Construction of basis vectors for symmetric irreducible representations of O(5) (Formula Presented.) O(3)

Pan, Feng, Bao, Lina, Zhang, Yao-Zhong and Draayer, Jerry P. (2014) Construction of basis vectors for symmetric irreducible representations of O(5) (Formula Presented.) O(3). European Physical Journal Plus, 129 8: 1-27. doi:10.1140/epjp/i2014-14169-0


Author Pan, Feng
Bao, Lina
Zhang, Yao-Zhong
Draayer, Jerry P.
Title Construction of basis vectors for symmetric irreducible representations of O(5) (Formula Presented.) O(3)
Formatted title
Construction of basis vectors for symmetric irreducible representations of O(5) ⊃ O(3)
Journal name European Physical Journal Plus   Check publisher's open access policy
ISSN 2190-5444
Publication date 2014-08-01
Year available 2014
Sub-type Article (original research)
DOI 10.1140/epjp/i2014-14169-0
Open Access Status DOI
Volume 129
Issue 8
Start page 1
End page 27
Total pages 27
Place of publication Heidelberg, Germany
Publisher Springer
Language eng
Subject 3100 Physics and Astronomy
Abstract A recursive method for the construction of symmetric irreducible representations of O(2l+1) in the O(2l+1) superset of O(3) basis for identical boson systems is proposed. The formalism is realized based on the group chain U(2l + 1) superset of U(2l - 1) circle times U(2), of which the symmetric irreducible representations are simply reducible. The basis vectors for symmetric irreducible representations of the O(2l + 1) superset of O(2l - 1) circle times U(1) can easily be constructed from those of U(2l + 1) superset of U(2l - 1) circle times U(2) superset of O(2l - 1) circle times U(1) with no l-boson pairs, namely with the total boson number exactly equal to the seniority number in the system, from which one can construct symmetric irreducible representations of O(2l+1) in the O(2l-1) circle times U(1) basis when all symmetric irreducible representations of O(2l-1) are known. As a starting point, basis vectors of symmetric irreducible representations of O(5) are constructed in the O-1(3) circle times U(1) basis, where O-1(3) equivalent to O(2l - 1), when l = 2, which is generated not by the angular momentum operators of the d-boson system, but by the operators constructed from d-boson creation (annihilation) operators d(mu)(+) (d(mu)) with mu = 1, 0, -1. Matrix representations of O(5) superset of O-1(3) circle times U(1), together with the elementary Wigner coefficients, are presented. After the angular momentum projection, a three-term relation in determining the expansion coefficients of the O(5) superset of O(3) basis vectors, where the O(3) group is generated by the angular momentum operators of the d-boson system, in terms of those of O-1(3) circle times U(1) is derived. The eigenvectors of the projection matrix with zero eigenvalues constructed according to the three-term relation completely determine the basis vectors of O(5) superset of O(3). Formulae for evaluating the elementary Wigner coefficients of O(5) superset of O(3) are derived explicitly. Analytical expressions of some elementary Wigner coefficients of O(5) superset of O(3) for the coupling (tau 0)circle times(1 0) with resultant angular momentum quantum number L = 2 tau+2-k for k = 0, 2, 3,..., 6 with a multiplicity 2 case for k = 6 are presented.
Formatted abstract
A recursive method for the construction of symmetric irreducible representations of O(2l+1) in the O(2l+1)⊃O(3) basis for identical boson systems is proposed. The formalism is realized based on the group chain U(2l+1)⊃U(2l−1)⊗U(2) , of which the symmetric irreducible representations are simply reducible. The basis vectors for symmetric irreducible representations of the O(2l+1)⊃O(2l−1)⊗U(1) can easily be constructed from those of U(2l + 1) ⊃ U(2l - 1) ⊗ U(2) ⊃ O(2l - 1) ⊗ U(1) with no l -boson pairs, namely with the total boson number exactly equal to the seniority number in the system, from which one can construct symmetric irreducible representations of O(2l+1) in the O(2l−1)⊗U(1) basis when all symmetric irreducible representations of O(2l - 1) are known. As a starting point, basis vectors of symmetric irreducible representations of O(5) are constructed in the O1(3)⊗U(1) basis, where O1(3)≡O(2l−1) , when l = 2 , which is generated not by the angular momentum operators of the d -boson system, but by the operators constructed from d -boson creation (annihilation) operators d†μ ( dμ with μ=1 , 0, -1 . Matrix representations of O(5)⊃O1(3)⊗U(1) , together with the elementary Wigner coefficients, are presented. After the angular momentum projection, a three-term relation in determining the expansion coefficients of the O(5)⊃O(3) basis vectors, where the O(3) group is generated by the angular momentum operators of the d -boson system, in terms of those of O1(3)⊗U(1) is derived. The eigenvectors of the projection matrix with zero eigenvalues constructed according to the three-term relation completely determine the basis vectors of O(5)⊃O(3) . Formulae for evaluating the elementary Wigner coefficients of O(5)⊃O(3) are derived explicitly. Analytical expressions of some elementary Wigner coefficients of O(5)⊃O(3) for the coupling (τ0)⊗(10) with resultant angular momentum quantum number L = 2 τ + 2 - k for k=0,2,3,…,6 with a multiplicity 2 case for k = 6 are presented.
Keyword Physics, Multidisciplinary
Physics
Q-Index Code C1
Q-Index Status Confirmed Code
Grant ID OCI-0904874
11175078
DP110103434
20102136110002
2013020091
9961
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
Official 2015 Collection
 
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