Born’s rule for arbitrary Cauchy surfaces

  • Matthias LienertEmail author
  • Roderich Tumulka


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.


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

Mathematics Subject Classification

81P05 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.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Fachbereich MathematikEberhard-Karls-UniversitätTübingenGermany

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