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

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.

💡 Research Summary

The paper presents a unified geometric framework that connects the one‑dimensional free particle and the harmonic oscillator through projective time, Cayley transformations, and the Schwarzian derivative. By treating the time variable as a projective coordinate on the real projective line RP¹, the authors expose the SL(2,ℝ) ≅ Sp(2,ℝ) conformal sector of the Schrödinger–Jacobi symmetry as a global Möbius action on time.

In Section 2 a complexified canonical transformation is introduced via the 2 × 2 Cayley matrix
(C=\frac{1}{\sqrt2}\begin{pmatrix}1&-i\-i&1\end{pmatrix}).
C maps the real phase‑space basis ((q,p)^T) to the complex basis ((a_+, -i a_-)^T) and belongs simultaneously to Sp(2,ℂ) and SU(2). Conjugation by C rotates the standard sl(2,ℝ) generators ((H_+, D, H_-)) into the SU(1,1) generators, effecting a Wick rotation of the corresponding flows. This matrix also implements the classical Cayley map that sends the upper half‑plane (\mathbb H^+) (the projective model of time) to the unit disc (\mathbb D), thereby identifying the ideal boundary RP¹ with the circle S¹.

Section 3 reviews the free‑particle dynamics, emphasizing that time translations, dilations, and special conformal transformations form an sl(2,ℝ) triple acting projectively on time, and together with the Heisenberg algebra generate the full Schrödinger (Jacobi) algebra.

In Section 4 the authors quantize the Cayley transformation. The canonical map lifts to the metaplectic double cover Mp(2,ℝ), and the resulting unitary operator is shown to coincide with the Bargmann transform. Hence the passage from the coordinate (Schrödinger) representation to the holomorphic Bargmann–Fock representation is precisely the quantum Cayley map.

Section 5 constructs the Conformal Bridge Transformation (CBT) at the level of stationary Schrödinger equations. By applying the same metaplectic operator that implements the quantum Cayley map, the free‑particle eigenfunctions are mapped to oscillator eigenfunctions, providing a spectral counterpart of the TDSE Cayley–Niederer map.

Section 6 reformulates the Cayley–Niederer correspondence for the oscillator in the compact Newton–Hooke realization, where time becomes an angular variable on S¹ and the sl(2,ℝ) sector is realized by quadratic phase‑space generators.

Section 7 treats simultaneous coordinate and time reparametrizations. The authors demonstrate that wavefunctions must be regarded as (±½)-densities; under a general time change (t\mapsto \tau(t)) a metaplectic half‑density factor appears, guaranteeing the preservation of the symplectic structure.

Section 8 introduces a Schwarzian term generated by arbitrary time reparametrizations. The Schwarzian cocycle ({ \tau,t}) appears universally as a correction to the Hamiltonian, producing an additional quadratic potential ( \propto { \tau,t}, q^2). This shows that the Schwarzian derivative is the projective invariant governing how time reparametrizations induce oscillator‑type terms.

Section 9 extends the analysis to generic reparametrizations, deriving a factorized metaplectic operator controlled by the Ermakov–Pinney amplitude. This operator explicitly implements the map between the free particle and the oscillator for any smooth time change, making the role of the Schwarzian transparent.

The appendices collect technical material: compatible complex structures on ((\mathbb R^2,\omega)) and their parametrization by (\mathbb H^+) (Appendix A); a review of the metaplectic representation (B); Jordan states and analyticity in energy (C); connections between Sp(2,ℝ) factorizations and ABCD optics (D); derivation of squeezed‑vacuum Gaussian formulas (E); a catalogue of Schrödinger‑group Gaussian packets (F); a compendium of Schwarzian identities and the cocycle property (G); and comments on exact versus semiclassical quantization of canonical transformations (H).

Overall, the paper demonstrates that (i) the time‑dependent Schrödinger‑equation Cayley–Niederer map, (ii) the stationary‑equation Conformal Bridge Transformation, and (iii) the universal Schwarzian correction arising from time reparametrizations are all manifestations of a single projective‑geometric and metaplectic structure. This unification not only clarifies the deep symmetry linking the free particle and the harmonic oscillator but also connects to modern contexts where the Schwarzian action appears, such as the low‑energy sector of the Sachdev‑Ye‑Kitaev model and two‑dimensional Jackiw–Teitelboim gravity.