1993 Fiscal Year Final Research Report Summary
Extension of quantum theory of angular momentum and Yang-Baxter relations
| Project/Area Number |
03804020
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
物理学一般
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| Research Institution | University of Tokyo |
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Principal Investigator |
NOMURA Masao Associate Professor, College of Arts and Science, University of Tokyo, 教養学部, 助教授 (10012402)
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| Project Period (FY) |
1991 – 1993
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| Keywords | Quantum groups / Yang-Baxter relations / Creation-annihilation operators / Covariant forms / Quantum theory of angular momentum / Tensors / Unitary groups of order n / Bogoliubov transformation |
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| 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).
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