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If they are not sufficient one needs extra (or missing) 2 26 Algebras Table 2-5 Classification of antisymmetric states jN' (Flowers, 1952) j = 1/2 j = 3/2 5/2 U(2) J [0] [1] 0 [1,1] _ [0] 0 U(4) Sp(4) J [0] (0,0) 0 [1] (1,0) 3/2 [1,1] (0,0) 0 (1,1) 2 [1,1,1] - [1] [1,1,1,1] [0] (1,0) (0,0 3/2 U(6) Sp(6) J [0] (0,0,0) 0 [1] (1,0,0) 5/2 [1,1] (0,0,0) 0 (1,1,0) (1,0,0) (1,1,1) (0,0,0) 2,4 0) 2 4 1/2 [1,1,1] [1,1,1,11 [1,1] (1 [1,1,1,1,1] - [1] [1,1,1,1,1,1] = [01 , 1 , (1,0,0) (0,0,0) 0 5/2 3/2,9/2 0 , 5/2 0 labels, v, giving rise to the classification scheme U(n) J Sp(n) D SU(2) I 1 1 {NF } (nl, n2, ...
5) The representations of Spin BF(6) can be obtained by taking the outer product of the representations of OB(6) and SUF(4). It is convenient to use here the SU(4) labels since the multiplication rules for unitary groups are simpler. For NF = 1 one must consider the product E ... E ® _ ... E ® ... 5), we obtain . ,1or 2; c2=2; c3=-2. 8) The representations (o , 2, 2) and (c1, 2, - 2) are, as mentioned in Sect. 6, equivalent. 8) to c1 = N -t 2, N - N - 2, ... , 2; o 2 = 1v3 1 = 2. 9) The representations of SpinBF(5) are labelled by two quantum numbers (r1,T2).
3-1 A typical spectrum with SpinBF(6) (III) symmetry for NB = 2, NF = 1. The Spin BF(6) quantum numbers are shown on top, the SpinBF(5) quantum numbers to the left and the angular momentum J to the right of each level. The quantity E01 does not contribute to the excitation energies but only to the binding energies. 16) is shown in Fig. 1. 15). 2) as follows: C2(SpinBF6) = 2G(2) G(2) + 4G(1) G(1) + 4G(3) G(3), C2(SpinBF5) = 4G(1) G(1) + 4G(3) G(3) C2(SpinBF3) = 20G(1) G(1). 4 Wave functions. Isoscalar factors In order to compute matrix elements of operators one needs wave functions obtained by coupling OB(6) and SUF(4) representations, since that is the level at which boson and fermion algebras are combined.