Mathematical Principles of Theoretical Physics

3.5.SU(N)REPRESENTATION INVARIANCE 161

Definition 3.30(SU(N)Tensors).Let T be given as

T={Tba 11 ······baji| 1 ≤ak,bl≤K=N^2 − 1 }.

We say that T is a(i,j)-type of SU(N)tensor, under the generator basis transformation as
(3.5.22), we have

T ̃a^1 ···ai

b 1 ···bj=y

a 1

c 1 ···y

ai

cix

d 1

b 1 ···x

dj

bjT

c 1 ···ci

d 1 ···dj,

where(xab)and(yab)are as in (3.5.22) and (3.5.23).

Based on Definition3.30, it is easy to see that theSU(N)gauge fields(A^1 μ,···,Aaμ)is a

contra-variantSU(N)tensor. In other words, under (3.5.22),(A^1 μ,···,Aaμ)transforms as

(3.5.25) A ̃μa=yabAbμ, Y= (yab)as in( 3. 5. 24 ).

This can be seen from the fact that the operatorAaμτain the differential operatorDμin (3.5.4)
is independent of generator basesτaofSU(N).
We now verify that the structure constantsλbcaofSU(N)constitute a (1,2)-type ofSU(N)
tensor. By the definition ofλbca,

[τb,τc] =τbτc†−τcτb†=iλbcaτa.

By (3.5.22),
[ ̃τb, ̃τc] =xabxdc[τa,τd] =ixabxdcλadfτf,

and by definition

[τ ̃b, ̃τc] =i ̃λbcaτ ̃a=i ̃λbcaxdaτd.

Then it follows that

(3.5.26) ̃λbca=xbfxgcyadλdfg,

which means that{λbca}is a (1,2)-typeSU(N)tensor.
Next, we introduce two second-order covariantSU(N)tensorsGabandgab, and later we
shall prove that they are equivalent.
LetA∈SU(N). Then the tangent spaceTASU(N)is given by (3.5.17). Let

(3.5.27) ω 1 ,···,ωK∈TASU(N),

be a generator basis ofSU(N)atA.

1) SU(N)tensorGab(A). By the basis (3.5.27) we can get a 2-covariantSU(N)tensor

defined by

(3.5.28) Gab(A) =

1

2

tr(ωaωb†), A∈SU(N),

whereωa( 1 ≤a≤K)are as in (3.5.27).