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Projective Time, Cayley Transformations and the Schwarzian Geometry of the Free Particle--Oscillator Correspondence

Andrey Alcala, Mikhail S. Plyushchay

TL;DR

This work develops a unified, projective-geometric framework for relating the one-dimensional free particle and harmonic oscillator. By embedding time into RP¹ and employing the rotated Cayley matrix, it links the TDSE Cayley–Niederer lens transform with the SSE conformal bridge transformation, all wrapped in a metaplectic formalism that also realizes the Bargmann transform as a canonical intertwinement. The Schwarzian derivative appears as a universal cocycle: it governs oscillator-type terms under general time reparametrizations and encodes the projective corrections in both Liouville-type and unitary settings. The framework yields exact (metaplectic) factorisations for constant Schwarzian cases and a constructive Ermakov–Pinney approach for general reparametrizations, offering a robust route to extend FP–HO dualities to time-dependent backgrounds and related areas in mathematical physics.

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 $\mathbb {RP}^1$ 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.

Projective Time, Cayley Transformations and the Schwarzian Geometry of the Free Particle--Oscillator Correspondence

TL;DR

This work develops a unified, projective-geometric framework for relating the one-dimensional free particle and harmonic oscillator. By embedding time into RP¹ and employing the rotated Cayley matrix, it links the TDSE Cayley–Niederer lens transform with the SSE conformal bridge transformation, all wrapped in a metaplectic formalism that also realizes the Bargmann transform as a canonical intertwinement. The Schwarzian derivative appears as a universal cocycle: it governs oscillator-type terms under general time reparametrizations and encodes the projective corrections in both Liouville-type and unitary settings. The framework yields exact (metaplectic) factorisations for constant Schwarzian cases and a constructive Ermakov–Pinney approach for general reparametrizations, offering a robust route to extend FP–HO dualities to time-dependent backgrounds and related areas in mathematical physics.

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 clarifies the 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.
Paper Structure (24 sections, 304 equations)

This paper contains 24 sections, 304 equations.

Table of Contents

  1. Introduction
  2. Cayley transform and complexified canonical transformation
  3. Free particle: Möbius and Schrödinger symmetries
  4. Quantum Cayley map: Bargmann transform and the complex-metaplectic operator
  5. Bargmann--Fock normalization ($\mathcal{B}\psi_0=1$).
  6. Schrödinger-side normalization ($U_C\chi_n=\psi_n$) and the factor $2^{1/4}$.
  7. Quantum Cayley transform and conformal bridge transformation
  8. Classical symmetries of the harmonic oscillator and Cayley--Niederer transform
  9. Stationary and time-dependent Schrödinger equations, and (+/-1/2)-weight densities
  10. Extended free particle and the Schwarzian-induced oscillator
  11. Quantum implementation and distributional collapse.
  12. Remark (sign of the chirp).
  13. General reparametrizations F and a factorized metaplectic form: the Ermakov--Pinney amplitude
  14. Summary, Discussion and Outlook
  15. Acknowledgments.
  16. ...and 9 more sections