  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.
