Time in quantum mechanics is peculiar: it is an observable that cannot be associated to an Hermitian operator. As a consequence it is impossible to explain dynamics in an isolated system without invoking an external classical clock, a fact that becomes particularly problematic in the context of quantum gravity. An unconventional solution was pioneered by Page and Wootters (PaW) in 1983. PaW showed that dynamics can be an emergent property of the entanglement between two subsystems of a static Universe. In this work we first investigate the possibility to introduce in this framework a Hermitian time operator complement of a clock Hamiltonian having an equally-spaced energy spectrum. An Hermitian operator complement of such Hamiltonian was introduced by Pegg in 1998, who named it "Age". We show here that Age, when introduced in the PaW context, can be interpreted as a proper Hermitian time operator conjugate to a "good" clock Hamiltonian. We therefore show that, still following Pegg's formalism, it is possible to introduce in the PaW framework bounded clock Hamiltonians with an unequally-spaced energy spectrum with rational energy ratios. In this case time is described by a POVM and we demonstrate that Pegg's POVM states provide a consistent dynamical evolution of the system even if they are not orthogonal, and therefore partially undistinguishables.
 P. Busch, The Time-Energy Uncertainty Relation, (2008) on this topic. In: Muga J., Mayato R. S., Egusquiza Í., Time in Quantum Mechanics. Lecture Notes in Physics, vol. 734. Springer, Berlin, Heidelberg.
 L. Maccone and K. Sacha, Phys. Rev. Lett. 124, 110402 (2020).
 D. T. Pegg, J. Phys. A: Math. Gen. 24, 3031-3040 (1991).
 H. Everett, The Theory of the Universal Wave Function. In: The Many Worlds Interpretation of Quantum Mechanics, Princeton University Press, Department of Physics, 1-140 (1957).
 K.V. Kuchar, Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, edited by G. Kunstatter, D. Vincent and J. Williams (World Scientific, Singapore, 1992).
 Caterina Foti, Alessandro Coppo, Giulio Barni, Alessandro Cuccoli, and Paola Verrucchi, "Time and classical equations of motion from quantum entanglement via the Page and Wootters mechanism with generalized coherent states", Nature Communications 12 1, 1787 (2021).
 Philipp A. Höhn, Alexander R. H. Smith, and Maximilian P. E. Lock, "Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings", Frontiers in Physics 9, 587083 (2021).
The above citations are from Crossref's cited-by service (last updated successfully 2021-09-23 14:58:27) and SAO/NASA ADS (last updated successfully 2021-09-23 14:58:28). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.