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  • Open Access

Time delocalization and causality across temporal quantum reference frames

Veronika Baumann1,2,* and Maximilian P. E. Lock2,1,†

  • 1Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, 1090 Vienna, Austria
  • 2Atominstitut, TU Wien, 1020 Vienna, Austria

  • *Contact author: veronika.baumann@oeaw.ac.at
  • †Contact author: maximilian.paul.lock@tuwien.ac.at
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Phys. Rev. D 114, 086005 – Published 5 October, 2026

DOI: https://doi.org/10.1103/9pl6-z9g4

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. 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, focusing 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.

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Phys. Rev. D 114, 086005– Published 5 October, 2026
  • Received 20 March 2026
  • Accepted 10 August 2026
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