非相对论性量子力学的纤维丛表述。IV。混合态与演化输运曲率
Fibre bundle formulation of nonrelativistic quantum mechanics. IV. Mixed states and evolution transport's curvature
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）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。本系列的第四部分主要致力于混合量子态的纤维丛描述。我们表明，传统的密度算子对应于一个独特的密度态射（沿路径），相应的运动方程推导。研究了不同运动图像下混合量子态的束描述。我们计算了演化输运的曲率，证明了它是曲率自由的当且仅当哈密顿算符的值在不同时刻交换。
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. It's 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. The present fourth part of this series is devoted mainly to the fibre bundle description of mixed quantum states. We show that to the conventional density operator there corresponds a unique density morphism (along paths) for which the corresponding equations of motion are derived. It is also investigated the bundle description of mixed quantum states in the different pictures of motion. We calculate the curvature of the evolution transport and prove that it is curvature free iff the values of the Hamiltonian operator at different moments commute.
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