Abstract
In 2017, Lienert and Tumulka proved Born’s rule on arbitrary Cauchy surfaces in Minkowski space-time assuming Born’s rule and a corresponding collapse rule on horizontal surfaces relative to a fixed Lorentz frame, as well as a given unitary time evolution between any two Cauchy surfaces, satisfying that there is no interaction faster than light and no propagation faster than light. Here, we prove Born’s rule on arbitrary Cauchy surfaces from a different, but equally reasonable, set of assumptions. The conclusion is that if detectors are placed along any Cauchy surface Sigma , then the observed particle configuration on Sigma is a random variable with distribution density |Psi _Sigma |^2, suitably understood. The main different assumption is that the Born and collapse rules hold on any spacelike hyperplane, i.e., at any time coordinate in any Lorentz frame. Heuristically, this follows if the dynamics of the detectors is Lorentz invariant.
Highlights
In its usual form, Born’s rule asserts that if we measure the positions of all particles of a quantum system at time t, the observed configuration has probability distribution with density |Ψt|2
We prove here the curved Born rule as a theorem; more precisely, we prove that the Born rule holds on arbitrary Cauchy surfaces assuming (i) that the Born rule holds on hyperplanes, i.e., on flat surfaces, (ii) that the collapse rule holds on hyperplanes, (iii) that the unitary time evolution contains no interaction terms between spacelike separated regions, and (iv) that wave functions do not spread faster than light
As one would in quantum electrodynamics or quantum chromodynamics, exclude states of negative energy; it remains for future work to extend our result in this direction
Summary
In its usual form, Born’s rule asserts that if we measure the positions of all particles of a quantum system at time t, the observed configuration has probability distribution with density |Ψt|2. One would expect that Born’s rule holds on arbitrary Cauchy surfaces Σ in Minkowski space-time M in the following sense: If we place detectors along Σ, the observed particle configuration has probability distribution with density |ΨΣ|2, suitably understood. We call the latter statement the curved Born rule; it contains the former statement as a special case in which Σ is a horizontal 3-plane in the chosen Lorentz frame.
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