Gravity and Strings

40 Noether’s theorems
The Rosenfeld energy–momentum tensor has the required properties. 11,12
To illustrate this point, let us go back to the massless vector field of the previous section.
Let us consider the effect of conformal transformations on its action. The conformal group
consists of transformations that leave the Minkowski metric invariant up to a global (possibly
local) factor: (infinitesimal) constant translations ˜δx µ = ξ µ ,Lorentz rotations ˜δx µ =
σ µ νx ν ≡ σ µ (these two generate the Poincaré group), dilatations ˜δx µ = σ x µ ≡ w µ , and
special conformal transformations (or conformal boosts) ˜δx µ = 2(ζ · x)x µ − x 2 ζ µ ≡ v µ .
The vector field transforms under these coordinate transformations according to the general
rule for (world) vectors (2.59) with ɛ µ = ξ µ + σ µ + w µ + v µ . The variation of the
action is, again, given by Eq. (2.60). Conformal transformations are generated by conformal
Killing vectors of Minkowski spacetime that satisfy
∂(µɛλ) ∝ ηµλ. (2.87)
The proportionality factor is zero for Poincaré transformations but non-zero for dilatations
and conformal boosts. Then, the variation of the action will be zero only if the energy–
momentum tensor is traceless. This happens only in d = 4 dimensions. On integrating by
parts, etc., we find that the Noether current has the form Eq. (2.77), always with the same
(Rosenfeld’s) energy–momentum tensor.
11 However, this is still confusing because we have two different symmetric, on-shell divergenceless energy–
momentum tensors for a scalar field (the canonical and the improved, which is traceless) and Rosenfeld’s
procedure seems to give a unique energy–momentum tensor. This is not true, though: when we covariantize
a special-relativistic action introducing a metric the result is unique up to curvature terms that vanish in
Minkowski spacetime. In the case of the scalar, a covariantization that preserves the scaling invariance is
S[ϕ,γ] = d d x
1
|γ |
2 (∂ϕ)2 ω
+
4(d − 1) ϕ2
R(γ ) , (2.84)
where R(γ ) is the Ricci scalar of the background metric. This action is invariant, in fact, under local Weyl
rescalings of the metric and local rescalings of the scalar, leaving the coordinates untouched:
Using the results of Section 4.2, we find
2 δS[ϕ,γ]
δγµν
ϕ ′ = (2−d)/2 (x)ϕ, γ ′ µν = 2 (x)γµν. (2.85)
= Tcan µν −
ω
2(d − 1)
∇ µ ∂ ν ϕ 2 − γ µν ∇ 2
ϕ +
ω
2(d − 1) ϕGµν (γ ), (2.86)
where G µν (γ ) is the Einstein tensor of the background metric. On setting γµν = ηµν,wefind precisely the
improved energy–momentum tensor Eq. (2.80). Something similar can be said of the vector field in d = 4.
If, in the presence of a curved metric, the vector field scales as in Minkowski spacetime, the vector field is
really a vector density and then its covariantization is different from the standard one and should lead to a
Rosenfeld energy–momentum tensor identical to the improved one.
12 When the field theory has a symmetry, it is desirable or necessary to have an energy–momentum tensor
that is also invariant under the same transformations. For instance, the Belinfante energy–momentum tensor
for the Maxwell field is gauge-invariant, as is the Maxwell action. It can be shown that, in general, symmetries
of a theory are also symmetries of the Rosenfeld energy–momentum tensor if the symmetries are
also symmetries of the same theory covariantized with an arbitrary background metric. The Maxwell action
in a curved background is still gauge-invariant and the gauge-invariance of the Belinfante–Rosenfeld
energy–momentum tensor follows.