2.4. GAUGE INVARIANCE 75
where
m=
m 1 0
..
.
0 mN
(2.4.31) is the the mass matrix,
(2.4.32) Dμ=∂μ+igGμaτa,
andτa( 1 ≤a≤K)are a set ofSU(N)generators as in Theorem2.31.
For theNspinorsΨin (2.4.29), consider the followingSU(N)gauge transformations
(2.4.33) Ψ ̃=ΩΦ ∀Ω=eiθ
aτa
∈SU(N),
whereθa( 1 ≤a≤K)are functions ofxμ∈M^4.
To ensure that (2.4.30)-(2.4.32) are covariant under the transformation (2.4.33), we need
to determine
(1) the transformation form of the gauge fieldsGμ, and(2) the action for the gauge fieldsGμ.First, consider (1). The covariance of Dirac equations (2.4.30) is equivalent to the covari-
ance of the derivativeDμΨas given by
(2.4.34) D ̃μΨ ̃=ΩDμΨ ∀Ωas in( 2. 4. 33 ).
The left-hand side of (2.4.34) can be directly computed as
D ̃μΨ ̃=(∂μ+igG ̃aμτa)ΩΨ=Ω∂μΨ+ (∂μΩ)Ψ+igG ̃aμτaΩΨ,and the right-hand side is
ΩDμΨ=Ω(∂μ+igGaμτa)Ψ=Ω∂μΨ+igGaμΩτaΨ.Hence, it follows from (2.4.34) that
(2.4.35) G ̃aτa=GaμΩτaΩ−^1 +
i
g(∂μΩ)Ω−^1 ,which is called theSU(N)gauge transformation of the gauge fieldsGaμ.
In addition, the mass matrix (2.4.31) satisfies that
(2.4.36) m ̃=ΩmΩ−^1.
In summary, the transformations (2.4.33), (2.4.35) and (2.4.36) constitute theSU(N)
gauge transformations defined as
(2.4.37)
Ψ ̃=ΩΨ ∀Ω=eiθaτa∈SU(N),G ̃aμτa=GaμΩτaΩ−^1 +i
g(∂μΩ)Ω−^1 ,m ̃=ΩmΩ−^1.