IRREDUCIBLE REPRESENTATIONS OF THE GRADED LIE ALGEBRA SU(m/n)

Using the definition of irreducible tensor operators of the Lie group SU(n), we ive a method of calculating the irreducible representations of the graded Lie algebra U(m/n). We also give some simple examples of irreducible representations of U(3/2) and SU(m/n).     

ON THE IRREDUCIBLE REPRESENTATIONS OF THE COMPACT SIMPLE LIE GROUPS OF RANK 2(I)

In this paper, we analyse the commutation relations of the infintesimal opera-tors of the group SU₃ and find that the eight infinitesimal operators of the groupSU₃ can be written as a scalar operator A, three angular momentum operators (L₁,L_o, L_-1,)and two sets of the irreducible tensor operators of rank 1/2, （T_±1/2,V±1/2）By means of the commutation relations of these operators, all irreducible represen-tations of the group SU₃ can be easily obtained. In this pape, the matrices corresponding to these operators in the irreduciblerepresentation（λμ）, are given; therefore the irreducible representation and its re-presentation space R^λμ are completely defined. Besides, a method for calculatingthe scalar factors of the reduction coefficients and the symmetric relations of thosefactors are also given. As examples, the scalar factors of the reduction coefficientsof （λμ）×（10）, （λμ）×（01）, （λμ）×（20） and （λμ）×（11） are calculated. In the last part of this paper, we define the irreducible tensor operators ofthe group SU₃ and prove the corresponding Wigner-Eckart theory. The method used in the discussion of the group SU₃ be extended to allof the compact simple Lie groups of rank 2 and we shall discuss them in two suc-ceeding papers.     

THE SU2×SU2 BASIS AND THE PHYSICAL BASES FOR THE STATE VECTORS OF d-BOSON SYSTEMS AND THE TRACELESS BOSON OPERATORS (Ⅰ)

The group chain U₅O₅SU₂×SU₂ used by K. T. Hecht (1965) and by the othersprovides an important representation for expressing the physical basis of d-boson sys-tems. However the methods which have been introduced for this SU₂×SU₂ representa-tion to construct a physical basis is poorer in comparison with those for the otherrepresentations. In view of this we try to find appropriate methods to obtain the SU₂×SU₂ representation wave functions of the existing physical bases constructed byChacon et al. and by Szpikowski et al., In the present paper we analyse the SU₂×SU₂ tensor properties of the,bosonoperators and Vilenkin's traceless boson operators and express succinctly the elementaryvectors of the SU₂×SU₂ basis, the |PP SU₂×SU₂ > vectors, in terms of the tracelessoperators. With the help of this form of the| PP SU₂ SU₂> vectors we derive a simpleformula for obtaining the SU₂×SU₂ -representation wave functions of a physical basisfrom its (n_μ)-representation wave functions. Thus the problem mentioned above is partlysolved. The other parts of the solution of the problem will be found in a coming paper.     

DISCRETE SYMMETRY AND SU(N) GRAND UNIFIED MODELS

Using discrete symmetry S, a systematic analysis of all possible GUT models based on the low rank group SU(N)×S is presented. It is found that only SU(7)×S and SU(8)×S are the satisfactory gauge groups of flavour grand unification. In addition, SU(7)×S and SU(8)×S models are discussed in detail. All these models can accommodate. four generations of ordinary fermions and preserve the asympotic freedom of SU_c(3).     

AN SU(6) GRAND UNIFIED MODEL

A model of grand unified theory based on SU(6) gauge group is proposed. It can accommodate two generations of ordinary fermions with V－A weak coupling and two generations of anomalous fermions with V+A weak coupling. In this model a new discrete symmetry is introduced which insures existence of fermions with lower masses when SU(6) gauge symmetry is spontaneously broken. We choose simple Higgs fields with appropriate vacuum expectation values so that the masses of anomalous fermions are heavier than those of ordinary fermions. This model also gives the same value of Weinberg angle, sin²θw=3/8, as in the usual SU(5) grand unified model at the grand unified scale.     

A POSSIBLE SU(9) GRAND UNIFIED MODEL

In this letter, we propose a Higgs mechanism which is different from the usualone at the first step of spontaneous symmetry breaking. Using this new Higgs mechanismwe discuss an SU(9) model which can accommodate three generations of light fer-mions and sustains the asymptotic freedom of SU(3)_c.     

A RECURSIVE FORMULA FOR THE CLEBSCH-GORDAN COEFFICIENTS OF SUn GROUP

It is proved that under the Gelfand-Biedenharn phase convention any SU_n singletfactor （SF） belongs to the type of the derivable or underivable,designated as （SU_nSF）d and （SU_r SF） respectively,and （SU_n SF）d=（SU_r SF）,n=r+1,r+2,….Therefore the calculation of any SU_n Clebsch-Gordan coefficients is reduced to thecalculation of a few underivable singlet factor （SU_r SF）.     

Quark-Chiral Field interaction in SU(3) Model

A quark-quark potential is derived from the linear expression of the Lagrangian Which is invariant under the SU(3)_L×SU(3)_R infinitesimal transformation,and further used to make a qualitative analysis on the Λ-N interaction.The result shows that the SU(3)chiral scalar fields are important in explaining the spin-dependence feature of the Λ-N interaction.     

