Time Observables in a Timeless Universe

Tommaso Favalli1,2 and Augusto Smerzi1

1QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy
2Universitá degli Studi di Napoli Federico II, Via Cinthia 21, I-80126 Napoli, Italy

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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.

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Cited by

[1] Philipp A. Hoehn, Alexander R. H. Smith, and Maximilian P. E. Lock, "Equivalence of approaches to relational quantum dynamics in relativistic settings", arXiv:2007.00580.

[2] Caterina Foti, Alessandro Coppo, Giulio Barni, Alessandro Cuccoli, and Paola Verrucchi, "There is only one time", arXiv:2006.12103.

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