# Born's Rule for Arbitrary Cauchy Surfaces

## TL;DR

This paper derives a generalized Born's rule for arbitrary Cauchy surfaces in Minkowski space-time, connecting wave functions on these surfaces to detection probabilities in relativistic quantum theories.

## Contribution

It provides a derivation of the curved Born rule from standard Lorentz frame measurements, linking multi-time wave functions to detection probabilities on arbitrary Cauchy surfaces.

## Key findings

- Proves the probability distribution matches |_ extsubscript{}^2 for an idealized detection process
- Shows the relation between multi-time wave functions and detection probabilities
- Establishes conditions under which the curved Born rule holds in relativistic quantum theory.

## 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.