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Application of Deformation Quantization theory to Geometry and Mathematical Physics

Research Project

Project/Area Number 13640088
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTokyo University of Science

Principal Investigator

YOSHIOKA Akira  Tokyo University of Science, Mathematics, Professor, 理学部, 教授 (40200935)

Co-Investigator(Kenkyū-buntansha) MAEDA Yoshiaki  Keio University, Mathematical science, Professor, 理工学部, 教授 (40101076)
HAR Tamio  Tokyo University of Science, Engineering, Assistant Professor, 工学部, 助教授 (10120205)
OMORI Hideki  Tokyo University of Science, Mathematics, Professor, 理工学部, 教授 (20087018)
MIYAZAKI Naoya  Keio University, Economics, Assistant Professor, 経済学部, 助教授 (50315826)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2002: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)
KeywordsDeformation Quantization / star products / non-commutative geometry / Hamiltonian mechanics / simplistic geometry / quantization / Deformation quantization / Hamiltonian mechanics / deformation quantization / 非可換幾何学 / 量子力学 / シンプレクティック幾何学
Research Abstract

In this research program we obtain the following results.
1. In the complex 2n Euclidian space we introduce the Modyal products, the normal product and the anti-normal *oduct. We set up a function space, for which these three products are convergent and have meaning. We deonte by F(p) the space of all entire functions of order p (p is nonnegative). Then for p equal to or less than 2, the space F(p) becomes a noncommutative topological algebra with repect to each product
2. By means of differential equations, we construct exponential functions of quadratic functions for these products respectively. Thank to the formula of the expoenetial functions, we investigate their singularities with resect to the time variable. We define the Laplace transform by means of the exponential and we construct several transcendental elements of the noncommutative algebra
3. We consider the Weyl algebra W of 2n generators over the complex number. The Weyl ordring, the normal ordering and the anti-normal ordering on the Weyl algebra give isomorphisms of W to the space of all polynomials on C, P(C, 2n). These isomorphisms naturally induces noncommutative, associative products on P(C, 2n). We remarked these produ*
Composing these isomorphisms gines intertwiners between the algebras on P(C, 2n) and the Freshet space F(p). Patching these algebra together by these intertwiner we construct a certain noncommutative manifold
We show the noncommutative manifold is not a manifold in the ordinary sense by has the garb structure

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (35 results)

All Other

All Publications (35 results)

  • [Publications] Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka: "Convergent star products on Frechet linear Poisson algebras of Heisenberg type"Contemporary Mathematics. 288. 391-395 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka: "Singular system of exponential functions"Mathemtical Physics Studies. 23. 169-186 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Tamio Hara: "Multiplicative SK invariants on Z_n manifolds with boundary"Rocky Mountain Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Akira Yoshioka: "Contact Weyl manifold over a symplectic manifold"Advanced Studies in pure mathematics. 37. 459-493 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hideki Omori: "Associativity breaks down in deformation quantization"Advanced Studies in pure mathematic. 37. 287-316 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka: "Star exponential functions for quadratic forms and polar elements"Contemporary Mathematics. 315. 25-38 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Naoya Miyazaki: "Mehler formula and localization in deformation quantization"数理研講究録. 1260. 126-147 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hideki Omori: "One must break symmetry in order to keep associativity"Banach center publications. 55. 153-163 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Michel Doubois-Violette, Andreas Kriegel, Yoshiaki Maeda, Peter W.Michor: "Smooth *-algebras"Progress of Theoretical Physics. 144. 54-78 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka: "Strange phenomena related to ordering problems"Journal of Lie thoery. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Toshio Matumoto, Akira Yoshioka: "Path Integral for Star exponential functions of quadratic forms"Geometry, Integrability and Quantization, Proceedings of Varna Conference 2002. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Yoshiaki Maeda, Satosi Watamura: "Noncommutative geometry and string theory(Progress of Theoretical Physics, supplement)144"日本物理学会、基礎物理研究所. 194 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka: "Convergent star products on Freshet linear Poisson algebras of Heisenberg type"Contemporary Mathematics. 288. 391-395 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka: "Singular system of exponential functions"Mathematical Physics Studies. 23. 169-186 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Tamio Hara: "Multiplicative SK invariants on Zn manifolds with boundary"Rocky Mountain Journal of Mathematics (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Akira Yoshioka: "Contact Weyl manifold over a symplectic manifold"Advanced Studies in pure mathematics. 37. 459-493 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hideki Omori: "Associativity breaks down in deformation quantization"Advanced Studies in pure mathematics. 37. 287-316 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka: "Star exponential functions for quadratic forms and polar elements"Contemporary Mathematics. 315. 25-38 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Yoshiaki Maeda and Satosi Watamura (editors),: "Noncommutative geometry and string theory"Progress of Theoretical Physics, supplement. 144. 1-194 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Naoya Miyazaki: "Mehler formula and localization in deformation quantization"Suriken Kokyu-Roku. 1260. 126-147 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hideki Omori: "One must break symmetry in order to keep associativity"Banach center publications. 55. 153-163 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Michel Doubois-Violette, Andreas Krieger, Yoshiaki Maeda: "Smooth *-algebras"Progress of Theoretical Physics, supplement. 144. 54-78 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka: "Strange phenomena related to ordering problems"Journal of Lie Theory (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Toshio Matumoto, Akira Yoshioka: "Path Integral for star exponential functions of quadratic forms"Geometry, Integrability and Quantization, Proceedings of Varna conference (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Akira Yoshioka: "Contact Weyl manifold over a Symplectic manifold"Advanced studies in pure mathematics. 37. 459-493 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hideki Omori: "Associativity breaks down in deformation quantization"Advanced studies in pure mathematics. 37. 287-316 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hideki Omori: "Star exponential functions for quadratic forms and polar elements"Contemporary Mathematics. 315. 25-38 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Naoya Miyazaki: "Mehler formula and localization in deformation quantization"数理研講究録. 1260. 126-147 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hideki Omori: "One must break symmetry in order to keep associativity"Banach center publications. 55. 153-163 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Toshio Matsumoto, Akira Yoshioka: "Path Integral for Star Exponential Functions of quadratic forms"the proceedings of Varna conference 2002. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Michel Dubois-Violette: "Smooth ^*-algebras"Progress of Theoretical Physics, supplement. 144. 54-78 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Yoshiaki Maeda: "Noncommutative geometry and string theory"日本物理学会、基礎物理研究所. 194 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 大森英樹: "Convergent star products on Frechet linear Poisson algebras of Heisenberg type"Contemporary Mathematics. 288. 391-395 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 大森英樹: "Singular systems of exponential functions"Mathematical Physics Studies. 23. 169-186 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 原 民夫: "Multiplicative SK invariants on Z_n manifolds with boundary"to appear in Rocky Mountain J.Math.

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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