Mathematical Principles of Theoretical Physics

156 CHAPTER 3. MATHEMATICAL FOUNDATIONS

It follows that the covector fieldsψj( 1 ≤j≤N)in (3.4.52) satisfy

(3.4.55) ∆ψj= 0 for 1≤j≤N,

where∆=dδ+δdis the Laplace-Beltrami operator as defined in (3.2.31). Hence, the equa-
tions in (3.4.55) can be equivalently rewritten in the form

DkDkψlj=−Rklψkj for 1≤j≤N, 1 ≤l≤n,

which are exactly the equations in (3.4.51). The proof is complete.

3.5 SU(N)Representation Invariance

3.5.1 SU(N)gauge representation

We briefly recapitulate theSU(N)gauge theory. In the general case, a set ofSU(N)gauge
fields consists ofK=N^2 −1 vector fieldsAaμandNspinor fieldsψj:

(3.5.1) A^1 μ,···,AKμ, Ψ= (ψ^1 ,···,ψN)T.

For the fields (3.5.1), a gauge invariant functional is the Yang-Mills action:

(3.5.2) LYM=

∫

[LG+LD]dx,

whereLGandLDare the gauge field section and Dirac spinor section, and are written as

(3.5.3)

LG=−

1

4

Fμ νaFμ νa,

Fμ νa =∂μAνa−∂νAaμ+gλbcaAbμAcμ,

and

(3.5.4)

LD=Ψ(iγμDμ−m)Ψ,

Dμ=∂μ+igAaμτa,

whereτa( 1 ≤a≤K=N^2 − 1 )are the generators ofSU(N).
The actions (3.5.2)-(3.5.4) are invariant under both the Lorentz transformation and the
SU(N)gauge transformation as follows

(3.5.5)

Ψ ̃=ΩΨ,

A ̃aμτa=ΩAaμτaΩ−^1 +i

g

(∂μΩ)Ω−^1 ,

andΩ⊂SU(N)can be expressed as

Ω=eiθ

aτa

, τais in( 3. 5. 4 ).