Projective Time, Cayley Transformations and the Schwarzian Geometry of the Free Particle--Oscillator Correspondence
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| Format: | Preprint |
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2026
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| _version_ | 1866917409258995712 |
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| author | Alcala, Andrey Plyushchay, Mikhail S. |
| author_facet | Alcala, Andrey Plyushchay, Mikhail S. |
| contents | We investigate the relation between the one--dimensional free particle and the harmonic oscillator from a unified viewpoint based on projective geometry, Cayley transformations, and the Schwarzian derivative. Treating time as a projective coordinate on $\mathbb {RP}^1$ clarifies the $SL(2,\mathbb R)\cong Sp(2,\mathbb R)$ conformal sector of the Schrödinger--Jacobi symmetry and provides a common framework for two seemingly different correspondences: the Cayley--Niederer (lens) map between the time--dependent Schrödinger equations and the conformal bridge transformation relating the stationary problems. We formulate these relations as canonical transformations on the extended phase space and as their metaplectic lifts, identifying the quantum Cayley map with the Bargmann transform. General time reparametrizations induce oscillator--type terms governed universally by the Schwarzian cocycle, connecting the present construction to broader appearances of Schwarzian dynamics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_06378 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Projective Time, Cayley Transformations and the Schwarzian Geometry of the Free Particle--Oscillator Correspondence Alcala, Andrey Plyushchay, Mikhail S. High Energy Physics - Theory Mathematical Physics Exactly Solvable and Integrable Systems Quantum Physics We investigate the relation between the one--dimensional free particle and the harmonic oscillator from a unified viewpoint based on projective geometry, Cayley transformations, and the Schwarzian derivative. Treating time as a projective coordinate on $\mathbb {RP}^1$ clarifies the $SL(2,\mathbb R)\cong Sp(2,\mathbb R)$ conformal sector of the Schrödinger--Jacobi symmetry and provides a common framework for two seemingly different correspondences: the Cayley--Niederer (lens) map between the time--dependent Schrödinger equations and the conformal bridge transformation relating the stationary problems. We formulate these relations as canonical transformations on the extended phase space and as their metaplectic lifts, identifying the quantum Cayley map with the Bargmann transform. General time reparametrizations induce oscillator--type terms governed universally by the Schwarzian cocycle, connecting the present construction to broader appearances of Schwarzian dynamics. |
| title | Projective Time, Cayley Transformations and the Schwarzian Geometry of the Free Particle--Oscillator Correspondence |
| topic | High Energy Physics - Theory Mathematical Physics Exactly Solvable and Integrable Systems Quantum Physics |
| url | https://arxiv.org/abs/2602.06378 |