DEFORMATION QUANTIZATION AND ITS APPLICATION

点击翻译按钮获取中文摘要

This researchment is two fold ; (i) geometric aspect of Deformation quantization via Weyl manifolds, (ii) investigation of convergent deformation quantization with repect to the deformation parameter h. We obtain the following. (i) The moduli space of Weyl manifolds are the formal power serires with coefficients in the 2nd cohomology classes of the base manifol. Using the cohomolgy corresponding to the Weyl manifold, we construct a contact Weyl manifold which contains Weyl manifold as a subbundle. On contact Weyl manifold, we also constuct a connection whose curvature form determines the cohomology class of the Weyl manifold. We show this connection is an extension of Fedosov connection and proved that the cohomolgy class given by the curvature coincides with the cohomology class of the Weyl manifold, hence we show the Poincare-Cartan class of Weyl manifold and the cohomology class of the curvature of Fedosov connection are the same thing. (ii) Using the Moyal product formula, we set certain Frechet space of certain holomophic functions on the multidimensional complex plane where the Moyal products are absolutely convergent. Singular exponent of holomorphic functions are introduced with respect which the star products breaks the associativity of product. We also investigate a star exponential functions of quadratic functions. Althoug the Frechet space does not contain the exponentials of the quadratic functions, the star product is well defined between the quadratic exponentials and holomorphic function having the exponent less than the singular one. Certain properties are investigated for the group generated by the quadratic exponential functions. Especially, the group is considered an extension of the special linear groups.