Time in quantum mechanics and the Page-Wootters formalism

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2023
Authors
De Rosa, Guido
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University College Cork
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Abstract
This work relates to the "problem of time", the difficulties of which represent a classic problem in the foundations of quantum mechanics. It can be traced back to the argument, by Wolfgang Pauli, on the impossibility of defining time as a quantum observable, due to contradictions related to the spectrum of energy (1933). A number of models, aimed at overcoming such difficulties, have been developed over the decades, and are reviewed in the present work. Particular attention is paid to the theoretical framework first proposed by Page and Wootters in 1983 (later improved by Lloyd, Giovannetti and Maccone and others), where time and unitary evolution only emerge in terms of entanglement between non-interacting subsystems of an otherwise stationary "universe'", and where one of the subsystems acts as a "clock'" for the `"rest" of it. Discrete clocks, within the framework, are implemented, using existing results related to quantum systems described by finite-dimensional Hilbert spaces. The formalism is then applied to some simple quantum systems, and a numerical comparison is performed between the Page--Wootters model and the predictions of "ordinary'" quantum mechanics. The Page and Wootters formalism is also applied to non-unitary systems, such as those modeled in absorption theories of time-of-arrival at a particle detector. A comparison is made between the predictions of the two models as well. The case of a differing result opens the way to the potential of an experimental verification. Future lines of research may cover a broader phenomenology, and include a relativistic extension, which is briefly introduced.
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Quantum , Time , Foundations
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De Rosa, G. 2023. Time in quantum mechanics and the Page-Wootters formalism. MSc Thesis, University College Cork.