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Projective time, Cayley transformations and the Schwarzian geometry of the free particle-oscillator correspondence

Andrey Alcala and Mikhail S. Plyushchay*

  • *Contact author: mikhail.plyushchay@usach.cl
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Phys. Rev. D 113, 086007 – Published 13 April, 2026

DOI: https://doi.org/10.1103/ks12-b2v5

Abstract

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 RP1 clarifies the SL(2,R)≅Sp(2,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.

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Phys. Rev. D 113, 086007– Published 13 April, 2026
  • Received 11 February 2026
  • Accepted 18 March 2026
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