Flow of time during energy measurements and the resulting time-energy uncertainty relations

Ismael L. Paiva1, Augusto C. Lobo2, and Eliahu Cohen1

1Faculty of Engineering and the Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 5290002, Israel
2Institute of Exact and Biological Sciences, Federal University of Ouro Preto, Ouro Preto, Minas Gerais 35400-000, Brazil

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Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.

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Cited by

[1] Ismael L. Paiva, Amit Te’eni, Bar Y. Peled, Eliahu Cohen, and Yakir Aharonov, "Non-inertial quantum clock frames lead to non-Hermitian dynamics", Communications Physics 5 1, 298 (2022).

[2] Eliahu Cohen, "Quantum clock frames: Uncertainty relations, non-Hermitian dynamics and nonlocality in time", Journal of Physics: Conference Series 2533 1, 012018 (2023).

[3] Michael Suleymanov and Eliahu Cohen, "Quantum frames of reference and the relational flow of time", The European Physical Journal Special Topics 232 20-22, 3325 (2023).

[4] Michael Suleymanov, Ismael L. Paiva, and Eliahu Cohen, "Nonrelativistic spatiotemporal quantum reference frames", Physical Review A 109 3, 032205 (2024).

[5] Matteo Fadel and Lorenzo Maccone, "Time-energy uncertainty relation for quantum events", Physical Review A 104 5, L050204 (2021).

[6] Leandro R. S. Mendes, Frederico Brito, and Diogo O. Soares-Pinto, "Non-linear equation of motion for Page-Wootters mechanism with interaction and quasi-ideal clocks", arXiv:2107.11452, (2021).

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