重温2x2矩阵光学：复向量、费米子组合学和拉格朗日不变量
Revisiting 2x2 matrix optics: Complex vectors, Fermion combinatorics, and Lagrange invariants
摘要：
通过几何代数分析，提出矩阵光学中的高度角光线矢量应该是复数的。我们还提出光线的2 × 2矩阵算符应该是右作用的，这样矩阵乘积的连续性将与光的从左到右传播相一致。我们表示的传播和折射运营商作为一个单位矩阵和虚数矩阵的费米子创建或湮灭矩阵成比例的总和。通过这种方式，我们将矩阵算子的乘积约化为生成-湮灭乘积组合的和。我们将ABCD光学系统分为四类：望远光学系统、傅里叶逆变换光学系统、傅里叶变换光学系统和成像光学系统。我们表明，这些系统中的每一个都有一个相应的拉格朗日定理表示在偏导数，只有望远镜和成像系统有拉格朗日不变量。
英文摘要：
We propose that the height-angle ray vector in matrix optics should be complex, based on a geometric algebra analysis. We also propose that the ray's 2x2 matrix operators should be right-acting, so that the matrix product succession would go with light's left-to-right propagation. We express the propagation and refraction operators as a sum of a unit matrix and an imaginary matrix proportional to the Fermion creation or annihilation matrix. In this way, we reduce the products of matrix operators into sums of creation-annihilation product combinations. We classify ABCD optical systems into four: telescopic, inverse Fourier transforming, Fourier transforming, and imaging. We show that each of these systems have a corresponding Lagrange theorem expressed in partial derivatives, and that only the telescopic and imaging systems have Lagrange invariants.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
