Born’s rule for arbitrary Cauchy surfaces
- 11 Downloads
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.
KeywordsDetection probability Particle detector Tomonaga–Schwinger equation Interaction locality Multi-time wave function Spacelike hypersurface
Mathematics Subject Classification81P05 81P15 81P16 81T99
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.
- 2.Cauchy surface. In: Wikipedia, the Free Encyclopedia. http://en.wikipedia.org/wiki/Cauchy_surface Accessed 2 Mar 2018
- 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
- 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
- 17.Goldstein, S., Taylor, J., Tumulka, R., Zanghì, N.: Fermionic wave functions on unordered configurations (2014). Preprint arXiv:1403.3705
- 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
- 33.Rademacher’s theorem. In: Wikipedia, the free encyclopedia. http://en.wikipedia.org/wiki/Rademacher_theorem. Accessed 2 Mar 2018
- 35.Schweber, S.: An Introduction to Relativistic Quantum Field Theory. Row, Peterson and Company (1961)Google Scholar
- 39.Tumulka, R.: Distribution of the time at which an ideal detector clicks (2016). Preprint arXiv:1601.03715
- 40.Tumulka, R.: Detection time distribution for the Dirac equation (2016). Preprint arXiv:1601.04571