Mathematical Principles of Theoretical Physics

2.4. GAUGE INVARIANCE 77

Then, thanks to (2.4.41), we deduce that

i

g

[

D ̃μ,D ̃ν

]

Ψ ̃=F ̃μ νaτaΩΨ

=

i

g

Ω

[

Dμ,Dν

]

Ψ (by( 2. 4. 39 ))

=Fμ νaΩτaΨ,

which implies that

(2.4.43) F ̃μ νaτa=Fμ νaΩτaΩ−^1.

Asμ,νare the indices of 4-D tensors, a Lorentz invariant can be constructed using the
following contraction

(2.4.44) F ̃μ νaτaF ̃μ νbτb†=F ̃μ νaF ̃μ νbτaτb†,

whereFμ νb=gμ αgν βFα β. By (2.4.43) we have

(2.4.45) F ̃μ νaτaF ̃μ νbτb†=Fμ νa(ΩτaΩ−^1 )Fμ νb(ΩτbΩ−^1 )†

=(byΩ†=Ω−^1 )

=Fμ νaFμ νbΩτaτb†Ω−^1.

Therefore we deduce from (2.4.44) and (2.4.45) that

(2.4.46) F ̃μ νaF ̃μ νbτaτb†=Fμ νaFμ νbΩτaτb†Ω−^1.

In (2.3.16) we have verified that

trA=tr(BAB−^1 ) ∀matricesAandB.

Hence it follows from (2.4.46) that

(2.4.47) GabF ̃μ νaF ̃μ νb=GabFμ νaFμ νb,

whereGab=^12 tr(τaτb†).
The equality (2.4.47) shows that

(2.4.48) F=GabFμ νaFμ νb=Gabgμ αgν βFμ νaFα βb

is invariant under both the Lorentz transformations and theSU(N)gauge transformations,
whereFμ νa are given by (2.4.42).
The function (2.4.48) is a unique form which is both Lorentz and theSU(N)gauge in-
variant, and contains up to first-order derivatives of the gauge fieldsGaμ. Hence,Fin (2.4.48)
is a unique candidate to be the Lagrange density.
In Section3.5, we have shown that{Gab}in (2.4.48) is a Riemannian metric ofSU(N),
and

(2.4.49) Gab=

1

2

tr(τaτ†b) =

1

4 N

λadcλcbd,