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

126 8 Unitary symmetries
π a b = ˆζ a b = ξa ξ b − 1 2 δa b ξc ξ c = 1 √
2
(σ i ) a b πi , (8.50)
where in the last term we have introduced the isovector π of components
normalized so that
π i = 1 √
2
Tr(πσ i ) , (8.51)
π a bπ b a = π i π i . (8.52)
In matrix form, taking into account the correspondence between the quantum
numbers of π ± , π 0 and those of the N N states, one gets
π =
( 1
2 (ξ1 ξ 1 − ξ 2 ξ 2 ) ξ 1 ξ 2
ξ 2 ξ 1
1
2 (ξ2 ξ 2 − ξ 1 ξ 1 )
and, by comparison with Eq. (8.51),
)
=
(
√2
1
π 0 π + )
π − − √ 1
2
π 0
, (8.53)
π ± = 1 √
2
(π 1 ∓ iπ 2 ) ,
π 0 = π 3 .
(8.54)
Finally, the πN system can be described on the basis of the decomposition
D (1) ⊗ D ( 1 2) = D ( 1 2) ⊕ D ( 3 2) . (8.55)
and the states can be classified in a doublet (I = 1 2 , I 3 = ± 1 2
) and a quadruplet
(I = 3 2 , I 3 = ± 3 2 , ±1 2 )
I = 1 2
I = 3 2
√
1
⎧
⎪⎨ I 3 = 1 2 3 pπ0 −
√
⎪⎩
I 3 = − 1 2
2 3 pπ− −
⎧
I 3 = 3 2
pπ +
√
2
3 nπ+
√
1
3 nπ0 ,
√ √
⎪⎨ I 3 = 1 2
2 3 pπ0 1
+
3 nπ+
√ √ ,
I 3 = − 1 1
2 3 pπ− 2
+
3 nπ0
⎪⎩
I 3 = − 3 nπ − 2
(8.56)
where use has been made of Eq. (8.33) and of the Tables of the Clebsch-Gordan
coefficients of Appendix A.
Following similar procedure, one can classify all hadrons into isospin multiplets
with any flavour quantum numbers. At first sight, all this appears as
a formal game. Physics enters when one assumes that strong interactions depend
on the total isospin I of the system and not on the third component I 3 ; in