2.4. GAUGE INVARIANCE 73
gis a coupling constant,τk( 1 ≤k≤N^2 − 1 )are the generators ofSU(N).
We recall thatSU(N)is the group consisting ofN×Nunitary matrices with unit deter-
minant:
(2.4.19) SU(N) ={Ω|ΩtheN×Nmatrix,Ω†=Ω−^1 ,detΩ= 1 }.
In theSU(N)gauge theory for (2.4.16)-(2.4.19), we encounter a mathematical concept, gen-
eratorsτk( 1 ≤k≤N^2 − 1 )ofSU(N). We now introduce the representation ofSU(N).
By (2.4.19) we see that each matrixΩ∈SU(N)satisfies
(2.4.20) Ω†=Ω−^1 , detΩ= 1 , Ω∈SU(N),
whereΩ†= (ΩT)∗is the complex conjugate of the transport ofΩ. Note that an exponenteiA
of a matrixiAis anN×Ncomplex matrix satisfying
(eiA)†=e−iA†
.It follows that
(eiA)†(eiA) =e−iA
†
eiA=ei(A−A†)
.Hence we see that
(2.4.21) (eiA)†= (eiA)−^1 if and only ifA=A†is Hermitian.
In addition, we have
(2.4.22) deteA=etrA,
which will be proved at the end of this subsection.
Based on (2.4.20)-(2.4.22), it is not difficult to understand the representation ofSU(N)in
a neighborhood of the unit matrix, stated in the following theorem.
Theorem 2.31(SU(N)Representation).The matricesΩin a neighborhood U⊂SU(N)of
the unit matrix can be expressed as
(2.4.23) Ω=eiA, A†=A, trA= 0.
Furthermore, the tangent space TeSU(N)of SU(N)at the unit matrix e=I is a K-dimensional
linear space(K=N^2 − 1 ), generated by K linear independent traceless Hermitian matrices
τk( 1 ≤k≤K):
(2.4.24) TeSU(N) =span{τ 1 ,···,τK} withτk†=τk, trτk= 0.
In particular, the matricesΩin (2.4.23) can be written as
(2.4.25) Ω=eiθ
kτk
, τk( 1 ≤k≤K)as in( 2. 4. 24 ),whereθk( 1 ≤k≤K)are real numbers, and
(2.4.26) [τk,τl] =τkτl−τlτk=iλkljτj,
whereλkljare called the structure constants of SU(N).