Masao Nomura

Relations for Clebsch–Gordan and Racah coefficients in suq(2) and Yang–Baxter equations

Relations are exploited among Clebsch–Gordan (CG) and Racah coefficients in the algebra suq(2), known as a deformation of su(2). These are used to show that the Yang–Baxter (YB) relation for the IRF (interaction round a face) model results from one of the symmetry relations for the 9‐j symbol specific to suq(2), and that in an asymptotic limit this YB relation becomes the YB relation for the two‐dimensional vertex model. The Racah coefficient, which has a particularly simple dependence on q, is efficiently used such that an asymptotic limit of the Racah coefficient is the CG coefficient and another limit gives the factorized S matrix of the vertex model.

An Alternative Description of the Quantum Group SUq(2) and the q-Analog Racah-Wigner Algebra

A set of generators of the quantum group S U q (2) is presented which yields the q -analog Racah-Wigner (RW) algebra. While the standard specification of S U q (2) requires separate treatment for different components of a tensor, the present approach gives unified treatment of tensor algebra. Clebsch-Gordan (CG) and Racah coefficients are extensively used. This unified treatment is crucial to formulate the q -analog irreducible tensor operator and Wigner-Eckart theorem. Wittens specification of S U q (2), exploited in rational conformal field theories (RCFT), is realized in the present approach.

The multiplicity of states in a single shell

Abstract Two methods are proposed to get the multiplicity of states with the total angular momentum J ( L ) in a j n ( l n ) configuration. One gives the multiplicity by means of a set of numbers specified by special algebraic relations. It is especially powerful for small values of J . The other method makes use of a special kind of sum rule about c.f.p. Some relations between multiplicities and Racah coefficients are derived. These two methods do not require any discussion of the z -component of angular momentum.

Yang-Baxter Relations in Terms of n-j Symbols of suq(2) Algebra

A trigonometric function parametrization of Yang-Baxter (Y-B) relations is described in terms of n - j symbols of su q (2), a deformation of su (2). The formalism starts with deduction of the Y-B relation for the IRF (interaction round a face) model of su q (2) from a general operator relation for the vertex model. The relation obtained for the IRF model is transformed into the Y-B relation for the vertex model using the property that the 3- j symbol is an asymptotic limit of the 6- j symbol. A class of very general solutions to Y-B equations is also presented. This solution of the IRF model with infinite rapidity parameters is linked to a symmetry of the 12- j symbol of the second kind.

Extension of Wick’s theorem by means of the state operator formalism

Reordering of operators in the second quantization formalism is described by using state operators as basic operators, where the state operator is either a cluster of creation operators or that of annihilation operators specified by a set of quantum numbers of any representation. The commutation relation between state operators is simply expressed in terms of the coefficient of fractional parentage (cfp), the factor specific to state operator (de) composition. Wick’s theorem, prevalently applied to a string of creation and annihilation operators, is extended to a string of state operators. Similar extension is done for the contraction theorem.

Recursion Relations for the Clebsch-Gordan Coefficient of Quantum Group SUq(2)

Various recursion relations are exploited for the Clebsch-Gordan (CG) coefficient of the quantum group S U q (2). These are associated with four kinds of explicit forms for the CG coefficient of S U q (2).

Relations amomg n-j Symbols in Forms of the Star-Triangle Relation

Novel solutions to the star-triangle equation for the IRF model are presented which are described in terms of 6- j and 9- j symbols. Additivity of spectral parameters does not hold in this case.

Propagation coefficients for fixed‐isospin (T,Tz) average and related spectroscopic sum rules

The reduction relation for the fixed‐isospin (T,Tz) average of a general operator in the model space of many fermions is described in two forms with and without recourse to factorization of isospin z components. Algebraic treatment is developed to deduce various types of expressions for each propagation coefficient that plays the role of the Green’s function in each form of the reduction relation. Propagation coefficients are described also in relation to sum rules as to fixed‐isospin spectroscopic factors. These results lead to novel identities among n‐j symbols and factorials.

CONCEPTS OF TENSORS IN UQ(SL(2)) AND A VAN DER WAERDEN METHOD FOR QUANTUM CLEBSCH-GORDAN COEFFICIENTS

Concepts of invariants, co-variant and contra-variant tensors of quantum group U q ( s l (2)) are settled in relationship to q -deformed representation matrices. The formalism is used to exploit a q -deformed van der Waerden method for quantum Clebsch-Gordan coefficients. Difficulties inherent in a q -deformed van der Waerden method given by Ruegg are overcome.

A soluble nonlinear bose field as a dynamical manifestation of symmetric group characters and young diagrams

Abstract A model of quantum nonlinear wave interaction is solved algebraically. While the hamiltonian has no kinematical connection with the symmetric group S n , the eigenstates are expressed in terms of symmetric group characters as quantum realizations of Schur functions i.e. Young diagrams. Class multiplication of S n is embedded in the dynamics.