Application of Deformation Quantization theory to Geometry and Mathematical Physics

在本研究方案中，我们得到了以下结果：在复2n欧几里德空间中引入模积、正规积和反正规积。我们建立了一个函数空间，对于这个函数空间，这三个乘积是收敛的，是有意义的。我们用F(p)表示所有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)就变成了一个关于每个乘积2的非交换拓扑代数。利用微分方程分别构造了这些乘积的指数函数和二次函数。利用指数函数的公式，我们研究了它们对时间变量的奇异性。我们用指数来定义拉普拉斯变换，构造了非交换代数的几个超越元。我们考虑复数上2n个生成元的Weyl代数W。Weyl代数上的Weyl序、正规序和反正规序给出了W在C， P（C, 2n）上所有多项式空间的同构。这些同构在P（C, 2n）上自然地推导出非交换的结合积。我们注意到P（C, 2n）和Freshet空间F(P)上的代数之间的同构关系。将这些代数通过这些缠结器拼接在一起，我们构造了一个非交换流形，我们证明了这个非交换流形不是一般意义上的流形

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