Gravity and Strings

2.2 Noether’s theorems 27
The variation of the coordinates is zero by hypothesis. Then the variation of the field
commutes with the derivatives. On integrating by parts to obtain an overall factor of δϕ,we
find
δS =
d d x
δS
∂L
δϕ + ∂µ
δϕ ∂∂µϕ
− ∂ν
where we have defined the first variation of the action δS/δϕ,
δS ∂L
≡
δϕ ∂ϕ
− ∂µ
∂L
∂∂µϕ
∂L
δϕ +
∂∂µ∂νϕ
∂L
∂∂µ∂νϕ ∂νδϕ
+··· , (2.3)
+ ∂µ∂ν
∂L
+ ···. (2.4)
∂∂µ∂νϕ
We now use Stokes’ theorem Eq. (1.141) to reexpress the integral of the total derivative
as an integral over the boundary ∂:
δS = d
d x δS
δϕ + (−1)d−1 d
δϕ ∂
d−1
∂L ∂L
µ − ∂ν δϕ
∂∂µϕ ∂∂µ∂νϕ
+ ∂L
∂∂µ∂νϕ ∂νδϕ
+ ··· . (2.5)
In theories without higher derivatives L(ϕ, ∂ϕ) it is enough to impose that the field variations
vanish over the boundary δϕ| ∂ = 0, to see that the boundary term vanishes. Then,
requiring that the action is stationary, δS = 0, under those variations we obtain the usual
Euler–Lagrange equations
δS ∂L ∂L
= − ∂µ
δϕ ∂ϕ ∂∂ µ
= 0. (2.6)
ϕ
If the Lagrangian contains higher derivatives of the field, it is necessary either to impose
boundary conditions for derivatives of the variation of the field or to introduce (if possible)
into the action boundary terms that do not change the equations of motion but eliminate the
∂δϕ term in the total derivative. In any of these cases we obtain the equations of motion
δS ∂L ∂L
= − ∂µ
δϕ ∂ϕ ∂∂ µ
+ ∂
ϕ
ν ∂ µ
∂L
∂∂ν∂ µ
− ···=0. (2.7)
ϕ
As we can see, the equations of motion are of degree higher than two in derivatives of
the field. Thus, to solve them completely it is also necessary to give boundary conditions
for the field, and for its first and higher derivatives.
If we add a total derivative term ∂µk µ (ϕ) to the Lagrangian, it is clear that the equations
of motion will not be modified as long as the boundary conditions for δϕ and its derivatives
make k µ (ϕ) = 0 on the boundary.
2.2 Noether’s theorems
Let us now consider the infinitesimal transformations of the coordinates and fields
˜δx µ and ˜δϕ:
˜δx µ = x ′ µ − x µ ,
˜δϕ(x) ≡ ϕ ′ (x ′ ) − ϕ(x), (2.8)