投射时间、凯莱变换与自由粒子-振子对应关系的施瓦茨算子几何
Projective Time, Cayley Transformations and the Schwarzian Geometry of the Free Particle--Oscillator Correspondence
AI总结：
本文从投射几何、凯莱变换和施瓦茨导数角度探讨自由粒子与谐振子关系，揭示SL(2,R)≅Sp(2,R)的共形部分，并建立两种不同对应关系的统一框架。
摘要：
我们从投射几何、凯莱变换和施瓦茨导数的角度，研究一维自由粒子与谐振子之间的关系。将时间视为$\mathbb {RP}^1$上的投射坐标，阐明了Schrödinger-Jacobi对称性的SL(2,R)≅Sp(2,R)共形部分，并为两种看似不同的对应关系提供统一框架：凯莱-尼德雷尔（透镜）映射将时间依赖的薛定谔方程与共形桥变换联系起来。我们将这些关系作为扩展相空间上的规范变换及其元谱提升，并将量子凯莱映射与巴尔曼变换联系起来。一般时间重参数化诱导出振子型项，由施瓦茨协环普遍支配，将当前构造与更广泛的施瓦茨动力学出现联系起来。
英文摘要：
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.
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