Born's Rule for Arbitrary Cauchy Surfaces

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.

53 pages LaTeX, 11 figures

References in corpus (2)

• Quantum Hamiltonians and Stochastic Jumps
• A relativistically interacting exactly solvable multi-time model for two mass-less Dirac particles in 1+1 dimensions

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