We present a study of the properties of Bargmann Invariants (BI) and Null Phase Curves (NPC) in the theory of the geometric phase for finite dimensional systems. A recent suggestion to exploit the Majorana theorem on symmetric SU(2) multispinors is combined with the Schwinger oscillator operator construction to develop efficient operator based methods to handle these problems. The BI is described using intrinsic unitary invariant angle parameters, whose algebraic properties as functions of Hilbert space dimension are analysed using elegant group theoretic methods. The BI-geometric phase connection, extended by the use of NPC's, is explored in detail, and interesting new experiments in this subject are pointed out.

中文翻译：

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有限维系统的几何相位 - Bargmann 不变量、零相位曲线和 Schwinger-Majorana SU(2) 框架的作用

我们在有限维系统的几何相位理论中研究了巴格曼不变量 (BI) 和空相位曲线 (NPC) 的特性。最近提出的利用对称 SU(2) 多旋量的 Majorana 定理的建议与 Schwinger 振荡器算子构造相结合，以开发基于算子的有效方法来处理这些问题。BI 使用固有的酉不变角参数来描述，其代数特性作为希尔伯特空间维数的函数使用优雅的群论方法进行分析。详细探讨了通过使用 NPC 扩展的 BI 几何相位连接，并指出了该主题中有趣的新实验。