136 The very early universe
charges are called “colors”). Lagrangian (4.9) is obviously invariant with respect to
theglobalgauge transformation generated by the unitary, spacetime-independent
matrixU:
ψ→Uψ,becauseψ ̄→ψ ̄U†andU†U= 1 .This is no longer true if we assume that matrix
Uis a function ofxα.As in (4.3), the derivative∂αinduces an extra term:
∂μψ→∂μ(Uψ)=U(
∂μ+U−^1 (∂μU))
ψ,which needs to be compensated for if we want to preserve gauge invariance. With
this purpose, let us introduce the gauge fieldsAμ,which are HermitianN×N
matrixes, and replace∂αby the “covariant derivative”
Dμ≡∂μ+igAμ, (4.10)wheregis the gauge coupling constant. If we assume that under gauge transforma-
tionsAμ→A ̃μ,we obtain
Dμψ→D ̃μ(Uψ)=U(
∂μ+igU−^1 A ̃μU+U−^1 (∂μU))
ψ.(Note that one must be careful with the order of multiplication because the matrices
do not generally commute.) Therefore, wepostulatethe transformation law
Aμ→A ̃μ=UAμU−^1 +i
g
(∂μU)U−^1. (4.11)Then
Dμψ→D ̃μ(Uψ)=UDμψ,and the Lagrangian
L=iψ ̄γμDμψ−mψψ ̄ (4.12)is invariant underU(N)local gauge transformations. To derive the Lagrangian for
the gauge fields, we note that
Fμν≡DμAν−DνAμ=∂μAν−∂νAμ+ig(AμAν−AνAμ) (4.13)transforms asFμν→F ̃μν=UFμνU−^1 ,and hence the simplest gauge-invariant
Lorentz scalar is tr
(
FμνFμν)
. The full Lagrangian is then
L=iψ ̄γμ∂μψ−mψψ ̄ −gψ ̄γμAμψ−1
2
tr(
FμνFμν)
, (4.14)
where we have used the standard normalization for the last term in cases where
N≥ 2.