We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: K\"ahler and non-K\"ahler. Traditional variational methods typically require the variational family to be a K\"ahler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-K\"ahler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.

中文翻译：

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变分方法的几何：封闭量子系统的动力学

我们提出一个系统的几何框架，研究了基于适当的选择变家庭封闭量子系统。为了（A）实时演化，（B）激发光谱，（C）光谱函数和（D）虚构时间演化，我们展示了几何方法如何强调区分两类流形的必要性： ahler和非K \“ ahler。传统的变分方法通常要求变分族为K'ahler流形，其中乘以虚数单位可保留切线空间。这涵盖了文献中研究的大多数情况。但是，最近提出的广义高斯状态类使得还必须包括非Kahler案例，这种情况已经偶然遇到过。我们通过一系列具体示例详细说明了我们的方法，其中所考虑的歧管的几何结构特别相关。这些从高斯状态和分组理论相干状态到广义高斯状态。