Mathematical Principles of Theoretical Physics

146 CHAPTER 3. MATHEMATICAL FOUNDATIONS

On the other hand, by (3.4.5) and

〈δF(u),v〉=

∫

Rn

δF(u)vdx,

we deduce that
∫

Rn

δF(u)vdx=

∫

Rn

(−∆u+f′(u))vdx ∀v∈H^1 (Rn).

Hence we obtain the derivative operatorδF(u)of (3.4.6) as

δF(u) =−∆u+f′(u).

3.4.2 Derivative operators of the Yang-Mills functionals

LetGaμ( 1 ≤a≤N^2 − 1 )be theSU(N)gauge fields. The Yang-Mills functional forGaμis
defined by

(3.4.7) F=

∫

M

[

−

1

4

Fμ νaFμ νa

]

dx,

whereMis the 4-dimensional Minkowski space, and

(3.4.8) Fμ νa =∂μGaν−∂νGaμ+gλabcGbμGcν.

In Section2.4.3, we have seen that the functional (3.4.7) is the scalar curvature part in the
Yang-Mills action (2.4.50).
Referring to derivative operators of functionals for the electromagnetic potential deduced
in Subsection2.5.3, we now derive the derivative operator for the Yang-Mills functional
(3.4.7):

d

dλ

∣

∣

∣

λ= 0

F(G+λG ̃) =−

1

2

∫

M

gμ αgν βFα βa

d

dλ

∣

∣

∣

λ= 0

Fμ νa(G+λG ̃)dx

=(by( 3. 4. 8 ))

=−

1

2

∫

M

Fμ νa

(

∂G ̃aν

∂xμ

−

∂G ̃aμ

∂xν

)

dx

−

1

2

∫

M

Fμ νagλabc(GbμG ̃cν+G ̃bμGcν)dx

=

∫

M

gμ αgν β

(

∂Fα βa

∂xμ

−gFα βcλcbaGbμ

)

G ̃aνdx.

By (3.4.5) we deduce the derivative operatorδFof (3.4.7) as

(3.4.9) δF=∂αFα βa −ggα μλcbaFα βc Gμb, β= 0 , 1 , 2 , 3 ,

whereλcba=λcba.