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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2512.23923 |
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| _version_ | 1866918324964687872 |
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| author | Küçük, Eren Volkan |
| author_facet | Küçük, Eren Volkan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23923 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric View of One-Dimensional Quantum Mechanics Küçük, Eren Volkan Quantum Physics History and Philosophy of Physics 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. |
| title | Geometric View of One-Dimensional Quantum Mechanics |
| topic | Quantum Physics History and Philosophy of Physics |
| url | https://arxiv.org/abs/2512.23923 |