Gravity and Strings

2.4 The special-relativistic energy–momentum tensor 35
again, up to terms that vanish on-shell. The second term in this expression can be eliminated
by the usual procedure. The spin-angular-momentum tensor has been absorbed into the
new angular-momentum tensor. We are left with the following conserved on-shell currents
associated with translations and Lorentz rotations, both of them expressed in terms of the
same energy–momentum tensor (the Belinfante tensor):
jN1 (ν) µ = T µ ρ ˜δ(ν)x ρ = T µ ρ,
jN1 (ρσ ) µ = T µ λ ˜δ(ρσ )x λ = 2T µ [ρxσ ].
(2.54)
It is worth stressing that the existence of these conserved currents is primarily due to the
invariance of the Minkowski metric that enters into special-relativistic Lagrangians and of
the Minkowski volume element under the Poincarégroup or, in other words, to the existence
of d(d + 1)/2 Killing vectors precisely of the form
˜δ(ν)x µ ∂µ = ∂ν, ˜δ(ρσ )x µ ∂µ =−2x[ρ∂σ ]. (2.55)
Acouple of simple examples of the symmetrization of the canonical energy–momentum
tensor are in order here.
The energy–momentum tensor of a vector field. The Lagrangian and canonical energy–
momentum tensor are given by
L =− 1
4 F 2 , Fµν = 2∂[µ Aν], Tcan µ ν = F µρ ∂ν Aρ − 1
4ηµ ν F 2 . (2.56)
Under Lorentz rotations we have
˜δ Aµ =−Aνσ ν µ ⇒ S µ ρσ = F µ [ρ Aσ ] ⇒ ρµ ν = F ρµ Aν, (2.57)
and, using the equations of motion ∂ρ F ρµ = 0,
T µ ν = Tcan µ ν + ∂ρ ρµ ν = F µρ Fνρ − 1
4ηµ ν F 2 , (2.58)
which is the standard, gauge-invariant, energy–momentum tensor of a vector field, coinciding
with the one derived via Rosenfeld’s prescription, which we are going to introduce in
Section 2.4.3, inspired by general relativity.
There is yet another way to obtain this energy–momentum tensor that is worth pointing
out: let us consider the transformations
˜δx µ = ɛ µ , ˜δ Aµ = ɛ λ ∂λ Aµ − Lɛ Aµ =−∂µɛ λ Aλ. (2.59)
Following the same steps as those we followed to prove the Noether theorem, we find
now
˜δS = d d x∂µɛ λ T µ ν, (2.60)
with T µ ν as above (the Belinfante tensor). This variation vanishes if
∂(µɛλ) = 0, (2.61)