ABSTRACT
Once the term with v ˜ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3897.tif"/> is present in (33.30) the O(5) group can no longer be used to classify eigenstates of the boson Hamiltonian. This term admixes states whose numbers of s-bosons differ by one. For instance, the state ( d + + d + ) μ ( 2 ) | 0 〉 ( τ = 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3898.tif"/> is admixed with the τ = 1 state s + d μ + | 0 〉 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3899.tif"/> . Hence, the seniority τ of d-bosons can no longer be a good quantum number. In that case, however, there is a possible dynamical symmetry associated with another subgroup of U(6). It appears if v ˜ ≠ 0 , v ˜ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3900.tif"/> , is proportional to it and ϵ and the c L ′ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3901.tif"/> have certain values. The subgroup from whose generators such a Hamiltonian can be constructed is the SU(3) group. In Section 30, the SU(3) group was considered in the case of fermion states. Here it appears as a symmetry group of s- and d-bosons. As we recall from Sections 6 and 30, the U(3) group has 9 generators, one of which commutes with all the others. Once it is removed, only 8 elements of the Lie algebra remain which generate transformations whose matrices have determinants equal to 1. The orthogonal group in 3 dimensions O(3) is a subgroup of U(3) as well as of SU(3). Its generators yield infinitesimal three dimensional rotations and hence, in the space of s- and d-bosons they are given by the components of J in (33.27). The other 7985 generators must be the components of a quadrupole operator which we will now consider.
