Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

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 ).
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