3.5.SU(N)REPRESENTATION INVARIANCE 163
3.5.4 Intrinsic Riemannian metric onSU(N)
By Theorem3.31,Gab(A)andgabare symmetric. We now show that bothGabandgabare
positive definite, and consequentlyGab(A)is an intrinsic Riemannian metric onSU(N).
Theorem 3.32.For the 2-order SU(N)tensors gabandGab, the following assertions hold
true:
1) The tensor{gab}given by (3.5.30) is positive definite; and2) For each point A∈SU(N), there is a coordinate system in whereGab(A) =gab. There-
foreGab(A)is a Riemannian metric on SU(N).Proof.First, we prove Assertion (1). As a 2-order covariant tensor,gabtransforms as
(3.5.35) ( ̃gab) =X(gab)XT, X= (xab)as in( 3. 5. 22 ).
Hence, if for a given basis{τa| 1 ≤a≤K}ofTeSU(N)we can verify thatgab= 41 Nλadcλcbd
is positive definition, then by (3.5.35) we derive Assertion (1). In the following, we proceed
first forSU( 2 )andSU( 3 ), then for the generalSU(N).
ForSU( 2 ), we take the Pauli matrices
(3.5.36) σ 1 =
(
0 1
1 0
)
, σ 2 =(
0 −i
i 0)
, σ 3 =(
1 0
0 − 1
)
as the generator basis ofSU( 2 ). The structure constantsλabc of (3.5.36) are as follows
(3.5.37) λabc= 2 εabc, εabc=
1 if(abc)is even,
−1 if(abc)is odd,
0 if otherwise.Based on (3.5.37), direct calculation shows that
gab=1
8
λadcλcbd=δab.Namely(gab) =Iis identity.
ForSU( 3 ), we can take the following Gell-Mann matrices as the generator basis of
SU( 3 ):
λ 1 =
0 1 0
1 0 0
0 0 0
, λ 2 =
0 −i 0
i 0 0
0 0 0
, λ 3 =
1 0 0
0 −1 0
0 0 0
,
λ 4 =
0 0 1
0 0 0
−1 0 0
, λ 5 =
0 0 −i
0 0 0
i 0 0
, λ 6 =
0 0 0
0 0 1
0 1 0
(3.5.38) ,
λ 7 =
0 0 0
0 0 −i
0 i 0
, λ 8 =√^1
3