Chinese Physics C > 1980, Vol. 4 > Issue(1): 73-92

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

Sun Hung-zhou

Peking University

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Abstract：
In this paper, we analyse the commutation relations of the infintesimal opera-tors of the group SU ₃and find that the eight infinitesimal operators of the groupSU ₃can be written as a scalar operator A, three angular momentum operators ( L ₁, 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 SU ₃can 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 ^λμare 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 SU ₃and prove the corresponding Wigner-Eckart theory. The method used in the discussion of the group SU ₃be extended to allof the compact simple Lie groups of rank 2 and we shall discuss them in two suc-ceeding papers.

References

[1]

G. Racah, Group Theory and Spectroscopy, Princeton,(1951).[2] R. E. Bchrends et al., Rev. Mod. Phys., 34 (1962),1.[3] A. Salam, Theoretical Physics, p. 173., Viena,(1963).[4] J. P. Elliott et al., Proc. Roy. Soc., A277 (1963), 557.[5] A. Arima et al., Nucl. Phys., A138 (1969),273., A162(1971),605.[6] K. T. Hecht, Nucl. Phys., 63 (1965), 177.[7] G. Gneuss et al., Nucl. Phys., A171 (1971),449.[8] B. H. Flowers et al., Proc. Phys. Soc., 84 (1964),139.[9] A. R. Edmonds, Angular Momentum in Quantum Mechanics, Priceton,(1957).[10] 孙洪洲等,物理学报21(1965), 56,[11] 杨国祯等,北京大学学报(自然科学版)10(1964), 269.[12] S. McDonald et al., J. Math. Phys.,14 (1973),1248.[13] J. P. Draayer et al., J. Math. Phys.,14 (1973),1904.[14] E. M. Haacke et al., J. Math. Phys.,17 (1976),2040.[15] 侯伯宇,中国科学,14(1965), 367.[16] J. J. De Swaat, Rev. Mod. Phys., 35(1963),916.[17] L. C. Biedenharn et a1, J. Math. Phys., 4(1963),1449.,4(1964),1723.,4 (1964),1730.[18] 陈金全等,高能物理与核物理,3(1979),216.

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Sun Hung-zhou. ON THE IRREDUCIBLE REPRESENTATIONS OF THE COMPACT SIMPLE LIE GROUPS OF RANK 2(I)[J]. Chinese Physics C, 1980, 4(1): 73-92.

Sun Hung-zhou. ON THE IRREDUCIBLE REPRESENTATIONS OF THE COMPACT SIMPLE LIE GROUPS OF RANK 2(I)[J]. Chinese Physics C, 1980, 4(1): 73-92. shu

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Received: 1978-12-08
Revised: 1900-01-01

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沈阳化工大学材料科学与工程学院沈阳 110142

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

Sun Hung-zhou

Peking University

Received Date:1978-12-08
Accepted Date:1900-01-01
Available Online:1980-02-05

Abstract:In this paper, we analyse the commutation relations of the infintesimal opera-tors of the groupSU₃and find that the eight infinitesimal operators of the groupSU₃can be written as a scalar operatorA, 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 groupSU₃can 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^λμare 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 SU₃and prove the corresponding Wigner-Eckart theory. The method used in the discussion of the groupSU₃be extended to allof the compact simple Lie groups of rank 2 and we shall discuss them in two suc-ceeding papers.