-time formalism. Ph.D. thesis, Mathematics Institute, Ludwig-Maximilians University, Munich, Germany (2015)Google Scholar
  24. 24.
    Lienert, M., Nickel, L.: A simple explicitly solvable interacting relativistic \(N\)-particle model. J. Phys. A: Math. Theor. 48, 325301 (2015). arXiv:1502.00917 ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Lienert, M., Petrat, S., Tumulka, R.: Multi-time wave functions. J. Phys.: Conf. Ser. 880, 012006 (2017). arXiv:1702.05282 zbMATHGoogle Scholar
  26. 26.
    Nickel, L., Deckert, D.A.: Consistency of multi-time Dirac equations with general interaction potentials. J. Math. Phys. 57, 072301 (2016). arXiv:1603.02538 ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    O’Neill, B.: Semi-Riemannian Geometry. Academic Press, San Diego (1983)zbMATHGoogle Scholar
  28. 28.
    Parthasarathy, K.S.: Introduction to Quantum Stochastic Calculus. Birkhäuser, Basel (1992)CrossRefGoogle Scholar
  29. 29.
    Petrat, S., Tumulka, R.: Multi-Time Schrödinger Equations Cannot Contain Interaction Potentials. J. Math. Phys. 55, 032302 (2014). arXiv:1308.1065 ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Petrat, S., Tumulka, R.: Multi-time wave functions for quantum field theory. Ann. Phys. 345, 17–54 (2014). arXiv:1309.0802 ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Petrat, S., Tumulka, R.: Multi-time equations, classical and quantum. Proc. R. Soc. A 470(2164), 20130632 (2014). arXiv:1309.1103 ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Petrat, S., Tumulka, R.: Multi-time formulation of pair creation. J. Phys. A: Math. Theor. 47, 112001 (2014). arXiv:1401.6093 ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Rademacher’s theorem. In: Wikipedia, the free encyclopedia. http://en.wikipedia.org/wiki/Rademacher_theorem. Accessed 2 Mar 2018
  34. 34.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol 1: Functional Analysis. Academic Press, San Diego (1980)zbMATHGoogle Scholar
  35. 35.
    Schweber, S.: An Introduction to Relativistic Quantum Field Theory. Row, Peterson and Company (1961)Google Scholar
  36. 36.
    Schwinger, J.: Quantum electrodynamics. I. A covariant formulation. Phys. Rev. 74(10), 1439–1461 (1948)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Thaller, B.: The Dirac Equation. Springer, Berlin (1992)CrossRefGoogle Scholar
  38. 38.
    Tomonaga, S.: On a relativistically invariant formulation of the quantum theory of wave fields. Prog. Theor. Phys. 1(2), 27–42 (1946)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Tumulka, R.: Distribution of the time at which an ideal detector clicks (2016). Preprint arXiv:1601.03715
  40. 40.
    Tumulka, R.: Detection time distribution for the Dirac equation (2016). Preprint arXiv:1601.04571
  41. 41.
    Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Fachbereich MathematikEberhard-Karls-UniversitätTübingenGermany

Personalised recommendations