PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

The group contains two one-dimensional and one two-dimensional irreducible representations.
The one-dimensional representations are given by [37]
The two-dimensional representation is given by
(
0 1
2 : S =
1 0
1 : S = 1, T = 1 (2.47)
1 ′ : S = −1, T = 1 . (2.48)
)
, T =
( )
ω 0
0 ω 2
. (2.49)
The character table is given in Table 2.2. Using the Table we can write down the rules
for the tensor products. For the one-dimensional irreducible representations we have
1×1 = 1, 1×1 ′ = 1 ′ , 1 ′ ×1 ′ = 1 . (2.50)
Tensor products between two doublets ψ = (ψ 1 ,ψ 2 ) T and φ = (φ 1 ,φ 2 ) T are given as
where
2×2 = 1+1 ′ +2 , (2.51)
1 ≡ ψ 1 φ 2 +ψ 2 φ 1 , (2.52)
1 ′ ≡ ψ 1 φ 2 −ψ 2 φ 1 , (2.53)
( )
ψ2 φ
2 ≡ 2
. (2.54)
ψ 1 φ 1
The complex conjugate doublet ψ ⋆ is given as 2 ⋆ for which the generators are S ⋆ and T ⋆ .
One can easily check that ψ ⋆ does not transform as doublet (2) of S 3 and therefore for
this case a meaningful way of writing the tensor products for the conjugate fields is by
defining
( )
ψ
ψ ′ ≡ σ 1 ψ ⋆ ⋆
= 2
ψ1
⋆ . (2.55)
Using the relations σ 1 S ⋆ σ 1 = S and σ 1 T ⋆ σ 1 = T one can show that ψ ′ transforms as a
doublet. Then the tensor products ψ ′ ×φ are given by Eq. (2.51) where
1 ≡ ψ ⋆ 1φ 1 +ψ ⋆ 2φ 2 , (2.56)
1 ′ ≡ ψ2φ ⋆ 2 −ψ1φ ⋆ 1 , (2.57)
( )
ψ
⋆
2 ≡ 1 φ 2
ψ2φ ⋆ . (2.58)
1
28