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Generally covariant quantum mechanics

Edwin Beggs Orcid Logo, Shahn Majid Orcid Logo

Letters in Mathematical Physics, Volume: 116, Issue: 1, Start page: 9

Swansea University Author: Edwin Beggs Orcid Logo

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DOI (Published version): 10.1007/s11005-025-02036-6

Abstract

We obtain generally covariant operator-valued geodesic equations on a pseudo-Riemannian manifold M as part of the construction of quantum geodesics on the algebra D(M) of differential operators. Geodesic motion arises here as an associativity condition for a certain form of first-order differential...

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Published in: Letters in Mathematical Physics
ISSN: 1573-0530
Published: Springer Nature 2026
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URI: https://cronfa.swan.ac.uk/Record/cronfa71249
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spelling 2026-01-14T10:29:49.5105983 v2 71249 2026-01-14 Generally covariant quantum mechanics a0062e7cf6d68f05151560cdf9d14e75 0000-0002-3139-0983 Edwin Beggs Edwin Beggs true false 2026-01-14 MACS We obtain generally covariant operator-valued geodesic equations on a pseudo-Riemannian manifold M as part of the construction of quantum geodesics on the algebra D(M) of differential operators. Geodesic motion arises here as an associativity condition for a certain form of first-order differential calculus on this algebra in the presence of curvature. The corresponding Schrödinger picture has wave functions on spacetime and proper time evolution by the Klein–Gordon operator, with stationary modes being solutions of the Klein–Gordon equation. As an application, we describe gravatom solutions of the Klein–Gordon equations around a Schwarzschild black hole, i.e. gravitationally bound states which far from the event horizon resemble atomic states with the black hole in the role of the nucleus. The spatial eigenfunctions exhibit probability density banding as for higher orbital modes of an ordinary atom, but of a fractal nature approaching the horizon. Journal Article Letters in Mathematical Physics 116 1 9 Springer Nature 1573-0530 Noncommutative geometry; Quantum mechanics; Black holes; Quantum spacetime; Quantum geodesics; Quantum gravity 1 2 2026 2026-02-01 10.1007/s11005-025-02036-6 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University Another institution paid the OA fee SM was supported by a Leverhulme Trust project grant RPG-2024-177. 2026-01-14T10:29:49.5105983 2026-01-14T10:23:00.4187177 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 71249__35990__622117aeff8848c3b8cf77dd1eede3a1.pdf 11005_2025_Article_2036.pdf 2026-01-14T10:23:00.4180202 Output 1584101 application/pdf Version of Record true © The Author(s) 2026. This article is licensed under a Creative Commons Attribution 4.0 International License. true eng http://creativecommons.org/licenses/by/4.0/
title Generally covariant quantum mechanics
spellingShingle Generally covariant quantum mechanics
Edwin Beggs
title_short Generally covariant quantum mechanics
title_full Generally covariant quantum mechanics
title_fullStr Generally covariant quantum mechanics
title_full_unstemmed Generally covariant quantum mechanics
title_sort Generally covariant 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 Letters in Mathematical Physics
container_volume 116
container_issue 1
container_start_page 9
publishDate 2026
institution Swansea University
issn 1573-0530
doi_str_mv 10.1007/s11005-025-02036-6
publisher Springer Nature
college_str Faculty of Science and Engineering
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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 obtain generally covariant operator-valued geodesic equations on a pseudo-Riemannian manifold M as part of the construction of quantum geodesics on the algebra D(M) of differential operators. Geodesic motion arises here as an associativity condition for a certain form of first-order differential calculus on this algebra in the presence of curvature. The corresponding Schrödinger picture has wave functions on spacetime and proper time evolution by the Klein–Gordon operator, with stationary modes being solutions of the Klein–Gordon equation. As an application, we describe gravatom solutions of the Klein–Gordon equations around a Schwarzschild black hole, i.e. gravitationally bound states which far from the event horizon resemble atomic states with the black hole in the role of the nucleus. The spatial eigenfunctions exhibit probability density banding as for higher orbital modes of an ordinary atom, but of a fractal nature approaching the horizon.
published_date 2026-02-01T05:34:50Z
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score 11.096295

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