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Quantum space, quantum time, and relativistic quantum mechanics

Singh, Ashmeet (2022) Quantum space, quantum time, and relativistic quantum mechanics. Quantum Studies: Mathematics and Foundations, 9 (1). pp. 35-53. ISSN 2196-5609. doi:10.1007/s40509-021-00255-9.

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We treat space and time as bona fide quantum degrees of freedom on an equal footing in Hilbert space. Motivated by considerations in quantum gravity, we focus on a paradigm dealing with linear, first-order Hamiltonian and momentum constraints that lead to emergent features of temporal and spatial translations. Unlike the conventional treatment, we show that Klein-Gordon and Dirac equations in relativistic quantum mechanics can be unified in our paradigm by applying relativistic dispersion relations to eigenvalues rather than treating them as operator-valued equations. With time and space being treated on an equal footing in Hilbert space, we show symmetry transformations to be implemented by unitary basis changes in Hilbert space, giving them a stronger quantum mechanical footing. Global symmetries, such as Lorentz transformations, modify the decomposition of Hilbert space; and local symmetries, such as U(1) gauge symmetry are diagonal in coordinate basis and do not alter the decomposition of Hilbert space. We briefly discuss extensions of this paradigm to quantum field theory and quantum gravity.

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Singh, Ashmeet0000-0002-4404-1416
Additional Information:© The Author(s) under exclusive license to Chapman University 2021. Received 05 February 2021; Accepted 05 September 2021; Published 22 September 2021. I would like to thank Sean Carroll, Charles Cao, Aidan Chatwin-Davies, Swati Chaudhary and Frank Porter for helpful discussions during the course of this project. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632, as well as by the Walter Burke Institute for Theoretical Physics at Caltech and the Foundational Questions Institute. On behalf of all authors, the corresponding author states that there is no conflict of interest.
Group:Walter Burke Institute for Theoretical Physics
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Department of Energy (DOE)DE-SC0011632
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Foundational Questions Institute (FQXI)UNSPECIFIED
Subject Keywords:Relativistic quantum mechanics; Emergent time; Quantum gravity
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Issue or Number:1
Record Number:CaltechAUTHORS:20200423-160749656
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Official Citation:Singh, A. Quantum space, quantum time, and relativistic quantum mechanics. Quantum Stud.: Math. Found. 9, 35–53 (2022).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:102760
Deposited By: Joy Painter
Deposited On:23 Apr 2020 23:14
Last Modified:19 Jan 2022 18:03

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