Optics, Quantisation and the Superintegrable Zernike System

量子化的思想支撑了现代物理学的许多结构。粗略地说，量子化在旧的数学结构的基础上建立了一个新的数学结构，以这种方式尊重旧理论的对称性。量子化过程是量子力学和量子场论中的基本结构，在抽象代数几何中得到了应用，在这个项目中，我们将讨论它们在描述纯经典物理中的用途。特别是在描述发生在激光腔中的射线光学和波动光学之间的转变时。这样的模型包括了量子力学的大部分结构(通过近轴近似)，同时保留了对研究有用的关键差异。我们希望对呈现典型病理特征(尤其是无界解、曲率和非线性的存在)的系统的光线光学量子化将导致对量子化理论及其解释的有用的洞察，并有助于当前光学几何学的蓬勃发展。

The idea of quantisation underpins much of the structure of modern physics. Roughly speaking, quantisation builds a new mathematical structure out of an old one, in such a way that the symmetries of the old theory are respected. Quantisation process are the foundational constructions in quantum mechanics and quantum field theory, have found applications in abstract algebraic geometry and, in this project, we will argue for their use in describing purely classical physics. In particular, in describing the transition between ray optics and wave optics occurring in laser cavities. Such a model includes much of the structure of quantum mechanics (via the paraxial approximation), while retaining crucial differences that make is useful to study. We hope the ray-optical quantisation of systems exhibiting typically pathological features (notably the presence of unbounded solutions, curvature and nonlinearites) will lead to useful insight into the theory of quantisation, its interpretation and contribute to the current flourishing of geometry in optics.