DEFORMATION QUANTIZATION AND ITS APPLICATION

变形量化及其应用

• 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 /

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.

这项研究有两个方面； (i) 通过 Weyl 流形进行变形量化的几何方面，(ii) 相对于变形参数 h 的收敛变形量化的研究。我们得到以下内容。 (i) 外尔流形的模空间是具有基流形第二上同调类系数的形式幂系列。利用Weyl流形对应的上同调，我们构造了一个包含Weyl流形作为子丛的接触Weyl流形。在接触Weyl流形上，我们还构造了一个连接，其曲率形式决定了Weyl流形的上同调类。我们证明这种联系是费多索夫联系的扩展，并证明曲率给出的上同调类与Weyl流形的上同调类一致，因此我们显示了Weyl流形的Poincare-Cartan类和Fedosov曲率的上同调类连接是同一件事。 (ii)利用Moyal乘积公式，在Moyal乘积绝对收敛的多维复平面上设置某些全息函数的特定Frechet空间。引入全纯函数奇异指数，星积打破了乘积的结合性。我们还研究了二次函数的星指数函数。尽管 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函数的全纯函数之间的星积得到了很好的定义。研究了由二次指数函数生成的群的某些属性。特别是，该群被认为是特殊线性群的延伸。

项目成果

大森英樹: "MONCOMMUTATIVE WORLD AND ITS GEOMETRICAL PICTURE"AMS TRANSLATION OF SUGAKO EXPOSITIONS. (2001)

大森秀树：“单对数世界及其几何图景”SUGAKO EXPOSITIONS 的 AMS 翻译（2001）。

大森英樹,前田吉昭,宮崎直哉,吉岡朗: "Anomalous quadratic exponentials in the star-products. Lie groups, geometric structures and differential equations-one hundred years after Sophus Lie (Japanese))."数理研講究録:ソーフィス・リー没後百年記念国際研究集会報告集. 1150. 128-132 (2000)

Hideki Omori、Yoshiaki Maeda、Naoya Miyazaki、Akira Yoshioka：“明星产品中的反常二次指数。李群、几何结构和微分方程 - Sophus Lie 一百年后（日语））。”纪念李逝世一百周年的会议 1150. 128-132 (2000)。

吉岡朗: "ワイル多様体のコンタクト構造とDEFORMATION QUANTIZATION"数理解析研究所講究録. 1119. 1-18 (1999)

Akira Yoshioka：“Weyl 流形的接触结构和变形量子化”数学分析研究所的 Kokyuroku 1119. 1-18 (1999)。

大森英樹,前田吉昭,宮崎直哉,吉岡朗: "Deformation quantization of Frechet-Poisson algebras : Convergence of the Moyal product"The proceedings of the Coference Moshe Flato 1999,Kluwer Academic Publishers in the series Mathematical Physics Studies,. 2. 233-246 (2000)

Hideki Omori、Yoshiaki Maeda、Naoya Miyazaki、Akira Yoshioka：“Frechet-Poisson 代数的变形量子化：Moyal 乘积的收敛”Moshe Flato 会议记录，1999 年，Kluwer 学术出版社的数学物理研究系列，2. 233。 -246 (2000)

吉岡朗: "CONTACT WEYL MANIFOLD OVER A SYMPLECTIC MANIFOLD"AdVANCED STUDIES IN PURE MATHEMATICS. (2001)

Akira Yoshioka：“在辛流形上接触 WEYL 流形”纯数学高级研究 (2001)。