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

Feng Pan; Lina Bao; Yao-Zhong Zhang; Jerry P. Draayer

doi:10.1140/epjp/i2014-14169-0

2024 Impact factor 2.9

EPJ PLUS - Condensed Matter and Complex Systems

Eur. Phys. J. Plus (2014) 129: 169
https://doi.org/10.1140/epjp/i2014-14169-0

Regular Article

Construction of basis vectors for symmetric irreducible representations of O(5) $\supset$ O(3)

Feng Pan¹^,2^,3^*, Lina Bao¹, Yao-Zhong Zhang³ and Jerry P. Draayer²

¹ Department of Physics, Liaoning Normal University, 116029, Dalian, China
² Department of Physics and Astronomy, Louisiana State University, 70803-4001, Baton Rouge, LA, USA
³ School of Mathematics and Physics, The University of Queensland, Qld 4072, Brisbane, Australia

^* e-mail: daipan@dlut.edu.cn

Received: 14 April 2014
Revised: 30 June 2014
Accepted: 9 July 2014
Published online: 13 August 2014

Abstract

A recursive method for the construction of symmetric irreducible representations of $ O(2l+1)$ in the $O(2l+1)\supset O(3)$ basis for identical boson systems is proposed. The formalism is realized based on the group chain $U(2l+1)\supset U(2l-1)\otimes U(2)$ , of which the symmetric irreducible representations are simply reducible. The basis vectors for symmetric irreducible representations of the $O(2l+1)\supset O(2l-1)\otimes U(1)$ can easily be constructed from those of U(2l + 1) $\supset$ U(2l - 1) ⊗ U(2) $\supset$ 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 $O(2l-1)\otimes U(1)$ 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 $O_{1}(3)\otimes U(1)$ basis, where $O_{1}(3)\equiv O(2l-1)$ , 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 $d^{\dagger}_{\mu}$ ( $d_{\mu}$ with $\mu=1$ , 0, -1 . Matrix representations of $O(5)\supset O_{1}(3)\otimes U(1)$ , together with the elementary Wigner coefficients, are presented. After the angular momentum projection, a three-term relation in determining the expansion coefficients of the $O(5)\supset O(3)$ basis vectors, where the O(3) group is generated by the angular momentum operators of the d -boson system, in terms of those of $O_{1}(3)\otimes U(1)$ is derived. The eigenvectors of the projection matrix with zero eigenvalues constructed according to the three-term relation completely determine the basis vectors of $O(5)\supset O(3)$ . Formulae for evaluating the elementary Wigner coefficients of $O(5)\supset O(3)$ are derived explicitly. Analytical expressions of some elementary Wigner coefficients of $O(5)\supset O(3)$ for the coupling $(\tau 0)\otimes(1 0)$ with resultant angular momentum quantum number L = 2 $\tau$ + 2 - k for $k=0,2,3,\ldots,6$ with a multiplicity 2 case for k = 6 are presented.