Application of Deformation Quantization theory to Geometry and Mathematical Physics

在本研究计划中，我们获得了以下结果.在复2n欧几里德空间中，我们引入了Modyal积，正规积和反正规积.我们建立了一个函数空间，在这个函数空间中，这三个乘积是收敛的并且是有意义的。我们用F（p）定义了所有p阶整函数的空间（p是非负的）.则当p ≤ 2时，空间F（p）成为关于每个乘积2的非交换拓扑代数。利用微分方程的方法，分别构造了这些乘积的二次函数的指数函数。借助于奇异函数的公式，我们研究了它们关于时间变量的奇异性。我们定义的拉普拉斯变换的指数，我们构造几个超越元素的非交换代数3。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到C，P（C，2n）上所有多项式空间的同构.这些同构自然地导出P（C，2n）上的非交换结合积。我们注意到这些乘积构成P（C，2n）上的代数与Freshet空间F（p）之间的同构。用这些交织线将这些代数拼接在一起，我们构造了一个非对易流形，并通过具有garb结构证明了非对易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