Time observables in a timeless universe

Year: 2020

Authors: Favalli T., Smerzi A.

Autors Affiliation: INO CNR, QSTAR, Largo Enrico Fermi 2, I-50125 Florence, Italy; LENS, Largo Enrico Fermi 2, I-50125 Florence, Italy; Univ Napoli Federico II, Via Cinthia 21, I-80126 Naples, Italy.

Abstract: 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 un-distinguishables.

Journal/Review: QUANTUM

Volume: 4      Pages from: 354  to: 354

More Information: We acknowledge funding from the project EMPIR-USOQS, EMPIR projects are co-funded by the European Unions Horizon2020 research and innovation programme and the EMPIR Participating States. We also acknowledge financial support from the H2020 QuantERA ERA-NET Cofund in Quantum Technologies projects QCLOCKS.
KeyWords: QUANTUM; GRAVITY
DOI: 10.22331/q-2020-10-29-354

Citations: 5
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