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Extension of quantum theory of angular momentum and Yang-Baxter relations

Research Project

Project/Area Number 03804020
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field 物理学一般
Research InstitutionUniversity of Tokyo

Principal Investigator

NOMURA Masao  Associate Professor, College of Arts and Science, University of Tokyo, 教養学部, 助教授 (10012402)

Project Period (FY) 1991 – 1993
Project Status Completed (Fiscal Year 1993)
Budget Amount *help
¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1993: ¥100,000 (Direct Cost: ¥100,000)
Fiscal Year 1992: ¥200,000 (Direct Cost: ¥200,000)
Fiscal Year 1991: ¥700,000 (Direct Cost: ¥700,000)
KeywordsQuantum groups / Yang-Baxter relations / Creation-annihilation operators / Covariant forms / Quantum theory of angular momentum / Tensors / Unitary groups of order n / Bogoliubov transformation / テンソル積 / ボブリューボフ変換 / YangーBaxter関係式 / 回転群表現行列
Research Abstract

We have widely extended the theory of angular momentum i.e.representation theory of rotation group, called Wigner-Racah algebras. Mathematical devices are devoted mainly to quantum (q-) group algebras. Physical space of concern is so-called q-space, a non-commutative one, which is an extension of the prevalent commutative space.
We have succeeded to specify all the quantities and their relations in the framework of q-covariant/contravariant forms. It is along the line of our postulate that any observable quantity (transition probability) should not rely on a certain linear coordinate transformation (q-linear transformation) of the q-space. According to the q-covariance, all the quantities are properly classified as q-tensors (q-scalar, q-vector, etc.). We have q-Wigner-Eckart theorem, which is an extension of well-known central theorem by Wigner and Eckart, and notions of q-unit tensors in a very natural way.
We have investigated kinds of basis functions, such as q-rotation functions, q-Clebsh-Gordan coefficients, q-Racah coefficients (or, generally q-n-j symbols), and have established various relationship among them. Emphasis has been put on the point that these q-functions form a class of basis functions constituting Yang-Baxter relations of face models and of vertex models.
Further, we have found a systematic way to specify q-commutation relations among extended creation-annihilation operators of q-bosons and q-fermions. There are several ways to assign the q-creation and annihilation operators to form spinors in the q-covariant theory. For example, we describe a q-spinor in terms of a pair of creation and annihilation operators to obtain q-analog BCS Bogoliubov formalism. Using this result, we have succeeded to extend so-called symplecton algebras.
We have generalized a part of the above consideration to the case of U_q(n). We have found also very new kinds of q-extended Young diagrams (Schur functions).

Report

(4 results)
  • 1993 Annual Research Report   Final Research Report Summary
  • 1992 Annual Research Report
  • 1991 Annual Research Report
  • Research Products

    (21 results)

All Other

All Publications (21 results)

  • [Publications] 野村正雄: "Concepts of tensors in UqSl(2) and van der Waerden method for quantum Clebsh-Gordan coefficient." Journal of the Physical Society Japan. 60. 789-797 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 野村正雄: "Recoupling and braiding of angular momenta in theories of UqSl(2)." Journal of the Physical Society Japan. 60. 1906-1916 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 野村正雄: "Co- and contra-variant tensors of a general rank in quantum matrix algebras SUq(2) expressed by creation-annihilation operators I" Journal of the Physical Society Japan. 60. 3260-3270 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 野村正雄: "Co- and Contra-variant tensors of a general rank in quantum matrix algebras SUq(2) expressed by creation-annihilation operators II" Journal of the Physical Society Japan. 60. 4060-4070 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 野村正雄: "Covariant exchange algebras and quantum groups(I)" Journal of the Physical Society Japan. 61. 1485-1494 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 野村正雄,J.C.Biedenharn: "Ons q-symplecton realization of the quantum group SUq(2)" Journal of Mathematical Physics. 33. 3636-3648 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Nomura: "Concepts of tensors in U_qsl(2) and van der Waerden method for quantum Clebsch-Gordan coefficient." Journal of the Physical Society Japan. 60. 789-797 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Nomura: "Recoupling and braiding of angular momenta in theories of U_qsl(2)." Journal of the Physical Society Japan. 60. 1906-1916 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Nomura: "Co- and contra-variant tensors of a general rank in quantum matrix algebras su_q(2) expressed by creation-annihilation operators I." Journal of the Physical Society Japan. 60. 3260-3270 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Nomura: "Co- and contra-variant tensors of a general rank in quantum matrix algebras su_q(2) expressed by creation-annihilation operators II." Journal of the Physical Society Japan. 60. 4060-4070 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Nomura: "Covariant exchange algebras and quantum groups (I)." Journal of the Physical Society Japan. 61. 1485-1494 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Nomura and L.C.Biedenharn: "On the q-symplecton realization of the quantum group SU_q(2)." Journal of Mathematical Physics. 33. 3636-3648 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 野村正雄: "On quantized quantum-analog rotation functions" Journal of the Physical Society of Japan. 62. 36-45 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 野村正雄: "Covariant and quantized representation functions of quantum group SUq(2)" Geometric Methods in Theoretical Physics. 298-301 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 野村正雄: "Covariant q-boson operator description of Uq(n)basis matrices" Symmetries in Science. 6. 537-552 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 野村正雄: "Boson-fermion operators for description of Young diagrams and related topics" Symmetries in Science. 7. 445-456 (1994)

    • Related Report
      1993 Annual Research Report
  • [Publications] 野村 正雄: "Covariant exchange algebras and quantum groups.I" Journal of the Physical Society of Japan. 61. 1485-1494 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] 野村 正雄: "On the q-symplecton realization of the quantum group SUq(2)" Journal of Mothematical Physics. 33. 3636-3648 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] 野村 正雄: "On quantized quantum-analog votation functions" Journal of the Physical Society of Japan. 62. 36-45 (1993)

    • Related Report
      1992 Annual Research Report
  • [Publications] 野村 正雄: "Coーand ContraーVariant Tensors of a General rank in Quantum Matrix Algebras suf(2)Expressed by CreationーAnnihilation Operators.I" Journal of the Physical Society of Japan. 60. 3260-3270 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] 野村 正雄: "Coーand ContraーVariant Tensors of a General rank in Quantum Matrix Algebras Suq(2)Expressed by CreationーAnnihilation Operators.II" Journal of The Physical Society of Japan. 60. 4060-4070 (1991)

    • Related Report
      1991 Annual Research Report

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Published: 1991-04-01   Modified: 2016-04-21  

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