Research on Noncommutative Geometry by deformation quantization

基于变形量化的非交换几何研究

• 批准号: 17540096
• 负责人: YOSHIOKA Akira
• 金额: $ 1.73万
• 依托单位: Tokyo University of Science
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540096/
• 关键词: star product; deformation quantization; noncommutative geometry; symplectic geometry; Poisson geometry; deformation quantization; 量子力学

By means differential equations, we extend the Moyal product to exponential functions for quadratic functions with coefficients in complex numbers. With respect to the product, we obtain certain anomalous phenomena such as associativity breaking, doubled valuedness of star exponential functions. Further we define a star product, which is an extension of the Moyal product, by using complex symmetric matrices. An infinitesimal transformation of expression of star products, we obtain a nonlinear connection on the space of functions. We show that the extended transformation of orderings gives a transformation of Cayley type for complex matrices, and we give the explicit formula of the transformation for functions. By studying the parallel transform of the phase functions and the amplitude functions, we made a model of branching of star exponential functions. A commutative product of Moyal type gives several interesting functions such as elliptic functions in the space of exponential functi … More ons of quadratic forms. We also show that several identities for these special functions are given by the star product and simple relation of exponential functions.We call the extended Moyal products K-ordering products and we give a geometrical description of these products: we have an algebraic bundle over the space of all nxn complex symmetric matrices, whose fibers consist of Weyl algebras. Intertwines among K-ordering expressions give a flat connection of this bundle. Flat sections of the bundle are regarded as elements of the Weyl algebra. In this framework, we consider several transcendental elements such as star exponential functions. By means of the star exponential functions of linear forms we can define noncommutative Fourier transform and Laplace transform. By these transforms, we consider noncommutative version of elementary functions, e.g., Gamma functions, delta functions. We also study star exponential functions of quadratic forms on this bundle. These exponential functions give a grebe over the base space and the grebe is described by the bundle and the connection. Less

By means differential equations, we extend the Moyal product to exponential functions for quadratic functions with coefficients in complex numbers. With respect to the product, we obtain certain anomalous phenomena such as associativity breaking, doubled valuedness of star exponential functions. Further we define a star product, which is an extension of the Moyal product, by using complex symmetric matrices. An infinitesimal transformation of expression of star products, we obtain a nonlinear connection on the space of functions. We show that the extended transformation of orderings gives a transformation of Cayley type for complex matrices, and we give the explicit formula of the transformation for functions. By studying the parallel transform of the phase functions and the amplitude functions, we made a model of branching of star exponential functions. A commutative product of Moyal type gives several interesting functions such as elliptic functions in the space of exponential functi … More ons of quadratic forms. We also show that several identities for these special functions are given by the star product and simple relation of exponential functions.We call the extended Moyal products K-ordering products and we give a geometrical description of these products: we have an algebraic bundle over the space of all nxn complex symmetric matrices, whose fibers consist of Weyl algebras. Intertwines among K-ordering expressions give a flat connection of this bundle. Flat sections of the bundle are regarded as elements of the Weyl algebra. In this framework, we consider several transcendental elements such as star exponential functions. By means of the star exponential functions of linear forms we can define noncommutative Fourier transform and Laplace transform. By these transforms, we consider noncommutative version of elementary functions, e.g., Gamma functions, delta functions. We also study star exponential functions of quadratic forms on this bundle. These exponential functions give a grebe over the base space and the grebe is described by the bundle and the connection. Less

项目成果

Toward Geometric Quantum theory

走向几何量子理论

• 发表时间: 2007
• 期刊: Progress in mathematics 252
• 作者: Hideki Omori;Yoshiaki Maeda;Naoya Miyazaki;Akira Yoshioka;Naoya Miyazaki;Hideki Omori
• 通讯作者: Hideki Omori

On the integrability of deformation quantized Toda lattice.

关于形变量子化Toda晶格的可积性。

• 发表时间: 2006
• 期刊: Acta Appl. Math. 92
• 作者: Y.Maeda;H.Omori;et al.;N.Miyazaki
• 通讯作者: N.Miyazaki

A splitting theorem for proper complex equivocal submanifolds

真复模歧子流形的分裂定理

• 发表时间: 2006
• 期刊: Tohoku Mathematical Journal 58
• 作者: Yoshiaki Maeda;Peter Michor;Takushiro Ochiai;Akira Yoshioka;Naoya Miyazaki;Naoyuki Koike;Naoyuki Koike;Naoya Miyazaki;Naoyuki Koike
• 通讯作者: Naoyuki Koike

Geometric objects in an approach to quantum geometry

量子几何方法中的几何对象

• 期刊: Geometric objects in an approach to quantum geometry (To appear)
• 作者: Y.Maeda;H.Omori 他
• 通讯作者: H.Omori 他

From Geometry to Quantum mechanics, In Honor of Hideki Omori

从几何到量子力学，纪念大森秀树

• 发表时间: 2007
• 期刊: Birkhauser
• 作者: Yoshiaki Maeda;Peter Michor;Takushiro Ochiai;Akira Yoshioka
• 通讯作者: Akira Yoshioka