Einstein causality of quantum measurements in the Tomonaga-Schwinger picture

Abstract

We investigate a generalisation to L\"uders' rule \`a la Aharonov-Albert in those globally hyperbolic spacetimes which allow unitarily equivalent Hilbert spaces to be defined along Cauchy hypersurfaces, thus relying on the existence of an interaction picture \`a la Tomonaga-Schwinger. We show that under this rule and under the additional assumptions of the integrability and unitarity of the Tomonaga-Schwinger dynamics and the foliation-independence of rays on acausal Cauchy hypersurfaces, selective quantum measurements satisfy a state-independent anyonic commutation relation over spacelike-separated precompact regions. We highlight that this propagates to positive operator-valued measures, where the commutation is necessarily bosonic. In the instantaneous-measurement idealisation, this implies quantum no-signalling for non-selective measurements. We then examine Sorkin's impossible measurements and show that immediate contradictions can be averted as long as collapse-inducing measurements are irreversible. These results reaffirm the consistency of the Tomonaga-Schwinger picture of relativistic quantum theory, for which unitarity, integrability and foliation-independence of the states exclude superluminal signalling despite the ``instantaneity" of a side-cone measurement collapse rule. We finish by discussing the possibility of extending such results beyond the interaction picture.