Mathematical Principles of Theoretical Physics

3.5.SU(N)REPRESENTATION INVARIANCE 165

whereσk( 1 ≤k≤ 3 )are as in (3.5.36) andλj( 3 ≤j≤ 8 )as in (3.5.38). Corresponding to
(3.5.40),gabare as follows

(3.5.41) gab=

1

4 N

λadcλcbd=δab.

Hence the 2-orderSU(N)tensor{gab}is positive definition.

Now, we prove Assertion (2). For eachA∈SU(N)we take the matrices

(3.5.42) ωa=Aτa for 1≤a≤K,

whereτa( 1 ≤a≤K)form a basis ofTeSU(N). It is clear thatωasatisfy the properties
(3.5.31). Hence the matrices{ωa| 1 ≤a≤K}constitute a basis ofTASU(N). On the other
hand, we see that

Gab(A) =

1

2

tr(ωaωb†)

=

1

2

tr(Aτaτb†A†) (by( 3. 5. 41 ))

=

1

2

(3.5.43) tr(τaτ†b) (by( 2. 3. 16 )).

If we take (3.5.40) as the basisτa( 1 ≤a≤K), then we have

1

2

tr(τaτb†) =δab.

It follows from (3.5.43) that with the basis (3.5.40) ofTASU(N),

Gab(A) =Gab(I) =δab=gab (by( 3. 5. 42 )).

Assertion (2) and the theorem are proved.

3.5.5 Representation invariance of gauge theory

In this subsection, we consider the representation invariance for theSU(N)gauge theory. In
Subsection3.5.1we see that the classical Yang-Mills action (3.5.2)-(3.5.4) will change under
the transformation of generator bases ofSU(N). The modified version of the Yang-Mills
action obeying the representation invariance is given by

(3.5.44)

LYM=LG+LD,

LG=−

1

4

GabFμ νaFμ νb,

LD=Ψ

[

iγμ(∂μ+igAaμτa)−m

]

Ψ,

whereGabis as in (3.5.28).
It is clear that the action density (3.5.44) is invariant under the transformation ofTASU(N):

(3.5.45) ̃τa=xbaτb.