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

TL;DR

This paper explores the deep geometric and algebraic connections between the free particle and harmonic oscillator in quantum mechanics, using projective geometry, Cayley transformations, and Schwarzian derivatives.

Contribution

It unifies the free particle and oscillator frameworks through projective geometry and conformal transformations, revealing new insights into their symmetries and quantum mappings.

Findings

Identifies the quantum Cayley map with the Bargmann transform.

Shows time reparametrizations induce oscillator-like terms via the Schwarzian cocycle.

Provides a unified geometric framework for Schrödinger--Jacobi symmetry and correspondences.

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 Schr\"odinger--Jacobi symmetry and provides a common framework for two seemingly different correspondences: the Cayley--Niederer (lens) map between the time--dependent Schr\"odinger 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.