Noncommutative functional identites with non formal deformation quantization and its application

• Project/Area Number: 24540097
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo University of Science
• Principal Investigator: YOSHIOKA Akira 東京理科大学, 理学部, 教授 (40200935)
• Co-Investigator(Renkei-kenkyūsha): OMORI Hideki 東京理科大学, 理工学部, 教授 (20087018) ; MAEDA Yoshiaki 東北大学, 理学部, 教授 (40101076) ; MIYAZAKI Naoya 慶應義塾大学, 経済学部, 教授 (50315826)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥5,200,000 (Direct Cost: ¥4,000,000、Indirect Cost: ¥1,200,000); Fiscal Year 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
• Keywords: 変形量子化 / シンプレクティック幾何学 / ポアソン幾何学 / 非可換幾何学 / 量子化 / 力学 / 接触幾何学 / 複素解析幾何 / シンプレクティック・ポアソン幾何学 / 力学的幾何 / 複素解析

Outline of Final Research Achievements

We consider non-formal deformation quantizaztion. In the obtained star product aligebra, we consider star exponntaials. Using the star exponentials, we can construct certain noncommutative functional identites. As an application, we investigate a concrete MIC-Kepler problem, its eigenvalues, and also its noncommutative symplectic reduction, with non formal star products. Based on the investitation on the MIC-Kepler problem,we extend the noncommutative reduction to general situation with S1 symmetry group.
Using star exponentials,we also obtain vacumms, with which we can obtain
representation of non-formal star product algebras.

Research Products

• [Journal Article] Deformation of expressions for elements of an algebra (2014)
  • Author(s): Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka
  • Journal Title: Symplectic, Poisson, and Noncommutative Geometry, Mathematical Sciences Research Institute Publications Volume: 62 Pages: 170-209
  • Peer Reviewed
• [Journal Article] An S^1 reduction of non-formal star product (2014)
  • Author(s): Tomoyo Kanazawa, Akira Yoshioka
  • Journal Title: Geometry, integrability and quantization Volume: 15
  • DOI: 10.7546/giq-15-2014-162-172
  • Peer Reviewed
• [Journal Article] An S^1-reduction of non-formal star product (2014)
  • Author(s): Tomoyo Kanazawa, Akira Yoshioka
  • Journal Title: Proceedings of fourteenth international conference Geometry, Integrability and Quantization Volume: 14
  • Peer Reviewed
• [Journal Article] Green's function, wavefunction and Wigner function of the MIC-Kepler problem (2013)
  • Author(s): Tomoyo Kanazawa
  • Journal Title: Journal of geometry and symmetry in physics Volume: 30 Pages: 63-73
  • Peer Reviewed
• [Journal Article] Star products and certain star functions (2013)
  • Author(s): Mari Iida, Chishu Tsukamoto, Akira Yoshioka
  • Journal Title: Geometric methods in Physics. XXXI Workshop 2012, Trends in Mathematics Volume: 1
  • Peer Reviewed
• [Presentation] Star product and Weyl manifold (2015)
  • Author(s): Akira Yoshioka
  • Organizer: Current Problems in Theoretical Physics XXI Edition
  • Place of Presentation: Vietri, Italy
  • Year and Date: 2015-03-28 – 2015-03-31
  • Invited
• [Presentation] Star product and application (2015)
  • Author(s): 吉岡 朗
  • Organizer: (非)可換代数とトポロジー
  • Place of Presentation: 松本市，信州大学
  • Year and Date: 2015-02-13 – 2015-02-15
  • Invited
• [Presentation] Star functions, example and application (2014)
  • Author(s): Akira Yoshioka
  • Organizer: XXXIII Workshop on geometric methods in physics
  • Place of Presentation: Bialowieza, Poland
  • Year and Date: 2014-06-29 – 2014-07-05
  • Invited
• [Presentation] Star product and star functions (2014)
  • Author(s): Akira Yoshioka
  • Organizer: The sixteenth international conference on Geometry, integrability and quantization
  • Place of Presentation: Varna, Bulgaria
  • Year and Date: 2014-06-06 – 2014-06-11
  • Invited
• [Presentation] Star product and its application (2014)
  • Author(s): Akira Yoshioka
  • Organizer: International Conference Symmetry Methods, Applications, and Related Fields celebrating the work of Professor George W. Bluman
  • Place of Presentation: Vancouver, Canada, Univeristy of British Columbia
  • Year and Date: 2014-05-13 – 2014-05-16
  • Invited
• [Presentation] Star products and star exponentials (2013)
  • Author(s): Akira Yoshioka
  • Organizer: The fifteenth international conference on Geometry, Integrability and Quantization
  • Place of Presentation: Varna, Bulgaria
• [Presentation] Star products, star exponentials and applications (2013)
  • Author(s): Akira Yoshioka
  • Organizer: XXXII Workship on geometric methods in physics
  • Place of Presentation: Bialowieza, Poland
• [Presentation] Star積について (2013)
  • Author(s): 吉岡 朗
  • Organizer: 場の数理とトポロジー
  • Place of Presentation: 信州大学
  • Invited
• [Presentation] Star products-formal and nonformal (2012)
  • Author(s): Akir Yoshioka
  • Organizer: XIVth international conference, Geometry, Integrability and Quantization
  • Place of Presentation: Varna, Bulgaria
• [Presentation] Star products-formal extension and nonformal extension (2012)
  • Author(s): Akira Yoshioka
  • Organizer: XXXI Workshop on Geometric methods in Physics
  • Place of Presentation: Bialowieza, Poland
• [Presentation] Star products and star functions (2012)
  • Author(s): Akira Yoshioka
  • Organizer: International conference: Harmonic analysis on homegeneous spaces and quantization
  • Place of Presentation: Tambov, Russia
  • Invited
• [Book] Proceedings of fifteenth international conference: Geometry, Integrablity and Quantization (2014)
  • Author(s): Ivilo Mladenov, Andrei Ludu, Akira Yoshioka
  • Total Pages: 321
  • Publisher: Avangard Prima, Sofia, Bulgaria
• [Book] Proceedings of fourteenth international conference; Geometry, Integrability and Quantization 14 (2013)
  • Author(s): Ivailo Mladenov, Andrei Ludu, Akira Yoshioka ed.
  • Total Pages: 271
  • Publisher: AvangardPrima, Sofia, Bulgaria