A STRUCTURE MODEL OF STRATONS AND LEPTONS

A subquark model is proposed. According to this model, stratons and leptons are consisted of rishons T and V, which are multiplets of group SU_H(3)×SU_C(3)×SU_G(2)×SU(2)×U(1). Three generations of both stratons and leptons are obtained naturally. In this model, proton decay modes are μ^－e⁺e⁺ or e^－μ⁺μ⁺.     

ON THE SU(4) MONOPOLE AND DYON SOLUTIONS

A simple method is proposed to obtain the classical monopole and dyon solutions in the case of SU(4) symmetry, which can be systematically generalized to the general case of SU(N). The solutions obtained in the SU(4) case correspond to magnetic charges of ±1/g,±√3/(2g),±1/(√2g).     

THE MULTISYNCHRO-SPHERICAL SOLUTIONS OF THE SU(N)GAUGE THEORY

We study the monopole solutions of the SU(N) gauge group systematically,generalize the current synchro-spherical symmetrical monopole to the multi synchro-spherical symmetrical case,and give the complete classification of the monopole so-lutions.As examples,we have calculated the SU(3) and SU(4) monopoles in detail,and discuss some problems such as angular momenta and topological quantum num-bers.     

THE IRREDUCIBLE REPRESENTATIONS AND THEIR OCCUPATION NUMBER IN DIRECT MULTIPLICATION OF IRREDUCIBLE REPRESENTATIONS OF SU3 GROUP

In this paper, the C-G series of SU₃ group is studied using Racah-Speier theorem. A general formula which can be used to calculate irreducible representations and their occupation number in C-G series of SU₃ group is derived. The general formula is very straight forward and effective for actual calculation One can apply the formula to analyse the distribution rule of occupation number in C-G series of SU₃ group and to define the null space of Winger operator of SU₃ group, the latter is important to the calculation of Wigner coefficients when there is a multiplicity.     

A VARIATIONAL METHOD IN LATTICE GAUGE THEORY BASED ON HÖLDER INTEGRAL INEQUALITY

A variational method based on Hölder integral inequality is presented. Taking the parameters in integral inequality as variational parameters, we can calculate the free energy in lattice gauge system. The plaquatte energy of Z₂ 、U(1)、 SU(2) gauge system is found and consistent with that given by Monte Carlo method. But, the higher phase transitions for U(1), SU(2) gauge systems are not found.     

COMMENT ON LIGHT-FERMION MASS HIERARCHY PROBLEM IN SU(9) GRAND UNIFIED MODELS

According to the idea of reference [3] for solving light-fermion mass hierarchy problem, all possible SU(9) grand unified models are systematically examined. The conclusion is negative, there is no hope to realize (α²,α,1) pattern in SU(9) models.     

ON THE PROBLEM OF THE DUAL CHARGE (MAGNETIC MONOPOLE) OF NON-ABELLAN GROUPS (Ⅱ)

The method of relating the gauge potential to the connection on a spherical surface developed in a previous paper is generalized to the discussion of the potential of an SU(2) magnetic monopole with O(5) symmetry. Expressions for the potential of the monopole with double string singularity or single string singularity are obtained. Finally, by a combined coordinate-gauge transformation, we obtain a string free expression for an SU(2) magnetic monopole with O(5) symmetry.     

A BAG MODEL WITH PSEUDOSCALAR MESON CLOUD

Using SU(3)×SU(3) symmetry the bag model with pion cloued suggested in re-cent years is generalized to the bag model containing K, η, η′meson cloud. The massspectra of hadrons are calculated and the origin of pion cloud is discussed.     

ON THE PHASE AND THE REPRESENTATION TRANSFORMATIONS OF SUn BARYON AND MESON WAVE FUNCTIONS

The baryon and meson wave functions of SU_n symmetry under the Baird-Bie-denharn phase convention are given explicitly.The phase confusion is clarified andthe matrices of transformations between I-spin,U-spin and V-spin representationsare given for the IR（8）,（27）,（35） of SU₃.     

THE SU(6) “UNITARY ENERGY BAND” STRUCTURE IN THE TRIBARYON SYSTEM

A picture of the tribaryon system is proposed based on the papers [1—3]. It is considered that proton, neutron and A hyperon from the basic particles of SU(3) group and form the basic particles of SU(6) group on account for spin. It turns out that there exist "unitary energy band" structure in the tribaryon system. From the mass formula of T₃³ breaking, one can find that the SU(6) symmetry gives the mass relations between different spin levels of the same strangeness analog states. Finally, simple calculations are made and tbere is no discrepancy between tbe results and the experimental data.     

CONTINUOUS VARIABLE REPRESENTATION OF ELLIOTT SU(3) MODEL

The continuous variable representation and the geometrical interpretation of the Elliott SU(3) model are given by virtue of the generator coordinate method. It is pointed out that the carrier subspace of collective states of the dynamical group SU(3) in the Elliott model is spanned by the β and γ vibration states as well as the rotation states.