442
M. Group Theory
(
)
ˆ
K 2 = −i x
2 ∂
∂
+ x
0
(M.48)
∂x 0
∂x 2
(
)
∂
∂
ˆ
K 3 = −i x
3
+ x
0
.
(M.49)
∂x 0
∂x 3
The corresponding algebra is
ˆ ˆ
ˆ
[X i , X j ] = i∈ ijk X k
(M.50)
ˆ ˆ
ˆ
[X i , K j ] = i∈ ijk K k
(M.51)
ˆ ˆ
ˆ
[K i , K j ] = −i∈ ijk X k .
(M.52)
Note the minus sign on the right-hand side of (M.52) as compared with (M.39).
M.4.5 SU(3)
A general infinitesimal SU(3) transformation may be written as (cf (12.71)
and (12.72))
(
)
(
)
q 1
′
(
)
q 1
( q 2 )
1
= 1 + i η · λ ( q 2 ) ,
(M.53)
2
q 3
q 3
where there are now 8 of these η’s, η = (η 1 , η 2 , . . . , η 8 ), and the λ-matrices
are the Gell-Mann matrices
(
)
(
)
(
)
0 1 0
0
)
−i 0
1 0 0
λ 1 = ( 1 0 0
, λ 2 = ( i 0 0 ) , λ 3 = ( 0 −1 0 ) (M.54)
0 0 0
0 0 0
0 0 0
(
)
(
)
(
)
0 0 1
0 0
i
0 0 0
λ 4 = ( 0 0 0 )
−
, λ 5 = ( 0 0 0 ) , λ 6 = ( 0 0 1 ) (M.55)
1 0 0
i 0 0
0 1 0
(
)
(
)
1
0
0
0 0 0
√
|
3
λ 7 = ( 0 0 −i )
1
|
, λ 8 = ( 0
√
0
.
(M.56)
3
)
0 i 0
0
0 − √ 2
3
ˆ
In this parametrization the first three of the eight generators G r (r = 1, 2, . . . , 8)
ˆ
are the same as X 1
′ ˆ ˆ
, X 2
′ , X 3
′ of (M.29)–(M.30). The others may be constructed
as usual from (M.10); for example,
(
)
(
)
i
∂
∂
i
∂
∂
ˆ
ˆ
G 5 =
q 3
− q 1
, G 7 =
q 3
2
∂q 1
∂q 3
2
∂q 2
− q 2
.
(M.57)
∂q 3
The SU(3) algebra is found to be
ˆ ˆ
ˆ
[G a , G b ] = if abc G c ,
(M.58)