Gravity and Strings

28 Noether’s theorems
where x ′ and x stand for the coordinates of the same point in the two different coordinate
systems. The transformation of the fields may contain terms associated with the coordinate
transformations and also with other “internal” transformations (see footnote 3 in Chapter 1).
We want to find the consequences of the invariance, possibly up to a total derivative that
depends on the variations, of the action Eq. (2.1) under the above infinitesimal changes of
the field and the coordinates (which are, then, symmetry transformations). We express this
invariance as follows:
˜δS = d d x ∂µs µ (˜δ). (2.9)
Let us now perform directly the variation of the action explicitly, 2
˜δS =
˜δd d x L + d d
x ˜δL . (2.10)
We have
˜δd d x = d d x ∂µ ˜δx µ ,
˜δL = δL + ˜δx µ ∂µL, (2.11)
δL = ∂L ∂L
δϕ +
∂ϕ ∂∂µϕ δ∂µϕ + ∂L
∂∂µ∂νϕ δ∂µ∂νϕ + ···,
where δ stands for the variation of the field at two different points whose coordinates are
the same in the two different coordinate systems considered,
δϕ(x) ≡ ϕ ′ (x) − ϕ(x), (2.12)
and we have used the field-operator identity
˜δ = δ + ˜δx µ ∂µ. (2.13)
δ and ∂µ commute since δ does not involve any change of coordinates. Thus
˜δS = d
d
x ∂µ ˜δx µ L + ˜δx µ ∂µL + ∂L ∂L
δϕ +
∂ϕ ∂∂µϕ ∂µδϕ + ∂L
∂∂µ∂νϕ ∂µ∂νδϕ
+ ··· .
(2.14)
On integrating by parts as many times as necessary, we obtain
˜δS = d
d
x ∂µ L˜δx µ
∂L
+
∂∂µϕ
− ∂ν
∂L
δϕ +
∂∂µ∂νϕ
∂L
∂∂µ∂νϕ ∂νδϕ
+ ··· + δS
δϕ δϕ
.
(2.15)
On reexpressing δϕ in terms of ˜δϕ inside the total derivative, and equating the result with
Eq. (2.9), we arrive at
d d
x ∂µj µ
N1 (˜δ) + δS
δϕ δϕ
= 0, (2.16)
2 We follow here [795].