Application of Deformation Quantization theory to Geometry and Mathematical Physics

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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