ABSTRACT
We now consider another limit of the U(6) Hamiltonian which leads to dynamical symmetry. In the U(5) case discussed in the preceding section, we encountered the O(5) group which is a subgroup of U(6) and contains O(3) as its subgroup. In that case, U(5) is an intermediate group between O(5) and U(6). There is, however, another such intermediate group, namely O(6)—the group of real orthogonal transformations in the 6-dimensional space of s- and d-bosons (Arima and Iachello 1978). The number of generators of O(5), given by (34.2) is 10 (= 5 × 4/2) whereas the number of O(6) generators is 15(= 6 × 5/2). The components of another quadrupole operator should be added to the generators (34.2) to obtain the O(6) Lie algebra. Since the s-boson should be included, the only k = 2 tensor among the U(6) generators is one of those in (33.25). It is customary to define the O(6) Lie algebra by the generators (34.2), ( d + × d + ) κ ( k ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3812.tif"/> , k = 1, 3 and the components of () Q ′ = s + d ˜ + d + s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3813.tif"/>
