74 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS
We remark here that the basis{τk| 1 ≤k≤K}of the tangent space (2.4.24) is usually
called the generators ofSU(N), and (2.4.25) is called the generator representation ofSU(N).
We are now in position to give a proof of (2.4.22). By the classical Jordan theorem, for a
given matrixAthere is a non-degenerate matrixBsuch that
(2.4.27) BAB−^1 =
λ 1 ∗
..
.
0 λN
,
is an upper triangular matrix, whereλ 1 ,···,λNare the eigenvalues ofA. Recall thateAis
defined by
eA=∞
∑
k= 0Ak
k!.
By (2.4.27) we have
BeAB−^1 =∞
∑
k= 01
k!(BAB−^1 )k=∞
∑
k= 01
k!
λ 1 ∗
..
.
0 λN
k=
eλ^1 ∗
..
.
0 eλN
.
Hence we get
(2.4.28) deteA=det(BeAB−^1 ) =eλ^1 +···+λN.
It is known that
trA=λ 1 +···+λN.
Thus, the formula (2.4.22) follows from (2.4.28).
2.4.3 Yang-Mills action ofSU(N)gauge fields
AnSU(N)gauge theory mainly deals with the invariance of the Dirac equations forNspinors
underSU(N)gauge transformations. The physical meaning ofSU(N)gauge invariance is
that in a system ofNfermions we cannot distinguish one particle from others by the weak or
strong interaction.
Based on theSU(N)representation Theorem2.31, we now introduce theSU(N)gauge
theory.
ConsiderNDirac spinorsψk( 1 ≤k≤N)andK= (N^2 − 1 )Lorentz vector fieldsGaμ( 1 ≤
a≤K), called the gauge fields, given by
(2.4.29) Ψ= (ψ^1 ,···,ψN)T, Gμ= (G^1 μ,···,GKμ).
The Dirac equations for the fermions are:
(2.4.30)
[
iγμDμ−mc
̄h