Geometric View of One-Dimensional Quantum Mechanics

• URL: http://arxiv.org/abs/2512.23923v1
• Date: Tue, 30 Dec 2025 00:48:29 GMT
• Title: Geometric View of One-Dimensional Quantum Mechanics
• Authors: Eren Volkan Küçük,
• Abstract summary: We apply De Haro's Geometric View of Theories to one of the simplest quantum systems.<n>The classical phase space M = T*Q is taken as the base of a trivial Hilbert bundle E M x H.<n>The position and momentum representations are realised as different global trivialisations of this bundle.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We apply De Haro's Geometric View of Theories to one of the simplest quantum systems: a spinless particle on a line and on a circle. The classical phase space M = T*Q is taken as the base of a trivial Hilbert bundle E ~ M x H, and the familiar position and momentum representations are realised as different global trivialisations of this bundle. The Fourier transform appears as a fibrewise unitary transition function, so that the standard position-momentum duality is made precise as a change of coordinates on a single geometric object. For the circle, we also discuss twisted boundary conditions and show how a twist parameter can be incorporated either as a fixed boundary condition or as a base coordinate, in which case it gives rise to a flat U(H)-connection with nontrivial holonomy. These examples provide a concrete illustration of how the Geometric View organises quantum-mechanical representations and dualities in geometric terms.

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