ABSTRACT
Within the nuclear shell model with nucleons occupying several ℓ orbits ℓ1, ℓ2, . . ., as already mentioned in Chapter 6, it is possible to define a isoscalar plus isovector pairing Hamiltonian, as given by Eq. (6.29), carrying SO(8) symmetry. The commutators for the 28 operators P †µ, Pµ, D
† µ, Dµ, S
1 µ, T
(στ)1,1µ,µ′ and Q0 defined by Eqs. (6.27) - (6.30) establish that they form a closed algebra and this is the SO(8) algebra [246]. The commutators are as follows,[
P †µ Pµ′ ] = (−1)µ′+1
√ 2 〈1µ 1− µ′ | 1µ− µ′〉 T 1µ−µ′ + δµµ′ Q0 ,[
P †µ Dµ′ ] = (−1)µ′ (στ)1,1−µ′,µ ,[
D†µ Dµ′ ] = (−1)µ′+1√2 〈1µ 1− µ′ | 1µ− µ′〉 S1µ−µ′ + δµµ′ Q0 ,[
P †µ Q0 ] = −P †µ ,
[ D†µ Q0
] = −D†µ ,[
] = 0 ,
[ P †µ T
] = −
√ 2 〈1µ 1µ′ | 1µ+ µ′〉 P †µ+µ′ ,[
D†µ T 1µ′ ] = 0 ,
] = −
√ 2 〈1µ 1µ′ | 1µ+ µ′〉 D†µ+µ′ ,[
] = δµ,−µ′′ (−1)1+µD†µ′ ,[
] = δµ,−µ′ (−1)1+µ P †µ′′ ,[
] = −
√ 2 〈1µ 1µ′ | 1µ+ µ′〉 S1µ+µ′ ,[
] = −
√ 2 〈1µ 1µ′ | 1µ+ µ′〉 (στ)1,1µ+µ′ ,µ′′ ,
Models[ T 1µ T
] = −√2 〈1µ 1µ′ | 1µ+ µ′〉 T 1µ+µ′ ,[
] = −
√ 2 〈1µ 1µ′′ | 1µ+ µ′′〉 (στ)1,1µ′ ,µ+µ′′ ,[
] = δµ1,−µ3 (−1)1+µ1
√ 2 〈1µ2 1µ4 | 1µ2 + µ4〉T 1µ2+µ4
+δµ2,−µ4 (−1)1+µ2 √ 2 〈1µ1 1µ3 | 1µ1 + µ3〉S1µ1+µ3 .
