量子演化系统微分几何概念札记(II)：标架场、自旋联络、旋量与挠率

A Note on the Differential Geometry Concepts in Quantum Evolutional Systems (II): Vielbein, Spin-Affine Connection, Spinors and Torsion

摘要: 一个物理体系，如果相互作用哈密顿量算符含有演化参数，那么其可以被称为量子演化系统。这样的(参数)演化系统可携带几何或整体特性，体现其拓扑效应。研究量子演化系统的前人理论方法包括Lewis-Riesenfeld不变量理论及基于此理论的一种么正变换方法。我们以最简单的二态体系旋量(可用于手工计算)为例，旋量可以用于构造此么正变换，我们探究该么正变换方法的物理含义，发现其可以用于表示规范群空间标架场，构造Yang-Mills规范势。由此我们可以定义非阿贝尔规范群空间的平直与弯曲度规，研究该规范群空间中的复Levi-Civita联络以及自旋联络，也可以计算其存在的扭率和挠率。本文从探讨非厄密哈密顿量系统的一般性质出发，研究了非厄密薛定谔方程和非齐次薛定谔方程的求解方法，然后以二态体系哈密顿量、Lewis-Riesenfeld不变量及其本征态作为基本素材，构造规范标架场、联络、曲率等，再将规范群空间与高维引力规范理论内空间对接起来。我们把非阿贝尔规范场论写成复流形微分几何，而高维引力规范理论内空间正是一个复流形，此为理解和建立引力场和规范场的统一理论奠定数学基础。本文内容的应用并不仅限于规范理论和引力理论。如果将此套方案用于量子力学，可以定义所谓的薛定谔联络。本文得到结论：凡是线性微分方程组，无论是否齐次，均可以定义其数学空间内的“旋量”、标架场、联络、曲率和挠率；即使对于单一的微分方程，只要它含有两个或两个以上自变量(如二维和三维空间中的电磁学和声学Helmholtz方程)且无法用分离变量法求解，那么其中必然会呈现出等效的规范势，而该等效的规范势(联络)也可用标架场表示出来，从而度规、曲率和扭率或挠率均可以定义和计算。本文专题对量子力学、量子光学及电磁理论中部分问题的理解亦有参考价值。

Abstract: A physical system could be referred to as a quantum evolutional one if its interaction Hamiltonian operator contains some evolutional parameters. Such an evolutional system can carry geometric or global characteristics, reflecting its topological effects. Theoretical methods for studying quantum evolutional systems include Lewis-Riesenfeld invariant theory and an invariant-based unitary transformation approach. Taking a kind of simplest two-state system spinors as an example (which can be used for manual calculation), the spinors can be adopted for constructing a unitary transformation. We explore the physical meanings of such a unitary transformation method and find that it can be used to express the mathematical form of Yang-Mills gauge potentials. Then we can define flat and curved metrics of its non-Abelian gauge group space, and study complex Le-vi-Civita connections and spin-affine connections, where the existing contortion and torsion can also be calculated. In this paper, general properties of non-Hermitian Hamiltonian systems will be discussed, and the methods for solving non-Hermitian Schrödinger equation and inhomogeneous Schrödinger equation will be investigated. The Hamiltonian operators, Lewis-Riesenfeld invariants and their eigenstates of such two-state systems will therefore be utilized as elementary tools aiming at constructing the relevant vielbein fields, affine connections, curvatures and so on for the present gauge group space, which is then connected with inner spaces of a higher-dimensional gravitational gauge theory. The non-Abelian gauge field theory can be rewritten as complex manifold differential geometry, and since the inner space in a higher-dimensional gravitational gauge theory is just such a complex manifold, it lays a mathematical foundation for understanding and establishing the theory of unification of gravitational field and gauge field. The application of this paper may not be limited to gauge theory and gravitation. If the scheme is employed in quantum mechanics, a so-called Schrödinger connection can be defined. Some conclusions in this paper can be drawn as follows: In all linear differential equations, whether homogeneous or not, the “spinor” field, vielbein, connection, curvature and torsion in their mathematical (group theoretical) spaces can be defined; even for a single differential equation, which contains two or more independent variables (such as electromagnetic and acoustic Helmholtz equations in two- and three-dimensional spaces) and it is unable to be solved by using the method of separation of variables, an equivalent gauge potential will unavoidably emerge, and such an emergent equivalent potential (connection) can also be expressed in terms of the vielbein fields. Therefore, the metric, curvature and contortion or torsion can be defined. The topics in this paper would also be of interest for understanding some issues in quantum mechanics, quantum optics and electromagnetic field theory.