Time delocalization and causality across temporal quantum reference frames

TL;DR

This paper investigates how different choices of quantum clocks affect the perception of time and causality in relational quantum dynamics, revealing conditions for consistent causal relations and scenarios with indefinite causal order.

Contribution

It introduces a second approach to modeling interventions in relational quantum dynamics that accounts for temporal delocalization and indefinite causal order, advancing understanding of quantum causality.

Findings

Multiple clocks can agree on event localization under certain conditions.

Only the second intervention approach ensures consistent causal relations across clocks.

Temporal delocalization naturally leads to scenarios with indefinite causal order.

Abstract

In relational quantum dynamics, evolution emerges via the correlations between some system of interest and a clock system, which plays the role of a temporal reference frame. Their combined state satisfies a Wheeler-de Witt-like constraint equation, and therefore does not evolve, leading to a ``block universe'' picture. Here we investigate the interplay of two aspects, namely temporal localization and causal relations, when comparing emergent dynamics with respect to different choices of clock. We first explore the extent to which two clocks can agree on the temporal localization of events. Then, focussing on the operational notion of causality, we require a clearly defined notion of interventions, i.e. quantum operations, and consider two different approaches to modeling these operations within relational dynamics. The first considers their application via the choice of solutions to the constraint equation, i.e.~the choice of which ``history'' is considered. The second approach incorporates the operations into the constraint equation itself and thereby into its solutions, giving a dynamical picture of the interventions. From the perspective of a single clock, both approaches allow for a notion of operational causality in relational dynamics. However, for multiple clocks, only the second approach gives a consistent picture regarding causal relations, while necessarily manifesting some degree of temporal delocalization between frames. Moreover, this second approach, when considering certain cases of temporal delocalization, naturally describes scenarios with indefinite causal order, a well-known quantum feature of operational causality.