164 CHAPTER 3. MATHEMATICAL FOUNDATIONS
The structure constants are
λabc = 2 fabc, 1 ≤a,b,c≤ 8 ,andfabcare anti-symmetric, given by
f 123 = 1 , f 147 =f 246 =f 257 =f 345 =1
2
,
f 156 =f 367 =−1
2
, f 458 =f 678 =√
(3.5.39) 3 / 2 ,
fabc= 0 for others.By (3.5.39) we can deduce that
λadcλcbd={
0 ifa 6 =b,
12 ifa=b.Hence we get
gab=1
12
λadcλcbd=δab.Namely, under the Gell-Mann representation (3.5.38),(gab) =I.
We are now in position to consider generalSU(N). In fact, for allN≥2, there exists
a generator basis{τa| 1 ≤a≤N^2 − 1 }ofSU(N)such that(gab) = 41 N(λadcλcbd) =I. The
generatorsτaare given by the following traceless Hermitian matrices:
(3.5.40)
τ
( 1 )
1 =(
σ 1 0
0 0)
, τ
( 1 )
2 =(
σ 2 0
0 0)
, τ
( 1 )
3 =(
σ 3 0
0 0)
,
τ
( 2 )
1 =(
λ 4 0
0 0)
, τ
( 2 )
2 =(
λ 5 0
0 0)
, τ
( 2 )
2 =(
λ 6 0
0 0)
,
τ
( 2 )
4 =(
λ 7 0
0 0)
, τ
( 2 )
5 =(
λ 8 0
0 0)
,
..
.
τ 1 (N−^1 )=
0 ··· 1
..
.
..
.
1 ··· 0
, τ
(N− 1 )
2 =
0 ··· −i
..
...
.
i ··· 0
,
τ
(N− 1 )
3 =
0 0 ··· 0
0 0 ··· 1
..
.
..
.
..
.
0 1 ··· 0
, τ
(N− 1 )
4 =
0 0 ··· 0
0 0 ··· −i
..
...
.
..
.
0 i ··· 0
,
..
.
τ 2 (NN−−^11 )=