Comment on "Born's rule for arbitrary Cauchy surfaces"

Abstract

A recent article has raised the question of how to generalize the Born rule from non-relativistic quantum theory to curved spacetimes and claimed to answer it for the special-relativistic case (Lienert and Tumulka, Lett. Math. Phys. 110, 753 (2019)). The proposed generalization originated in prior works on `hypersurface Bohm-Dirac models' as well as approaches to relativistic quantum theory developed by Bohm and Hiley. In this comment, we raise three objections to the rule and the broader theory in which it is embedded. In particular, to address the underlying assertion that the Born rule is naturally formulated on a spacelike hypersurface, we provide an analytic example showing that a spacelike hypersurface need not remain spacelike under proper time evolution -- even in the absence of curvature. We finish by proposing an alternative `curved Born rule' for the one-body case on general spacetimes, which overcomes these objections, and in this instance indeed generalizes the one Lienert and Tumulka attempted to justify. The respective mathematical theory is almost analogous for the conservation of charge and mass, being two additional examples of physical quantities obtained from integrating a scalar field over particular hypersurfaces. Our approach can also be generalized to the many-body case, which shall be the subject of a future work.Comment: 12 pages, 3 figures; Keywords: Integral conservation laws, continuity equation, Born rule, detection probability, multi-time wave function, spacelike hypersurfac