Gravity and Strings

34 Noether’s theorems
According to the general result Eq. (2.22), we find the following set of d(d − 1)/2 conserved
currents labeled by a pair of antisymmetric indices: 6
jN1 (ρσ ) µ = Tcan µ λ ˜δ(ρσ )x λ + ∂L
∂∂µϕα ˜δ(ρσ )ϕ α = 2Tcan µ [ρxσ ] + ∂L αβϕ Ɣr Mρσ
∂∂µϕα β . (2.46)
The first contribution to this current is the orbital-angular-momentum tensor and the
second is the spin-angular-momentum tensor, S µ ρσ,
S µ ρσ ≡ 1
2
∂L
Ɣr
∂∂µϕα αβϕ Mρσ
β . (2.47)
Only the total angular-momentum current is conserved.
The d(d − 1)/2 conserved charges are the components of a two-index antisymmetric
tensor: the angular-momentum tensor Mµν,
Mµν = Q(µν) =
Vt
d d−1 xj 0 N1 (µν) . (2.48)
It is instructive to take the divergence of the above current. Since in the theories we are
dealing with we always have ∂µTcan µ ν = 0, one finds
∂µ jN1 (ρσ ) µ
∂L
=−2Tcan [ρσ] + ∂µ
αβϕ Mρσ
β
, (2.49)
Ɣr
∂∂µϕα which should vanish on-shell according to the general formalism. This means that, except
for scalars, Tcan µν is not symmetric and the antisymmetric part is given by
Tcan [ρσ] = ∂µS µ ρσ, (2.50)
up to terms vanishing on-shell. This formula suggests that we can symmetrize the canonical
energy–momentum tensor, exploiting the ambiguities of Noether currents mentioned
earlier, i.e. adding to it a term of the form
∂µ µρ σ , µρ σ =− ρµ σ , (2.51)
whose divergence is automatically zero, which in this case would be given by the spin–
energy potential
µρ σ =−S µρ σ + S ρµ σ + Sσ µρ , (2.52)
and also removing all the antisymmetric terms that vanish on-shell. The resulting symmetric
energy–momentum tensor is usually considered as the energy–momentum tensor to which
gravity couples 7 [939] and we will denote it simply by T µ ν.Itisalso called the Belinfante
tensor [103]. Using it, the conserved current associated with Lorentz rotations is
jN1 (ρσ ) µ = 2T µ λµ
[ρxσ ] + ∂λ [ρxσ ] , (2.53)
6 Here we concentrate on theories without higher derivatives.
7 This can be justified in the framework of the Cartan–Sciama–Kibble (CSK) theory of gravity. As we will see
in Section 4.4, the Belinfante tensor has to coincide with the Rosenfeld energy–momentum tensor, whose
definition is based precisely on the coupling to gravity. It is also worth mentioning that, in the CSK theory,
the spin–energy potential also couples to gravity through the torsion (the energy–momentum tensor couples
through the metric).