量子演化系统微分几何概念札记(I)：几何相联络、规范势、度规与曲率张量
A Note on the Differential Geometry Concepts in Quantum Evolutional Systems (I): Geometric-Phase Connection, Gauge Potentials, Metric and Curvature Tensor
摘要: 凡是相互作用哈密顿量算符含演化参数的体系或模型，可以被称为量子演化系统。量子演化系统携带拓扑整体特性，呈现几何效应与现象。本文以量子力学与电磁规范理论为源起，研究量子系统演化过程中所涉及的若干微分几何概念的应用。回顾了量子演化系统与几何相位研究简史，以几何相位和规范势(联络)为重点，把量子演化系统看作一个规范么正变换体系，对么正变换性质、作用与标架场作了比较，介绍了计算量子演化系统态函数和几何相位的方法以及蕴含在其内的度规和曲率。本文证明，Lewis- Riesenfeld不变量本征态可以用于构造量子系统演化参数空间中的标架场，求解Schrödinger方程的么正变换算符正是这样的标架场，故而与几何相联络有关的规范群空间内的度规和联络(包括复Levi-Civita和“自旋仿射联络”)以及此二联络所对应的曲率张量也可以分别定义。本文作为一篇教研札记，希冀其内容可能对经典电磁光学、量子光学、约束体系量子力学以及诸种规范场论等专题和交叉问题的理解有一定的借鉴意义。
Abstract: Quantum evolutional systems can be defined as systems/models of interacting Hamiltonian operators with certain evolutional parameters. Quantum evolutional systems carry global or topological characteristics, exhibiting some geometric effects and phenomena. Based on quantum mechanics and electromagnetic gauge theory, we study some topics of the applications of differential geometry concepts in such systems. The brief history of quantum evolutional systems and geometric effects are reviewed with emphasis on the geometric phase and gauge potential (affine connection). The properties of the unitary transformation (related to the Lewis-Riesenfeld invariant formalism) are compared with vielbein fields in a manifold of differential geometry, and it can be found that such a unitary transformation operator can be identified as a vielbein field in gauge group space. The methods of calculating the state functions and geometric phases of the quantum evolutional systems as well as the manifold metric of parameter space in the evolutional Hamiltonian systems will be addressed. As a pedagogical note, the content of this paper would find an application in understanding the topics relevant to classical electromagfnetism, quantum optics, constrained quantum system dynamics, various gauge field theories and interdisciplinary researches.
文章引用：沈建其. 量子演化系统微分几何概念札记(I)：几何相联络、规范势、度规与曲率张量[J]. 现代物理, 2019, 9(6): 289-323. https://doi.org/10.12677/MP.2019.96029

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