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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
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa65249
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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 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.
College: Faculty of Science and Engineering
Issue: 1

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