ABSTRACT
As explained in Section 33, subgroups of U(6) may furnish a classification scheme for states of s- and d-bosons. The simplest subgroup of U(6) that we consider is the group U(5) whose generators are the 25 operators in (33.24). More precisely, the subgroup we consider is U(5)⊗ U(1) where the unitary transformations of U(5) transform states of a d-boson into themselves and U(1) trivially transforms the state of an s-boson into itself. Clearly, the generators (33.24) form a Lie algebra, as also follows formally from their commutation relations (33.34). A boson Hamiltonian constructed from only the generators of U(5) ⊗ U(1) is obtained from (33.29) by putting v ˜ 0 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3707.tif"/> and v ˜ 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3708.tif"/> . The conservation of N was used to derive (33.30). We now consider Hamiltonians which are constructed only from U(5) generators. Starting from (33.30) and putting v ˜ 0 = v ˜ 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3709.tif"/> we obtain a Hamiltonian with U(5) symmetry and will therefore refer simply to the U(5) subgroup of U(6).
