AI 中文总结
本研究将$\u0026#x1D4A9;$-bein几何框架拓展至简并量子系统参数空间，定义非阿贝尔双态QGT与类挠率张量，构造规范不变可观测量，并在耦合谐振子系统中验证其应用价值。
AI 中文摘要
近期，我们引入了一种与嘉当形式体系中正交标架类似的几何对象来研究量子系统的参数空间，将其命名为$\u0026#x1D4A9;$-bein，其中$\u0026#x1D4A9;$为表征量子系统的参数数量。作为量子几何张量（QGT）的“平方根”，$\u0026#x1D4A9;$-bein使我们能够定义新的张量，以深化对量子力学参数空间底层结构的理解。在本工作中，我们将围绕$\u0026#x1D4A9;$-bein的这一数学框架拓展至分析具有简并能谱的量子系统的参数空间。与非简并情形类似，我们定义了一个非阿贝尔双态QGT，用于识别两次连续参数变化后简并态之间可能发生的跃迁。此外，我们利用Wilczek-Zee联络，引入了一种类挠率张量，作为$\u0026#x1D4A9;$-bein的协变导数。该挠率刻画了连续参数变化的非对易性，且与双态QGT的反对称部分一致。我们还提出了一种使用微分形式的几何表述，并讨论了新定义张量的物理意义。此外，我们从$\u0026#x1D4A9;$-bein及其导数出发构造了若干规范不变可观测量，以凸显新张量的实用性。最后，为说明该形式体系的便捷性与应用场景，我们将这一理论框架应用于一个处于电场中的耦合谐振子系统。谐振子之间的耦合会形成简并系统，因此借助新的形式体系，我们发现了由新不变量量化的量子态之间的关联。
英文摘要
Recently, we introduced a geometric object analogous to an orthonormal frame in the Cartan formalism to study the parameter space of quantum systems; we called it $\mathcal{N}$-bein, with $\mathcal{N}$ being the number of parameters that characterize the quantum system. Acting as the ``square root'' of the quantum geometric tensor (QGT), the $\mathcal{N}$-bein allows us to define new tensors to improve our understanding of the structure beneath the parameter space of quantum mechanics. In this work, we extend this mathematical framework surrounding the $\mathcal{N}$-bein to analyze the parameter space of quantum systems with degenerate spectra. As in the non-degenerate case, we define a non-Abelian two-state QGT to identify possible transitions between degenerate states after two consecutive parameter variations. Additionally, using the Wilczek-Zee connection, we introduce a torsion-like tensor as the covariant derivative of the $\mathcal{N}$-bein. This torsion captures the noncommutativity of successive parameter variations and coincides with the antisymmetric part of the two-state QGT. We also present a geometrical formulation using differential forms and discuss the physical implications of the newly defined tensors. Furthermore, we construct several gauge-invariant observables from the $\mathcal{N}$-bein and its derivatives to highlight the utility of the new tensors. Finally, to illustrate the convenience and applications of this formalism, we apply the theoretical framework to a system of coupled harmonic oscillators immersed in an electric field. The coupling between the oscillators results in a degenerate system. Thus, using the new formalism, we found correlations among the quantum states quantified by the new invariants.
Comments33 pages, 7 figures. Accepted version (Journal of Physics A: Mathematical and Theoretical)
Journal refJorge Romero et al 2026 J. Phys. A: Math. Theor. 59 355305
