Time Observables in a Timeless Universe
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 quan...
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2020-10-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2020-10-29-354/pdf/ |
| _version_ | 1798025264343547904 |
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| author | Tommaso Favalli Augusto Smerzi |
| author_facet | Tommaso Favalli Augusto Smerzi |
| author_sort | Tommaso Favalli |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T18:15:53Z |
| format | Article |
| id | doaj.art-02b82dc5c8d643e385c3924cb54057a2 |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-04-11T18:15:53Z |
| publishDate | 2020-10-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-02b82dc5c8d643e385c3924cb54057a22022-12-22T04:09:53ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2020-10-01435410.22331/q-2020-10-29-35410.22331/q-2020-10-29-354Time Observables in a Timeless UniverseTommaso FavalliAugusto SmerziTime 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.https://quantum-journal.org/papers/q-2020-10-29-354/pdf/ |
| spellingShingle | Tommaso Favalli Augusto Smerzi Time Observables in a Timeless Universe Quantum |
| title | Time Observables in a Timeless Universe |
| title_full | Time Observables in a Timeless Universe |
| title_fullStr | Time Observables in a Timeless Universe |
| title_full_unstemmed | Time Observables in a Timeless Universe |
| title_short | Time Observables in a Timeless Universe |
| title_sort | time observables in a timeless universe |
| url | https://quantum-journal.org/papers/q-2020-10-29-354/pdf/ |
| work_keys_str_mv | AT tommasofavalli timeobservablesinatimelessuniverse AT augustosmerzi timeobservablesinatimelessuniverse |