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Conditions for unitarity in timeless quantum theory

Simone Rijavec*

  • *Contact author: simoner@tauex.tau.ac.il
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Phys. Rev. D 114, 045012 – Published 17 August, 2026

DOI: https://doi.org/10.1103/kpzt-jvnk

Abstract

Quantum timeless approaches solve the problem of time by recovering the usual unitary evolution of quantum theory relative to a clock in a stationary quantum universe. For some Hamiltonians of the universe, such as those including an interaction term with the clock, the dynamics is substantially altered and can be nonunitary. This work derives necessary and sufficient conditions for the relative dynamics to be unitary and finds the general form of the unitary evolution operator. A physical interpretation of these conditions is given in terms of the clock’s rate. Unitary dynamics is associated with rates that are constant in time and independent of the clock’s internal structure.

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Phys. Rev. D 114, 045012– Published 17 August, 2026
  • Received 4 May 2025
  • Accepted 18 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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