Gravity and Strings

where
2.2 Noether’s theorems 29
j µ
N1 (˜δ) =−s µ (˜δ) + Tcan µ ν ˜δx ν − ∂L
∂ρϕ∂ν ˜δx
ρ
∂∂µ∂νϕ
∂L ∂L
+ − ∂ν
˜δϕ +
∂∂µϕ ∂∂µ∂νϕ
∂L
∂∂µ∂νϕ ∂ν ˜δϕ + ···, (2.17)
where, in turn,
Tcan µ ν = η µ νL − ∂L
∂∂µϕ ∂νϕ − ∂L
∂∂µ∂ρϕ ∂ν∂ρϕ
∂L
+ ∂ρ ∂νϕ + ···. (2.18)
∂∂µ∂ρϕ
Tcan µ ν is the canonical energy–momentum tensor and is the only piece of j µ
N1 that survives
(apart from s µ ) when we consider constant ˜δx µ s.
It is worth stressing that the total-derivative term will not vanish in general after use of
Stokes’ theorem because the variations ˜δx µ and ˜δϕ do not vanish on the boundary.
Now we want to derive conservation laws from this identity. We see that, in the general
case, if the equations of motion δS/δϕ = 0 are satisfied, then we can conclude that j µ
N1 (˜δ)
is a conserved vector current (Noether current), i.e. satisfies the continuity equation
∂µj µ
N1 (˜δ) = 0. (2.19)
Thus, for a theory that is exactly invariant under constant translations, the canonical
energy–momentum tensor is the associated Noether conserved current.
Strictly speaking j µ
N1 (˜δ) is a vector density. In the presence of a metric, we can define a
vector current j µ
N1 (˜δ) = √ |g| j µ
N1 (˜δ) and write the continuity equation in general-covariant
form:
∇µ j µ
N1 (˜δ) = 0. (2.20)
In Minkowski spacetime this distinction is unnecessary. Such terms are called “conserved”
because they are used to define quantities (charges) that are conserved in time, as we will
see next.
This is the best we can do if the transformations are global, i.e. when they take the form
˜δx µ ≡ σ I ˜δI x µ , ˜δϕ ≡ σ I ˜δI ϕ, (2.21)
where ˜δI x µ and ˜δI ϕ are given functions of the coordinates and ϕ and the σ I , I = 1,...,n,
are the constant transformation parameters. Then, we find n on-shell conserved currents
j µ
N1 I independent of the parameters σ I and they are given by
j µ
N1 I =−sµ (˜δI ) + Tcan µ ν ˜δI x ν − ∂L
∂∂µ∂νϕ ∂ρϕ∂ν ˜δI x ρ
∂L ∂L
+ − ∂ν
˜δI ϕ +
∂∂µϕ ∂∂µ∂νϕ
∂L
∂∂µ∂νϕ ∂ν ˜δI ϕ + ···. (2.22)
If the transformations are local, i.e. they depend on n local parameters σI (x), the generic
result of the on-shell conservation of the current j µ
N1 is still true,3 but wecan do more.
3 In principle there is one current for each value of the local parameters. This gives an infinite number of
on-shell conserved currents. However, only for a certain asymptotic behavior of the σI (x)s will the integrals
defining the conserved charges converge. These asymptotic behaviors are usually associated with the global
invariances of the vacuum configuration.