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DEFORMATION QUANTIZATION AND ITS APPLICATION

变形量化及其应用

基本信息

批准号：
11640095
负责人：
YOSHIOKA Akira
金额：
$ 2.3万
依托单位：
SCIENCE UNIVERSITY OF TOKYO
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640095/
关键词：
DEFORMATION QUANTIZATION STAR PRODUCT NONCOMMOTATIVE GEOMETRY QUANTIZATION ASYHPTOTIC ANALYSIS SYMPLECTIC GEOMETRY HAMILTONIAN MECHANICS Deformation quantization star product noncommutative geometry quantization asymptotic analysis /

项目摘要

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.

本文的研究分为两个方面：（i）通过Weyl流形的形变量子化的几何方面，（ii）关于形变参数h的收敛形变量子化的研究。我们得到以下结果。(i)Weyl流形的模空间是其基流形的第二上同调类中系数的形式幂级数。利用与Weyl流形对应的上同调，构造了一个包含Weyl流形作为子丛的切触Weyl流形.在切触Weyl流形上，我们还构造了一个联络，它的曲率形式决定了Weyl流形的上同调类。证明了这一联络是Fedosov联络的推广，并证明了由曲率给出的上同调类与Weyl流形的上同调类是一致的，从而证明了Weyl流形的Poincare-Cartan类与Fedosov联络的曲率的上同调类是相同的. (ii)利用Moyal积公式，在多维复平面上建立了一类全纯函数的Frechet空间，在该空间中Moyal积是绝对收敛的.引入了全纯函数的奇异指数，证明了星星乘积打破了乘积的结合性。我们还研究了二次函数的一个星星指数函数。虽然Frechet空间不包含二次函数的指数，但二次指数与指数小于奇异指数的全纯函数之间的星星积是很好的定义.研究了由二次指数函数生成的群的某些性质。特别地，该群被认为是特殊线性群的推广。