Application of Deformation Quantization theory to Geometry and Mathematical Physics

形变量子化理论在几何与数学物理中的应用

• 批准号: 13640088
• 负责人: YOSHIOKA Akira
• 金额: $ 1.86万
• 依托单位: Tokyo University of Science
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640088/
• 关键词: Deformation Quantization; star products; non-commutative geometry; Hamiltonian mechanics; simplistic geometry; quantization; Deformation quantization; Hamiltonian mechanics; deformation quantization; 非可換幾何学; 量子力学; シンプレクティック幾何学

In this research program we obtain the following results.1. In the complex 2n Euclidian space we introduce the Modyal products, the normal product and the anti-normal *oduct. We set up a function space, for which these three products are convergent and have meaning. We deonte by F(p) the space of all entire functions of order p (p is nonnegative). Then for p equal to or less than 2, the space F(p) becomes a noncommutative topological algebra with repect to each product2. By means of differential equations, we construct exponential functions of quadratic functions for these products respectively. Thank to the formula of the expoenetial functions, we investigate their singularities with resect to the time variable. We define the Laplace transform by means of the exponential and we construct several transcendental elements of the noncommutative algebra3. We consider the Weyl algebra W of 2n generators over the complex number. The Weyl ordring, the normal ordering and the anti-normal ordering on the Weyl algebra give isomorphisms of W to the space of all polynomials on C, P(C, 2n). These isomorphisms naturally induces noncommutative, associative products on P(C, 2n). We remarked these produ*Composing these isomorphisms gines intertwiners between the algebras on P(C, 2n) and the Freshet space F(p). Patching these algebra together by these intertwiner we construct a certain noncommutative manifoldWe show the noncommutative manifold is not a manifold in the ordinary sense by has the garb structure

在本研究中我们取得了以下成果： 1．在复杂的2n欧几里得空间中我们引入了Modyal积、正规积和反正规*积。我们设置了一个功能空间，这三个产品在这个功能空间中是收敛的并且有意义的。我们用 F(p) 解构所有 p 阶函数的空间（p 是非负的）。那么对于 p 等于或小于 2，空间 F(p) 就成为关于每个乘积 2 的非交换拓扑代数。通过微分方程，我们分别构造了这些乘积的二次函数的指数函数。由于指数函数的公式，我们研究了它们相对于时间变量的奇异性。我们通过指数定义拉普拉斯变换，并构造非交换代数的几个超越元素。我们考虑复数上 2n 个生成元的韦尔代数 W。 Weyl 序环、Weyl 代数上的正态序和反正态序给出了 W 与 C、P(C, 2n) 上所有多项式的空间的同构。这些同构自然会导致 P(C, 2n) 上的非交换关联积。我们注意到这些积*组合这些同构使得 P(C, 2n) 上的代数和 Freshet 空间 F(p) 之间交织在一起。通过这些交织器将这些代数拼凑在一起，我们构造了一定的非交换流形我们通过具有 garb 结构证明非交换流形不是普通意义上的流形

项目成果

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka: "Convergent star products on Freshet linear Poisson algebras of Heisenberg type"Contemporary Mathematics. 288. 391-395 (2001)

Hideki Omori、Yoshiaki Maeda、Naoya Miyazaki、Akira Yoshioka：“海森堡型 Freshet 线性泊松代数上的收敛明星积”当代数学。

Hideki Omori: "One must break symmetry in order to keep associativity"Banach center publications. 55. 153-163 (2002)

大森英树：“必须打破对称性才能保持结合性”巴拿赫中心出版物。

Michel Doubois-Violette, Andreas Kriegel, Yoshiaki Maeda, Peter W.Michor: "Smooth *-algebras"Progress of Theoretical Physics. 144. 54-78 (2002)

Michel Doubois-Violette、Andreas Kriegel、Yoshiaki Maeda、Peter W.Michor：“平滑*-代数”理论物理学进展。

Hideki Omori: "Associativity breaks down in deformation quantization"Advanced Studies in pure mathematic. 37. 287-316 (2002)

大森秀树：“结合性在变形量子化中失效”纯数学高级研究。

Tamio Hara: "Multiplicative SK invariants on Zn manifolds with boundary"Rocky Mountain Journal of Mathematics (to appear).

Tamio Hara：“带边界的 Zn 流形上的乘法 SK 不变量”落基山数学杂志（待发表）。