Fock Representations and Deformation Quantization of Kahler Manifolds
Fock Representations and Deformation Quantization of Kahler Manifolds
复制标题
卡勒流形的 Fock 表示和变形量化
DOI：
10.1007/s00006-016-0753-z
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发表时间：
2017
影响因子：
1.5
通讯作者：
Hiroshi Umetsu
中科院分区：
文献类型：
--
作者：
Akifumi Sako;Hiroshi Umetsu
本文的目的是构造非对易Kähler流形的Fock表示。本文所研究的非对易Kähler流形是由Karabegov提出的分离变量形变量子化方法构造的。非交换Kähler流形的代数包含Heisenberg-like代数。Kähler势的局部复坐标和偏导数满足产生算符和湮灭算符之间的对易关系。Fock空间由在真空上应用创造算符而得到的态构成，该真空被所有湮灭算符湮灭。非交换Kähler流形上的代数被表示为作用在Fock空间上的线性算子的代数。在这里研究的表示中，一般来说，创造算子和湮灭算子不是彼此的厄米共轭。因此，Fock空间的基向量不是对偶向量空间的基向量的厄米共轭。在这个意义上，我们称这种表示为扭曲福克表示。本文构造了任意非对易Kähler流形上由分离变量形变量子化给出的扭曲Fock表示，并给出了扭曲Fock表示与非对易Kähler流形上函数之间的具体转换字典.
The goal of this paper is to construct the Fock representation of noncommutative Kähler manifolds. Noncommutative Kähler manifolds studied here are constructed by deformation quantization with separation of variables, which was given by Karabegov. The algebra of the noncommutative Kähler manifolds contains the Heisenberg-like algebras. Local complex coordinates and partial derivatives of a Kähler potential satisfy the commutation relations between creation and annihilation operators. A Fock space is constituted by states obtained by applying creation operators on a vacuum which is annihilated by all annihilation operators. The algebras on noncommutative Kähler manifolds are represented as those of linear operators acting on the Fock space. In representations studied here, creation operators and annihilation operators are not Hermitian conjugates of one other, in general. Therefore, the basis vectors of the Fock space are not the Hermitian conjugates of those of the dual vector space. In this sense, we call the representation the twisted Fock representation. In this presentation, we construct the twisted Fock representations for arbitrary noncommutative Kähler manifolds given by deformation quantization with separation of variables, and give a dictionary to translate between the twisted Fock representations and functions on noncommutative Kähler manifolds concretely.