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Quantum geodesics in quantum mechanics

Edwin Beggs Orcid Logo, Shahn Majid Orcid Logo

Journal of Mathematical Physics, Volume: 65, Issue: 1

Swansea University Author: Edwin Beggs Orcid Logo

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DOI (Published version): 10.1063/5.0154781

Abstract

We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus $\Omega^1$ depending on the Hamiltonian $p^2/2m + V(x)$, and a flat quantum connection $\nabla$ with torsion such that a previous quantum-geometric formulation of flow along autoparallel c...

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Published in: Journal of Mathematical Physics
ISSN: 0022-2488 1089-7658
Published: AIP Publishing 2024
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URI: https://cronfa.swan.ac.uk/Record/cronfa65249
first_indexed 2023-12-07T21:02:55Z
last_indexed 2024-11-25T14:15:40Z
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spelling 2024-04-10T11:11:14.0721235 v2 65249 2023-12-07 Quantum geodesics in quantum mechanics a0062e7cf6d68f05151560cdf9d14e75 0000-0002-3139-0983 Edwin Beggs Edwin Beggs true false 2023-12-07 MACS We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus $\Omega^1$ depending on the Hamiltonian $p^2/2m + V(x)$, and a flat quantum connection $\nabla$ with torsion such that a previous quantum-geometric formulation of flow along autoparallel curves (or `geodesics') is exactly Schr\"odinger's equation. The connection $\nabla$ preserves a non-symmetric quantum metric given by the canonical symplectic structure lifted to a rank (0,2) tensor on the extended phase space where we adjoin a time variable. We also apply the same approach to obtain a novel flow generated by the Klein Gordon operator on Minkowski spacetime with a background electromagnetic field, by formulating quantum `geodesics' on the relativistic Heisenberg algebra with proper time for the external geodesic parameter. Examples include quantum geodesics that look like a relativistic free particle wave packet and a hydrogen-like atom. Journal Article Journal of Mathematical Physics 65 1 AIP Publishing 0022-2488 1089-7658 5 1 2024 2024-01-05 10.1063/5.0154781 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University SU Library paid the OA fee (TA Institutional Deal) 2024-04-10T11:11:14.0721235 2023-12-07T20:52:33.3564189 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Edwin Beggs 0000-0002-3139-0983 1 Shahn Majid 0000-0003-1657-5434 2 65249__29453__15eb29ae3a2d4a029fedb8b1870f4e34.pdf 012101_1_5.0154781.pdf 2024-01-17T16:28:44.6586946 Output 5902837 application/pdf Version of Record true © 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license true eng http://creativecommons.org/licenses/by/4.0/).
title Quantum geodesics in quantum mechanics
spellingShingle Quantum geodesics in quantum mechanics
Edwin Beggs
title_short Quantum geodesics in quantum mechanics
title_full Quantum geodesics in quantum mechanics
title_fullStr Quantum geodesics in quantum mechanics
title_full_unstemmed Quantum geodesics in quantum mechanics
title_sort Quantum geodesics in quantum mechanics
author_id_str_mv a0062e7cf6d68f05151560cdf9d14e75
author_id_fullname_str_mv a0062e7cf6d68f05151560cdf9d14e75_***_Edwin Beggs
author Edwin Beggs
author2 Edwin Beggs
Shahn Majid
format Journal article
container_title Journal of Mathematical Physics
container_volume 65
container_issue 1
publishDate 2024
institution Swansea University
issn 0022-2488
1089-7658
doi_str_mv 10.1063/5.0154781
publisher AIP Publishing
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
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description We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus $\Omega^1$ depending on the Hamiltonian $p^2/2m + V(x)$, and a flat quantum connection $\nabla$ with torsion such that a previous quantum-geometric formulation of flow along autoparallel curves (or `geodesics') is exactly Schr\"odinger's equation. The connection $\nabla$ preserves a non-symmetric quantum metric given by the canonical symplectic structure lifted to a rank (0,2) tensor on the extended phase space where we adjoin a time variable. We also apply the same approach to obtain a novel flow generated by the Klein Gordon operator on Minkowski spacetime with a background electromagnetic field, by formulating quantum `geodesics' on the relativistic Heisenberg algebra with proper time for the external geodesic parameter. Examples include quantum geodesics that look like a relativistic free particle wave packet and a hydrogen-like atom.
published_date 2024-01-05T16:07:16Z
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score 11.086808

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