Born's Rule for Arbitrary Cauchy Surfaces

Abstract

Suppose that particle detectors are placed along a Cauchy surface $Σ$ 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 $Σ$ has density $|ψ_Σ|^2$, where $ψ_Σ$ is a suitable wave function on $Σ$, 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 $Σ$ 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 $|ψ_Σ|^2$. 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 $ψ_Σ$ can be obtained from the Tomonaga--Schwinger equation, or from a multi-time wave function by inserting configurations on $Σ$. Thus, our result establishes in particular how multi-time wave functions are related to detection probabilities.