Symmetries and Group Theory in Particle Physics: An Introduction to ...

146 8 Unitary symmetries
8.5.5 I-spin, U-spin and V -spin
For the classification of hadrons, we made use of the SU(2) isospin subgroup
of SU(3). We want to point out that there are alternative ways, besides the
use of I and Y , of labelling the states of an SU(3) multiplet.
In fact, one can identify three different subgroups SU(2) in SU(3). In the
set of the 8 λ-matrices one can indeed find three subsets which generates
SU(2) subgroups; obviously these subsets do not commute among them.
We know already that the matrices
I i = 1 2 λ i (i = 1, 2,3) (8.112)
can be taken as the generators of the isospin group SU(2) I . One can immediately
check from Eq. (8.86) and Table 8.5 that they satisfy the commutation
relations
Let us define two other sets:
[I i , I j ] = iǫ ijk I k , (8.113)
[I i , Y ] = 0 . (8.114)
U 1 = 1 2 λ 6 ,
U 2 = 1 2 λ 7 ,
(8.115)
(√ )
U 3 = 1 4 3λ8 − λ 3 ,
and
V 1 = 1 2 λ 4 ,
V 2 = 1 2 λ 5 ,
(8.116)
(
V 3 = 1 4 λ3 + √ )
3λ 8 .
Using again Eq. (8.86) and Table 8.5 for the f ijk coefficients, we get
[U i , U j ] = iǫ ijk U k , (8.117)
[V i , V j ] = iǫ ijk V k , (8.118)
which show that also U i and V i generate SU(2) subgroups (denoted in the
following by SU(2) U and SU(2) V ). For this reason, besides the isospin I or I-
spin, one defines the U-spin and the V -spin; in analogy with the shift operators
I ± = I 1 ± iI 2 (Eq. (8.58)), we define also the operators U ± = U 1 ± iU 2 and
V ± = V 1 ± iV 2 . Their action is represented in Fig. 8.8.
What is the use of these sets of generators? Let us consider first the U-
spin. Its introduction is particularly useful, since the U i ’s commute with the
electric charge Q defined in (8.45)
[U i , Q] = 0 , (8.119)