Abstract

We explain that when quantising phase spaces with varying symplectic structures, the bundle of quantum Hilbert spaces over the parameter space has a natural unitary connection. We then focus on symplectic vector spaces and their fermionic counterparts. After reviewing how the quantum Hilbert space depends on physical parameters such as the Hamiltonian and unphysical parameters such as choices of polarisations, we study the connection, curvature and phases of the Hilbert space bundle when the phase space structure itself varies. We apply the results to the \(2\)-sphere family of symplectic structures on a hyper-Kähler vector space and to their fermionic analogue, and conclude with possible generalisations.

中文翻译：

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相空间族的量化

摘要

我们解释说，当量化具有不同辛结构的相空间时，参数空间上的量子希尔伯特空间束具有自然的connection连接。然后，我们集中于辛向量空间及其对应的费米离子空间。在回顾了量子希尔伯特空间如何依赖物理参数（例如哈密顿量）和非物理参数（例如极化选择）之后，我们研究了当相空间结构本身变化时希尔伯特空间束的连接，曲率和相位。我们将结果应用于超Kähler向量空间上辛结构的\（2 \）球族及其费米离子类似物，并得出可能的概括。