Another Proof of Born’s Rule on Arbitrary Cauchy Surfaces

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 $$\Sigma $$ Σ , then the observed particle configuration on $$\Sigma $$ Σ is a random variable with distribution density $$|\Psi _\Sigma |^2$$ | Ψ Σ | 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.

Boundary Conditions that Remove Certain Ultraviolet Divergences

“…A version of the Born rule appropriate for φ is the curved Born rule [23,26]: If detectors are placed along a (possibly curved) Cauchy surface Σ (i.e., a spacelike 3surface) in M , then the probability density (relative to the Riemannian volume measure) of finding the configuration (x 1 , . .…”