非相对论性量子力学的纤维丛表述。II。运动方程和观测量
Fibre bundle formulation of nonrelativistic quantum mechanics. II. Equations of motion and observables
摘要：
提出了一种新的非相对论量子力学系统纤维丛表述。新形式的理论与通常的理论是等价的，但它与理论物理学的现代趋势相协调，并可能在不同的方向上接受新的概括。在它的一个纯态的一些量子系统描述的状态部分（沿路径）的（希尔伯特）纤维丛。它的演化是通过束（类似于）薛定谔方程确定的。现在，动力学变量和密度算子通过丛态射（沿着路径）来描述。所提到的量由本工作中导出的一些关系连接。在本研究的第二部分中，我们导出了几种形式的束（模拟）薛定谔方程的状态截面的时间演化。我们证明，直到一个常数的矩阵束哈密顿量，进入束模拟的矩阵形式的传统薛定谔方程，符合矩阵的系数的演化输运。这允许将哈密顿量解释为规范场。在这里，我们也适用于束的方法来描述可观的。它表明，任何观察有对应于一个独特的厄密丛态射（沿路径），反之亦然。
英文摘要：
We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bundle. Its evolution is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and the density operator are described via bundle morphisms (along paths). The mentioned quantities are connected by a number of relations derived in this work. In the second part of this investigation we derive several forms of the bundle (analogue of the) Schrödinger equation governing the time evolution of state sections. We prove that up to a constant the matrix-bundle Hamiltonian, entering in the bundle analogue of the matrix form of the conventional Schrödinger equation, coincides with the matrix of coefficients of the evolution transport. This allows to interpret the Hamiltonian as a gauge field. Here we also apply the bundle approach to the description of observables. It is shown that to any observable there corresponds a unique Hermitian bundle morphism (along paths) and vice versa.
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